Incomplete Gamma and Related Functions - 8.14 Integrals

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.14.E1	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x\right)}{\Gamma\left(b\right)}\mathrm{d}x=\frac{(1+a)^{-b}}{a}}}$ \int_{0}^{\infty}e^{-ax}\frac{\incgamma@{b}{x}}{\EulerGamma@{b}}\diff{x} = \frac{(1+a)^{-b}}{a}	${\displaystyle{\displaystyle\Re a>0,\Re b>-1,\Re b>0}}$	int(exp(- a*x)*(GAMMA(b)-GAMMA(b, x))/(GAMMA(b)), x = 0..infinity) = ((1 + a)^(- b))/(a)	Integrate[Exp[- a*x]*Divide[Gamma[b, 0, x],Gamma[b]], {x, 0, Infinity}, GenerateConditions->None] == Divide[(1 + a)^(- b),a]	Successful	Aborted	-	Skipped - Because timed out
8.14.E2	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)\mathrm{d}x=\Gamma\left(b\right)\frac{1-(1+a)^{-b}}{a}}}$ \int_{0}^{\infty}e^{-ax}\incGamma@{b}{x}\diff{x} = \EulerGamma@{b}\frac{1-(1+a)^{-b}}{a}	${\displaystyle{\displaystyle\Re a>-1,\Re b>-1,\Re b>0}}$	int(exp(- a*x)*GAMMA(b, x), x = 0..infinity) = GAMMA(b)*(1 -(1 + a)^(- b))/(a)	Integrate[Exp[- a*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Gamma[b]*Divide[1 -(1 + a)^(- b),a]	Failure	Aborted	Successful [Tested: 12]	Skipped - Because timed out
8.14.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)\mathrm{d}x=-\frac{\Gamma\left(a+b\right)}{a}}}$ \int_{0}^{\infty}x^{a-1}\incgamma@{b}{x}\diff{x} = -\frac{\EulerGamma@{a+b}}{a}	${\displaystyle{\displaystyle\Re a<0,\Re\left(a+b\right)>0,\Re(a+b)>0,\Re b>0}}$	int((x)^(a - 1)* GAMMA(b)-GAMMA(b, x), x = 0..infinity) = -(GAMMA(a + b))/(a)	Integrate[(x)^(a - 1)* Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == -Divide[Gamma[a + b],a]	Failure	Aborted	Failed [3 / 3] Result: Float(infinity) Test Values: {a = -1.5, b = 2} Result: Float(infinity) Test Values: {a = -.5, b = 1.5} ... skip entries to safe data	Skip - No test values generated
8.14.E4	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a}}}$ \int_{0}^{\infty}x^{a-1}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a}	${\displaystyle{\displaystyle\Re a>0,\Re\left(a+b\right)>0,\Re(a+b)>0}}$	int((x)^(a - 1)* GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a)	Integrate[(x)^(a - 1)* Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a]	Successful	Successful	-	Successful [Tested: 12]
8.14.E5	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x\right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1+b;1/(1+s)\right)}}$ \int_{0}^{\infty}x^{a-1}e^{-sx}\incgamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{b(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+b}{1/(1+s)}	${\displaystyle{\displaystyle\Re s>0,\Re\left(a+b\right)>0,\Re(a+b)>0,\Re b>0}}$	int((x)^(a - 1)* exp(- s*x)*GAMMA(b)-GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(b*(1 + s)^(a + b))* hypergeom([1, a + b], [1 + b], 1/(1 + s))	Integrate[(x)^(a - 1)* Exp[- s*x]*Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],b*(1 + s)^(a + b)]* Hypergeometric2F1[1, a + b, 1 + b, 1/(1 + s)]	Failure	Aborted	Failed [36 / 36] Result: Float(infinity) Test Values: {a = -1.5, b = 2, s = 1.5} Result: Float(infinity) Test Values: {a = -1.5, b = 2, s = .5} ... skip entries to safe data	Skipped - Because timed out
8.14.E6	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x\right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1+a;s/(1+s)\right)}}$ \int_{0}^{\infty}x^{a-1}e^{-sx}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+a}{s/(1+s)}	${\displaystyle{\displaystyle\Re s>-1,\Re\left(a+b\right)>0,\Re a>0,\Re(a+b)>0}}$	int((x)^(a - 1)* exp(- s*x)*GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a*(1 + s)^(a + b))* hypergeom([1, a + b], [1 + a], s/(1 + s))	Integrate[(x)^(a - 1)* Exp[- s*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a*(1 + s)^(a + b)]* Hypergeometric2F1[1, a + b, 1 + a, s/(1 + s)]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out