Integrand size = 9, antiderivative size = 29 \[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {1}{2} x^2 \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {7}{2},a x\right )}{2 a^2} \] Output:
1/2*x^2*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))-1/2*((a*x)^ (5/2)*exp(-a*x)+5/2*(a*x)^(3/2)*exp(-a*x)+15/4*(a*x)^(1/2)*exp(-a*x)+15/8* Pi^(1/2)*erfc((a*x)^(1/2)))/a^2
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {1}{2} x^2 \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {7}{2},a x\right )}{2 a^2} \] Input:
Integrate[x*Gamma[3/2, a*x],x]
Output:
(x^2*Gamma[3/2, a*x])/2 - Gamma[7/2, a*x]/(2*a^2)
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7116}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx\) |
\(\Big \downarrow \) 7116 |
\(\displaystyle \frac {1}{2} x^2 \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {7}{2},a x\right )}{2 a^2}\) |
Input:
Int[x*Gamma[3/2, a*x],x]
Output:
(x^2*Gamma[3/2, a*x])/2 - Gamma[7/2, a*x]/(2*a^2)
Int[Gamma[n_, (b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*(Gamma[n, b*x]/(d*(m + 1))), x] - Simp[(d*x)^m*(Gamma[m + n + 1, b*x]/(b *(m + 1)*(b*x)^m)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03
method | result | size |
derivativedivides | \(\frac {4 \sqrt {\pi }\, \operatorname {erfc}\left (\sqrt {x a}\right ) x^{2} a^{2}-20 \left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}-15 \sqrt {\pi }\, \operatorname {erfc}\left (\sqrt {x a}\right )-30 \sqrt {x a}\, {\mathrm e}^{-x a}}{16 a^{2}}\) | \(59\) |
default | \(\frac {\sqrt {\pi }\, \left (\frac {x^{2} a^{2} \operatorname {erfc}\left (\sqrt {x a}\right )}{4}+\frac {-\frac {\left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}}{2}-\frac {3 \sqrt {x a}\, {\mathrm e}^{-x a}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{8}}{2 \sqrt {\pi }}\right )-\left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}-\frac {3 \sqrt {x a}\, {\mathrm e}^{-x a}}{2}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{4}}{a^{2}}\) | \(105\) |
parts | \(\frac {-\left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}-\frac {3 \sqrt {x a}\, {\mathrm e}^{-x a}}{2}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{4}}{a^{2}}+\frac {\sqrt {\pi }\, \left (\frac {x^{2} a^{2} \operatorname {erfc}\left (\sqrt {x a}\right )}{4}+\frac {-\frac {\left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}}{2}-\frac {3 \sqrt {x a}\, {\mathrm e}^{-x a}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{8}}{2 \sqrt {\pi }}\right )}{a^{2}}\) | \(110\) |
Input:
int(x*((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2))),x,method=_RET URNVERBOSE)
Output:
1/16*(4*Pi^(1/2)*erfc((x*a)^(1/2))*x^2*a^2-20*(x*a)^(3/2)*exp(-x*a)-15*Pi^ (1/2)*erfc((x*a)^(1/2))-30*(x*a)^(1/2)*exp(-x*a))/a^2
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {4 \, \sqrt {\pi } a^{2} x^{2} - \sqrt {\pi } {\left (4 \, a^{2} x^{2} - 15\right )} \operatorname {erf}\left (\sqrt {a x}\right ) - 10 \, {\left (2 \, a x + 3\right )} \sqrt {a x} e^{\left (-a x\right )}}{16 \, a^{2}} \] Input:
integrate(x*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, algo rithm="fricas")
Output:
1/16*(4*sqrt(pi)*a^2*x^2 - sqrt(pi)*(4*a^2*x^2 - 15)*erf(sqrt(a*x)) - 10*( 2*a*x + 3)*sqrt(a*x)*e^(-a*x))/a^2
Time = 0.60 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.93 \[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\begin {cases} \frac {\sqrt {\pi } x^{2} \operatorname {erfc}{\left (\sqrt {a x} \right )}}{4} - \frac {5 x \sqrt {a x} e^{- a x}}{4 a} - \frac {15 \sqrt {a x} e^{- a x}}{8 a^{2}} - \frac {15 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {a x} \right )}}{16 a^{2}} & \text {for}\: a \neq 0 \\\frac {\sqrt {\pi } x^{2}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x*((a*x)**(1/2)*exp(-a*x)+1/2*pi**(1/2)*erfc((a*x)**(1/2))),x)
Output:
Piecewise((sqrt(pi)*x**2*erfc(sqrt(a*x))/4 - 5*x*sqrt(a*x)*exp(-a*x)/(4*a) - 15*sqrt(a*x)*exp(-a*x)/(8*a**2) - 15*sqrt(pi)*erfc(sqrt(a*x))/(16*a**2) , Ne(a, 0)), (sqrt(pi)*x**2/4, True))
\[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\int { \frac {1}{2} \, {\left (\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}\right )} x \,d x } \] Input:
integrate(x*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, algo rithm="maxima")
Output:
1/2*integrate((sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))*x, x)
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.10 \[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {4 \, \sqrt {\pi } a x^{2} - \frac {\sqrt {\pi } {\left (4 \, a^{2} x^{2} \operatorname {erf}\left (\sqrt {a x}\right ) + \frac {2 \, {\left (2 \, \sqrt {a x} a x + 3 \, \sqrt {a x}\right )} e^{\left (-a x\right )} - 3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {a x}\right )}{\sqrt {\pi }}\right )}}{a} - \frac {4 \, {\left (2 \, {\left (2 \, \sqrt {a x} a x + 3 \, \sqrt {a x}\right )} e^{\left (-a x\right )} - 3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {a x}\right )\right )}}{a}}{16 \, a} \] Input:
integrate(x*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, algo rithm="giac")
Output:
1/16*(4*sqrt(pi)*a*x^2 - sqrt(pi)*(4*a^2*x^2*erf(sqrt(a*x)) + (2*(2*sqrt(a *x)*a*x + 3*sqrt(a*x))*e^(-a*x) - 3*sqrt(pi)*erf(sqrt(a*x)))/sqrt(pi))/a - 4*(2*(2*sqrt(a*x)*a*x + 3*sqrt(a*x))*e^(-a*x) - 3*sqrt(pi)*erf(sqrt(a*x)) )/a)/a
Timed out. \[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\int x\,\left (\frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{2}+{\mathrm {e}}^{-a\,x}\,\sqrt {a\,x}\right ) \,d x \] Input:
int(x*((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2)),x)
Output:
int(x*((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2)), x)
\[ \int x \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {-4 \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right ) a^{2} x^{2}+3 \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )+12 e^{a x} \sqrt {a}\, \left (\int \frac {\sqrt {x}}{e^{a x} x}d x \right )-20 \sqrt {x}\, \sqrt {a}\, a x -30 \sqrt {x}\, \sqrt {a}+4 \sqrt {\pi }\, e^{a x} a^{2} x^{2}}{16 e^{a x} a^{2}} \] Input:
int(x*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2))),x)
Output:
( - 4*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a))*a**2*x**2 + 3*sqrt(pi)*e**(a* x)*erf(sqrt(x)*sqrt(a)) + 12*e**(a*x)*sqrt(a)*int(sqrt(x)/(e**(a*x)*x),x) - 20*sqrt(x)*sqrt(a)*a*x - 30*sqrt(x)*sqrt(a) + 4*sqrt(pi)*e**(a*x)*a**2*x **2)/(16*e**(a*x)*a**2)