Integrand size = 13, antiderivative size = 80 \[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=\frac {2 a b^2 e^{-a} \sqrt {b x} \Gamma \left (-\frac {3}{2},b x\right )}{5 \sqrt {x}}+\frac {2 b^2 e^{-a} \sqrt {b x} \Gamma \left (-\frac {1}{2},b x\right )}{5 \sqrt {x}}-\frac {2 \Gamma (2,a+b x)}{5 x^{5/2}} \] Output:
2/5*a*b^2*(b*x)^(1/2)*(4/3*Pi^(1/2)*erfc((b*x)^(1/2))-4/3/(b*x)^(1/2)*exp( -b*x)+2/3/(b*x)^(3/2)*exp(-b*x))/exp(a)/x^(1/2)+2/5*b^2*(b*x)^(1/2)*(-2*Pi ^(1/2)*erfc((b*x)^(1/2))+2/(b*x)^(1/2)*exp(-b*x))/exp(a)/x^(1/2)-2/5*exp(- b*x-a)*(b*x+a+1)/x^(5/2)
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=\frac {2}{15} \left (-2 (-3+2 a) b^{5/2} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )+\frac {2 b e^{-a-b x} x (a+3 b x-2 a b x)-3 \Gamma (2,a+b x)}{x^{5/2}}\right ) \] Input:
Integrate[Gamma[2, a + b*x]/x^(7/2),x]
Output:
(2*((-2*(-3 + 2*a)*b^(5/2)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/E^a + (2*b*E^(-a - b*x)*x*(a + 3*b*x - 2*a*b*x) - 3*Gamma[2, a + b*x])/x^(5/2)))/15
Time = 0.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.79, 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.
| \(\displaystyle \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle -\frac {2}{5} b \int \frac {e^{-a-b x} (a+b x)}{x^{5/2}}dx-\frac {2 \Gamma (2,a+b x)}{5 x^{5/2}}\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle -\frac {2}{5} b \int \left (\frac {e^{-a-b x} a}{x^{5/2}}+\frac {b e^{-a-b x}}{x^{3/2}}\right )dx-\frac {2 \Gamma (2,a+b x)}{5 x^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{5} b \left (\frac {4}{3} \sqrt {\pi } a e^{-a} b^{3/2} \text {erf}\left (\sqrt {b} \sqrt {x}\right )-2 \sqrt {\pi } e^{-a} b^{3/2} \text {erf}\left (\sqrt {b} \sqrt {x}\right )-\frac {2 a e^{-a-b x}}{3 x^{3/2}}+\frac {4 a b e^{-a-b x}}{3 \sqrt {x}}-\frac {2 b e^{-a-b x}}{\sqrt {x}}\right )-\frac {2 \Gamma (2,a+b x)}{5 x^{5/2}}\) |
Input:
Int[Gamma[2, a + b*x]/x^(7/2),x]
Output:
(-2*b*((-2*a*E^(-a - b*x))/(3*x^(3/2)) - (2*b*E^(-a - b*x))/Sqrt[x] + (4*a *b*E^(-a - b*x))/(3*Sqrt[x]) - (2*b^(3/2)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/E ^a + (4*a*b^(3/2)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(3*E^a)))/5 - (2*Gamma[2, a + b*x])/(5*x^(5/2))
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 = 0.32 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.78
| method | result | size |
| meijerg | \(b^{\frac {5}{2}} {\mathrm e}^{-a} \left (-\frac {2 \left (-2 b x +1\right ) {\mathrm e}^{-b x}}{3 x^{\frac {3}{2}} b^{\frac {3}{2}}}+\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{3}\right )+{\mathrm e}^{-a} a \,b^{\frac {5}{2}} \left (-\frac {2 \left (\frac {4}{3} b^{2} x^{2}-\frac {2}{3} b x +1\right ) {\mathrm e}^{-b x}}{5 x^{\frac {5}{2}} b^{\frac {5}{2}}}-\frac {8 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{15}\right )+b^{\frac {5}{2}} {\mathrm e}^{-a} \left (-\frac {2 \left (\frac {4}{3} b^{2} x^{2}-\frac {2}{3} b x +1\right ) {\mathrm e}^{-b x}}{5 x^{\frac {5}{2}} b^{\frac {5}{2}}}-\frac {8 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{15}\right )\) | \(142\) |
| derivativedivides | \(2 \,{\mathrm e}^{-a} \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )\) | \(173\) |
| default | \(2 \,{\mathrm e}^{-a} \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )\) | \(173\) |
Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(7/2),x,method=_RETURNVERBOSE)
Output:
b^(5/2)*exp(-a)*(-2/3/x^(3/2)/b^(3/2)*(-2*b*x+1)*exp(-b*x)+4/3*Pi^(1/2)*er f(b^(1/2)*x^(1/2)))+exp(-a)*a*b^(5/2)*(-2/5/x^(5/2)/b^(5/2)*(4/3*b^2*x^2-2 /3*b*x+1)*exp(-b*x)-8/15*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))+b^(5/2)*exp(-a)*(- 2/5/x^(5/2)/b^(5/2)*(4/3*b^2*x^2-2/3*b*x+1)*exp(-b*x)-8/15*Pi^(1/2)*erf(b^ (1/2)*x^(1/2)))
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {\pi } {\left (2 \, a - 3\right )} b^{\frac {5}{2}} x^{3} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} + {\left (2 \, {\left ({\left (2 \, a - 3\right )} b^{2} x^{2} - a b x\right )} e^{\left (-b x - a\right )} + 3 \, \Gamma \left (2, b x + a\right )\right )} \sqrt {x}\right )}}{15 \, x^{3}} \] Input:
integrate(gamma(2,b*x+a)/x^(7/2),x, algorithm="fricas")
Output:
-2/15*(2*sqrt(pi)*(2*a - 3)*b^(5/2)*x^3*erf(sqrt(b)*sqrt(x))*e^(-a) + (2*( (2*a - 3)*b^2*x^2 - a*b*x)*e^(-b*x - a) + 3*gamma(2, b*x + a))*sqrt(x))/x^ 3
Time = 102.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.31 \[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=\left (- \frac {a \left (b x\right )^{\frac {7}{2}} \left (- \frac {8 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{15} - \frac {\left (- \frac {8 b^{2} x^{2}}{15} + \frac {4 b x}{15} - \frac {2}{5}\right ) e^{- b x}}{\left (b x\right )^{\frac {5}{2}}}\right )}{b x^{\frac {7}{2}}} - \frac {\left (b x\right )^{\frac {5}{2}} \cdot \left (\frac {4 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{3} - \frac {\left (\frac {4 b x}{3} - \frac {2}{3}\right ) e^{- b x}}{\left (b x\right )^{\frac {3}{2}}}\right )}{x^{\frac {5}{2}}} - \frac {\left (b x\right )^{\frac {7}{2}} \left (- \frac {8 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{15} - \frac {\left (- \frac {8 b^{2} x^{2}}{15} + \frac {4 b x}{15} - \frac {2}{5}\right ) e^{- b x}}{\left (b x\right )^{\frac {5}{2}}}\right )}{b x^{\frac {7}{2}}}\right ) e^{- a} \] Input:
integrate(uppergamma(2,b*x+a)/x**(7/2),x)
Output:
(-a*(b*x)**(7/2)*(-8*sqrt(pi)*erfc(sqrt(b*x))/15 - (-8*b**2*x**2/15 + 4*b* x/15 - 2/5)*exp(-b*x)/(b*x)**(5/2))/(b*x**(7/2)) - (b*x)**(5/2)*(4*sqrt(pi )*erfc(sqrt(b*x))/3 - (4*b*x/3 - 2/3)*exp(-b*x)/(b*x)**(3/2))/x**(5/2) - ( b*x)**(7/2)*(-8*sqrt(pi)*erfc(sqrt(b*x))/15 - (-8*b**2*x**2/15 + 4*b*x/15 - 2/5)*exp(-b*x)/(b*x)**(5/2))/(b*x**(7/2)))*exp(-a)
\[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{x^{\frac {7}{2}}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(7/2),x, algorithm="maxima")
Output:
integrate(gamma(2, b*x + a)/x^(7/2), x)
\[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{x^{\frac {7}{2}}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(7/2),x, algorithm="giac")
Output:
integrate(gamma(2, b*x + a)/x^(7/2), x)
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.55 \[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=-\frac {2\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}\,\left (3\,b\,x+3\,b^2\,x^2-6\,b^3\,x^3-2\,a\,b^2\,x^2+4\,a\,b^3\,x^3+3\,a\,b\,x-4\,\sqrt {\pi }\,{\mathrm {e}}^{b\,x}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,{\left (b\,x\right )}^{7/2}-4\,a\,\sqrt {\pi }\,{\mathrm {e}}^{b\,x}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,{\left (b\,x\right )}^{7/2}+10\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b\,x}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,{\left (b\,x\right )}^{5/2}\right )}{15\,b\,x^{7/2}} \] Input:
int((exp(- a - b*x)*(a + b*x + 1))/x^(7/2),x)
Output:
-(2*exp(-a)*exp(-b*x)*(3*b*x + 3*b^2*x^2 - 6*b^3*x^3 - 2*a*b^2*x^2 + 4*a*b ^3*x^3 + 3*a*b*x - 4*pi^(1/2)*exp(b*x)*erfc((b*x)^(1/2))*(b*x)^(7/2) - 4*a *pi^(1/2)*exp(b*x)*erfc((b*x)^(1/2))*(b*x)^(7/2) + 10*b*x*pi^(1/2)*exp(b*x )*erfc((b*x)^(1/2))*(b*x)^(5/2)))/(15*b*x^(7/2))
\[ \int \frac {\Gamma (2,a+b x)}{x^{7/2}} \, dx=\frac {\frac {4 \sqrt {x}\, e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x^{2}}d x \right ) a \,b^{2} x^{2}}{15}-\frac {2 \sqrt {x}\, e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x^{2}}d x \right ) b^{2} x^{2}}{5}+\frac {4 a b x}{15}-\frac {2 a}{5}-\frac {2 b x}{5}-\frac {2}{5}}{\sqrt {x}\, e^{b x +a} x^{2}} \] Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(7/2),x)
Output:
(2*(2*sqrt(x)*e**(b*x)*int(sqrt(x)/(e**(b*x)*x**2),x)*a*b**2*x**2 - 3*sqrt (x)*e**(b*x)*int(sqrt(x)/(e**(b*x)*x**2),x)*b**2*x**2 + 2*a*b*x - 3*a - 3* b*x - 3))/(15*sqrt(x)*e**(a + b*x)*x**2)