Integrand size = 13, antiderivative size = 80 \[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\frac {2}{5} x^{5/2} \Gamma (2,a+b x)-\frac {2 a e^{-a} \sqrt {x} \Gamma \left (\frac {7}{2},b x\right )}{5 b^2 \sqrt {b x}}-\frac {2 e^{-a} \sqrt {x} \Gamma \left (\frac {9}{2},b x\right )}{5 b^2 \sqrt {b x}} \] Output:
2/5*x^(5/2)*exp(-b*x-a)*(b*x+a+1)-2/5*a*x^(1/2)*((b*x)^(5/2)*exp(-b*x)+5/2 *(b*x)^(3/2)*exp(-b*x)+15/4*(b*x)^(1/2)*exp(-b*x)+15/8*Pi^(1/2)*erfc((b*x) ^(1/2)))/b^2/exp(a)/(b*x)^(1/2)-2/5*x^(1/2)*((b*x)^(7/2)*exp(-b*x)+7/2*(b* x)^(5/2)*exp(-b*x)+35/4*(b*x)^(3/2)*exp(-b*x)+105/8*(b*x)^(1/2)*exp(-b*x)+ 105/16*Pi^(1/2)*erfc((b*x)^(1/2)))/b^2/exp(a)/(b*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.46 \[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\frac {e^{-a} \left (15 (7+2 a) \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )-2 \sqrt {b} e^{-b x} \sqrt {x} \left (105+70 b x+28 b^2 x^2+8 b^3 x^3+a \left (30+20 b x+8 b^2 x^2\right )-8 b^2 e^{a+b x} x^2 \Gamma (2,a+b x)\right )\right )}{40 b^{5/2}} \] Input:
Integrate[x^(3/2)*Gamma[2, a + b*x],x]
Output:
(15*(7 + 2*a)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]] - (2*Sqrt[b]*Sqrt[x]*(105 + 70 *b*x + 28*b^2*x^2 + 8*b^3*x^3 + a*(30 + 20*b*x + 8*b^2*x^2) - 8*b^2*E^(a + b*x)*x^2*Gamma[2, a + b*x]))/E^(b*x))/(40*b^(5/2)*E^a)
Leaf count is larger than twice the leaf count of optimal. \(236\) vs. \(2(80)=160\).
Time = 0.56 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7119, 2629, 2009}
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.
| \(\displaystyle \int x^{3/2} \Gamma (2,a+b x) \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle \frac {2}{5} b \int e^{-a-b x} x^{5/2} (a+b x)dx+\frac {2}{5} x^{5/2} \Gamma (2,a+b x)\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle \frac {2}{5} b \int \left (b e^{-a-b x} x^{7/2}+a e^{-a-b x} x^{5/2}\right )dx+\frac {2}{5} x^{5/2} \Gamma (2,a+b x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} b \left (\frac {15 \sqrt {\pi } a e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{8 b^{7/2}}+\frac {105 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{16 b^{7/2}}-\frac {15 a \sqrt {x} e^{-a-b x}}{4 b^3}-\frac {105 \sqrt {x} e^{-a-b x}}{8 b^3}-\frac {5 a x^{3/2} e^{-a-b x}}{2 b^2}-\frac {35 x^{3/2} e^{-a-b x}}{4 b^2}+x^{7/2} \left (-e^{-a-b x}\right )-\frac {a x^{5/2} e^{-a-b x}}{b}-\frac {7 x^{5/2} e^{-a-b x}}{2 b}\right )+\frac {2}{5} x^{5/2} \Gamma (2,a+b x)\) |
Input:
Int[x^(3/2)*Gamma[2, a + b*x],x]
Output:
(2*b*((-105*E^(-a - b*x)*Sqrt[x])/(8*b^3) - (15*a*E^(-a - b*x)*Sqrt[x])/(4 *b^3) - (35*E^(-a - b*x)*x^(3/2))/(4*b^2) - (5*a*E^(-a - b*x)*x^(3/2))/(2* b^2) - (7*E^(-a - b*x)*x^(5/2))/(2*b) - (a*E^(-a - b*x)*x^(5/2))/b - E^(-a - b*x)*x^(7/2) + (105*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(16*b^(7/2)*E^a) + ( 15*a*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(8*b^(7/2)*E^a)))/5 + (2*x^(5/2)*Gamma [2, a + b*x])/5
Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Int[Gamma[n_, (a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Block[{$UseGamma = True}, Simp[(c + d*x)^(m + 1)*(Gamma[n, a + b*x]/(d*(m + 1))), x] + Simp[b/(d*(m + 1)) Int[(c + d*x)^(m + 1)*((a + b*x)^(n - 1)/E ^(a + b*x)), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ [n, 0] || IntegersQ[m, n]) && NeQ[m, -1]
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.68
| method | result | size |
| meijerg | \(\frac {{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (28 b^{2} x^{2}+70 b x +105\right ) {\mathrm e}^{-b x}}{28}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8}\right )}{b^{\frac {5}{2}}}+\frac {{\mathrm e}^{-a} a \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (10 b x +15\right ) {\mathrm e}^{-b x}}{10}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4}\right )}{b^{\frac {5}{2}}}+\frac {{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (10 b x +15\right ) {\mathrm e}^{-b x}}{10}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4}\right )}{b^{\frac {5}{2}}}\) | \(134\) |
| derivativedivides | \(2 \,{\mathrm e}^{-a} \left (-\frac {x^{\frac {3}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}}{b}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {x^{\frac {3}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}}{b}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {x^{\frac {5}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}}{b}\right )\) | \(188\) |
| default | \(2 \,{\mathrm e}^{-a} \left (-\frac {x^{\frac {3}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}}{b}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {x^{\frac {3}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}}{b}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {x^{\frac {5}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}}{b}\right )\) | \(188\) |
Input:
int(x^(3/2)*exp(-b*x-a)*(b*x+a+1),x,method=_RETURNVERBOSE)
Output:
1/b^(5/2)*exp(-a)*(-1/28*x^(1/2)*b^(1/2)*(28*b^2*x^2+70*b*x+105)*exp(-b*x) +15/8*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))+1/b^(5/2)*exp(-a)*a*(-1/10*x^(1/2)*b^ (1/2)*(10*b*x+15)*exp(-b*x)+3/4*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))+1/b^(5/2)*e xp(-a)*(-1/10*x^(1/2)*b^(1/2)*(10*b*x+15)*exp(-b*x)+3/4*Pi^(1/2)*erf(b^(1/ 2)*x^(1/2)))
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.30 \[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\frac {15 \, \sqrt {\pi } {\left (2 \, a + 7\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} + 2 \, {\left (8 \, b^{3} x^{2} \Gamma \left (2, b x + a\right ) - {\left (8 \, b^{4} x^{3} + 4 \, {\left (2 \, a + 7\right )} b^{3} x^{2} + 10 \, {\left (2 \, a + 7\right )} b^{2} x + 15 \, {\left (2 \, a + 7\right )} b\right )} e^{\left (-b x - a\right )}\right )} \sqrt {x}}{40 \, b^{3}} \] Input:
integrate(x^(3/2)*gamma(2,b*x+a),x, algorithm="fricas")
Output:
1/40*(15*sqrt(pi)*(2*a + 7)*sqrt(b)*erf(sqrt(b)*sqrt(x))*e^(-a) + 2*(8*b^3 *x^2*gamma(2, b*x + a) - (8*b^4*x^3 + 4*(2*a + 7)*b^3*x^2 + 10*(2*a + 7)*b ^2*x + 15*(2*a + 7)*b)*e^(-b*x - a))*sqrt(x))/b^3
Time = 28.69 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.06 \[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\left (- \frac {a x^{\frac {3}{2}} \left (- \sqrt {b x} \left (- b x - \frac {3}{2}\right ) e^{- b x} + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{4}\right )}{b \left (b x\right )^{\frac {3}{2}}} - \frac {x^{\frac {5}{2}} \left (\sqrt {b x} \left (b^{2} x^{2} + \frac {5 b x}{2} + \frac {15}{4}\right ) e^{- b x} + \frac {15 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{8}\right )}{\left (b x\right )^{\frac {5}{2}}} - \frac {x^{\frac {3}{2}} \left (- \sqrt {b x} \left (- b x - \frac {3}{2}\right ) e^{- b x} + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{4}\right )}{b \left (b x\right )^{\frac {3}{2}}}\right ) e^{- a} \] Input:
integrate(x**(3/2)*uppergamma(2,b*x+a),x)
Output:
(-a*x**(3/2)*(-sqrt(b*x)*(-b*x - 3/2)*exp(-b*x) + 3*sqrt(pi)*erfc(sqrt(b*x ))/4)/(b*(b*x)**(3/2)) - x**(5/2)*(sqrt(b*x)*(b**2*x**2 + 5*b*x/2 + 15/4)* exp(-b*x) + 15*sqrt(pi)*erfc(sqrt(b*x))/8)/(b*x)**(5/2) - x**(3/2)*(-sqrt( b*x)*(-b*x - 3/2)*exp(-b*x) + 3*sqrt(pi)*erfc(sqrt(b*x))/4)/(b*(b*x)**(3/2 )))*exp(-a)
\[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\int { x^{\frac {3}{2}} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate(x^(3/2)*gamma(2,b*x+a),x, algorithm="maxima")
Output:
integrate(x^(3/2)*gamma(2, b*x + a), x)
\[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\int { x^{\frac {3}{2}} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate(x^(3/2)*gamma(2,b*x+a),x, algorithm="giac")
Output:
integrate(x^(3/2)*gamma(2, b*x + a), x)
Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.65 \[ \int x^{3/2} \Gamma (2,a+b x) \, dx=-x^{5/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}-\frac {21\,\sqrt {x}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{4\,b^2}-\frac {7\,x^{3/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2\,b}-\frac {3\,a\,\sqrt {x}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2\,b^2}-\frac {a\,x^{3/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{b}-\frac {21\,x^{5/2}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{8\,{\left (b\,x\right )}^{5/2}}-\frac {3\,a\,x^{5/2}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{4\,{\left (b\,x\right )}^{5/2}} \] Input:
int(x^(3/2)*exp(- a - b*x)*(a + b*x + 1),x)
Output:
- x^(5/2)*exp(-a)*exp(-b*x) - (21*x^(1/2)*exp(-a)*exp(-b*x))/(4*b^2) - (7* x^(3/2)*exp(-a)*exp(-b*x))/(2*b) - (3*a*x^(1/2)*exp(-a)*exp(-b*x))/(2*b^2) - (a*x^(3/2)*exp(-a)*exp(-b*x))/b - (21*x^(5/2)*pi^(1/2)*exp(-a)*erfc((b* x)^(1/2)))/(8*(b*x)^(5/2)) - (3*a*x^(5/2)*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2 )))/(4*(b*x)^(5/2))
\[ \int x^{3/2} \Gamma (2,a+b x) \, dx=\frac {6 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right ) a +21 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right )-8 \sqrt {x}\, a b x -12 \sqrt {x}\, a -8 \sqrt {x}\, b^{2} x^{2}-28 \sqrt {x}\, b x -42 \sqrt {x}}{8 e^{b x +a} b^{2}} \] Input:
int(x^(3/2)*exp(-b*x-a)*(b*x+a+1),x)
Output:
(6*e**(b*x)*int(sqrt(x)/(e**(b*x)*x),x)*a + 21*e**(b*x)*int(sqrt(x)/(e**(b *x)*x),x) - 8*sqrt(x)*a*b*x - 12*sqrt(x)*a - 8*sqrt(x)*b**2*x**2 - 28*sqrt (x)*b*x - 42*sqrt(x))/(8*e**(a + b*x)*b**2)