Integrand size = 11, antiderivative size = 29 \[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {1}{3} x^3 \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {9}{2},a x\right )}{3 a^3} \] Output:
1/3*x^3*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))-1/3*((a*x)^ (7/2)*exp(-a*x)+7/2*(a*x)^(5/2)*exp(-a*x)+35/4*(a*x)^(3/2)*exp(-a*x)+105/8 *(a*x)^(1/2)*exp(-a*x)+105/16*Pi^(1/2)*erfc((a*x)^(1/2)))/a^3
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {1}{3} x^3 \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {9}{2},a x\right )}{3 a^3} \] Input:
Integrate[x^2*Gamma[3/2, a*x],x]
Output:
(x^3*Gamma[3/2, a*x])/3 - Gamma[9/2, a*x]/(3*a^3)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx\) |
\(\Big \downarrow \) 7116 |
\(\displaystyle \frac {1}{3} x^3 \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {9}{2},a x\right )}{3 a^3}\) |
Input:
Int[x^2*Gamma[3/2, a*x],x]
Output:
(x^3*Gamma[3/2, a*x])/3 - Gamma[9/2, a*x]/(3*a^3)
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 = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45
method | result | size |
derivativedivides | \(\frac {8 \sqrt {\pi }\, \operatorname {erfc}\left (\sqrt {x a}\right ) x^{3} a^{3}-56 \left (x a \right )^{\frac {5}{2}} {\mathrm e}^{-x a}-140 \left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}-105 \sqrt {\pi }\, \operatorname {erfc}\left (\sqrt {x a}\right )-210 \sqrt {x a}\, {\mathrm e}^{-x a}}{48 a^{3}}\) | \(71\) |
default | \(\frac {\sqrt {\pi }\, \left (\frac {x^{3} a^{3} \operatorname {erfc}\left (\sqrt {x a}\right )}{6}+\frac {-\frac {\left (x a \right )^{\frac {5}{2}} {\mathrm e}^{-x a}}{2}-\frac {5 \left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}}{4}-\frac {15 \sqrt {x a}\, {\mathrm e}^{-x a}}{8}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{16}}{3 \sqrt {\pi }}\right )-\left (x a \right )^{\frac {5}{2}} {\mathrm e}^{-x a}-\frac {5 \left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}}{2}-\frac {15 \sqrt {x a}\, {\mathrm e}^{-x a}}{4}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{8}}{a^{3}}\) | \(131\) |
parts | \(\frac {-\left (x a \right )^{\frac {5}{2}} {\mathrm e}^{-x a}-\frac {5 \left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}}{2}-\frac {15 \sqrt {x a}\, {\mathrm e}^{-x a}}{4}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{8}}{a^{3}}+\frac {\sqrt {\pi }\, \left (\frac {x^{3} a^{3} \operatorname {erfc}\left (\sqrt {x a}\right )}{6}+\frac {-\frac {\left (x a \right )^{\frac {5}{2}} {\mathrm e}^{-x a}}{2}-\frac {5 \left (x a \right )^{\frac {3}{2}} {\mathrm e}^{-x a}}{4}-\frac {15 \sqrt {x a}\, {\mathrm e}^{-x a}}{8}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{16}}{3 \sqrt {\pi }}\right )}{a^{3}}\) | \(136\) |
Input:
int(x^2*((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2))),x,method=_R ETURNVERBOSE)
Output:
1/48*(8*Pi^(1/2)*erfc((x*a)^(1/2))*x^3*a^3-56*(x*a)^(5/2)*exp(-x*a)-140*(x *a)^(3/2)*exp(-x*a)-105*Pi^(1/2)*erfc((x*a)^(1/2))-210*(x*a)^(1/2)*exp(-x* a))/a^3
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {8 \, \sqrt {\pi } a^{3} x^{3} - \sqrt {\pi } {\left (8 \, a^{3} x^{3} - 105\right )} \operatorname {erf}\left (\sqrt {a x}\right ) - 14 \, {\left (4 \, a^{2} x^{2} + 10 \, a x + 15\right )} \sqrt {a x} e^{\left (-a x\right )}}{48 \, a^{3}} \] Input:
integrate(x^2*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, al gorithm="fricas")
Output:
1/48*(8*sqrt(pi)*a^3*x^3 - sqrt(pi)*(8*a^3*x^3 - 105)*erf(sqrt(a*x)) - 14* (4*a^2*x^2 + 10*a*x + 15)*sqrt(a*x)*e^(-a*x))/a^3
Time = 1.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.69 \[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\begin {cases} \frac {\sqrt {\pi } x^{3} \operatorname {erfc}{\left (\sqrt {a x} \right )}}{6} - \frac {7 x^{2} \sqrt {a x} e^{- a x}}{6 a} - \frac {35 x \sqrt {a x} e^{- a x}}{12 a^{2}} - \frac {35 \sqrt {a x} e^{- a x}}{8 a^{3}} - \frac {35 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {a x} \right )}}{16 a^{3}} & \text {for}\: a \neq 0 \\\frac {\sqrt {\pi } x^{3}}{6} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*((a*x)**(1/2)*exp(-a*x)+1/2*pi**(1/2)*erfc((a*x)**(1/2))),x )
Output:
Piecewise((sqrt(pi)*x**3*erfc(sqrt(a*x))/6 - 7*x**2*sqrt(a*x)*exp(-a*x)/(6 *a) - 35*x*sqrt(a*x)*exp(-a*x)/(12*a**2) - 35*sqrt(a*x)*exp(-a*x)/(8*a**3) - 35*sqrt(pi)*erfc(sqrt(a*x))/(16*a**3), Ne(a, 0)), (sqrt(pi)*x**3/6, Tru e))
\[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\int { \frac {1}{2} \, {\left (\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}\right )} x^{2} \,d x } \] Input:
integrate(x^2*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, al gorithm="maxima")
Output:
1/2*integrate((sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))*x^2, x)
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.00 \[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {8 \, \sqrt {\pi } a x^{3} - \frac {\sqrt {\pi } {\left (8 \, a^{3} x^{3} \operatorname {erf}\left (\sqrt {a x}\right ) + \frac {2 \, {\left (4 \, \sqrt {a x} a^{2} x^{2} + 10 \, \sqrt {a x} a x + 15 \, \sqrt {a x}\right )} e^{\left (-a x\right )} - 15 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {a x}\right )}{\sqrt {\pi }}\right )}}{a^{2}} - \frac {6 \, {\left (2 \, {\left (4 \, \sqrt {a x} a^{2} x^{2} + 10 \, \sqrt {a x} a x + 15 \, \sqrt {a x}\right )} e^{\left (-a x\right )} - 15 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {a x}\right )\right )}}{a^{2}}}{48 \, a} \] Input:
integrate(x^2*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, al gorithm="giac")
Output:
1/48*(8*sqrt(pi)*a*x^3 - sqrt(pi)*(8*a^3*x^3*erf(sqrt(a*x)) + (2*(4*sqrt(a *x)*a^2*x^2 + 10*sqrt(a*x)*a*x + 15*sqrt(a*x))*e^(-a*x) - 15*sqrt(pi)*erf( sqrt(a*x)))/sqrt(pi))/a^2 - 6*(2*(4*sqrt(a*x)*a^2*x^2 + 10*sqrt(a*x)*a*x + 15*sqrt(a*x))*e^(-a*x) - 15*sqrt(pi)*erf(sqrt(a*x)))/a^2)/a
Timed out. \[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\int x^2\,\left (\frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{2}+{\mathrm {e}}^{-a\,x}\,\sqrt {a\,x}\right ) \,d x \] Input:
int(x^2*((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2)),x)
Output:
int(x^2*((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2)), x)
\[ \int x^2 \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {-8 \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right ) a^{3} x^{3}+15 \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )+90 e^{a x} \sqrt {a}\, \left (\int \frac {\sqrt {x}}{e^{a x} x}d x \right )-56 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}-140 \sqrt {x}\, \sqrt {a}\, a x -210 \sqrt {x}\, \sqrt {a}+8 \sqrt {\pi }\, e^{a x} a^{3} x^{3}}{48 e^{a x} a^{3}} \] Input:
int(x^2*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2))),x)
Output:
( - 8*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a))*a**3*x**3 + 15*sqrt(pi)*e**(a *x)*erf(sqrt(x)*sqrt(a)) + 90*e**(a*x)*sqrt(a)*int(sqrt(x)/(e**(a*x)*x),x) - 56*sqrt(x)*sqrt(a)*a**2*x**2 - 140*sqrt(x)*sqrt(a)*a*x - 210*sqrt(x)*sq rt(a) + 8*sqrt(pi)*e**(a*x)*a**3*x**3)/(48*e**(a*x)*a**3)