Integrand size = 13, antiderivative size = 80 \[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\frac {2 a b^3 e^{-a} \sqrt {b x} \Gamma \left (-\frac {5}{2},b x\right )}{7 \sqrt {x}}+\frac {2 b^3 e^{-a} \sqrt {b x} \Gamma \left (-\frac {3}{2},b x\right )}{7 \sqrt {x}}-\frac {2 \Gamma (2,a+b x)}{7 x^{7/2}} \] Output:
2/7*a*b^3*(b*x)^(1/2)*(-8/15*Pi^(1/2)*erfc((b*x)^(1/2))+8/15/(b*x)^(1/2)*e xp(-b*x)-4/15/(b*x)^(3/2)*exp(-b*x)+2/5/(b*x)^(5/2)*exp(-b*x))/exp(a)/x^(1 /2)+2/7*b^3*(b*x)^(1/2)*(4/3*Pi^(1/2)*erfc((b*x)^(1/2))-4/3/(b*x)^(1/2)*ex p(-b*x)+2/3/(b*x)^(3/2)*exp(-b*x))/exp(a)/x^(1/2)-2/7*exp(-b*x-a)*(b*x+a+1 )/x^(7/2)
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20 \[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\frac {2}{105} \left (4 (-5+2 a) b^{7/2} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )+\frac {2 b e^{-a-b x} x \left (5 b x (1-2 b x)+a \left (3-2 b x+4 b^2 x^2\right )\right )-15 \Gamma (2,a+b x)}{x^{7/2}}\right ) \] Input:
Integrate[Gamma[2, a + b*x]/x^(9/2),x]
Output:
(2*((4*(-5 + 2*a)*b^(7/2)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/E^a + (2*b*E^(-a - b*x)*x*(5*b*x*(1 - 2*b*x) + a*(3 - 2*b*x + 4*b^2*x^2)) - 15*Gamma[2, a + b*x])/x^(7/2)))/105
Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(80)=160\).
Time = 0.51 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.40, 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.
| \(\displaystyle \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle -\frac {2}{7} b \int \frac {e^{-a-b x} (a+b x)}{x^{7/2}}dx-\frac {2 \Gamma (2,a+b x)}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle -\frac {2}{7} b \int \left (\frac {e^{-a-b x} a}{x^{7/2}}+\frac {b e^{-a-b x}}{x^{5/2}}\right )dx-\frac {2 \Gamma (2,a+b x)}{7 x^{7/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{7} b \left (-\frac {8}{15} \sqrt {\pi } a e^{-a} b^{5/2} \text {erf}\left (\sqrt {b} \sqrt {x}\right )+\frac {4}{3} \sqrt {\pi } e^{-a} b^{5/2} \text {erf}\left (\sqrt {b} \sqrt {x}\right )-\frac {8 a b^2 e^{-a-b x}}{15 \sqrt {x}}+\frac {4 b^2 e^{-a-b x}}{3 \sqrt {x}}+\frac {4 a b e^{-a-b x}}{15 x^{3/2}}-\frac {2 b e^{-a-b x}}{3 x^{3/2}}-\frac {2 a e^{-a-b x}}{5 x^{5/2}}\right )-\frac {2 \Gamma (2,a+b x)}{7 x^{7/2}}\) |
Input:
Int[Gamma[2, a + b*x]/x^(9/2),x]
Output:
(-2*b*((-2*a*E^(-a - b*x))/(5*x^(5/2)) - (2*b*E^(-a - b*x))/(3*x^(3/2)) + (4*a*b*E^(-a - b*x))/(15*x^(3/2)) + (4*b^2*E^(-a - b*x))/(3*Sqrt[x]) - (8* a*b^2*E^(-a - b*x))/(15*Sqrt[x]) + (4*b^(5/2)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x] ])/(3*E^a) - (8*a*b^(5/2)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(15*E^a)))/7 - (2 *Gamma[2, a + b*x])/(7*x^(7/2))
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 = 0.35 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.08
| method | result | size |
| meijerg | \(b^{\frac {7}{2}} {\mathrm e}^{-a} \left (-\frac {2 \left (\frac {4}{3} b^{2} x^{2}-\frac {2}{3} b x +1\right ) {\mathrm e}^{-b x}}{5 x^{\frac {5}{2}} b^{\frac {5}{2}}}-\frac {8 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{15}\right )+{\mathrm e}^{-a} a \,b^{\frac {7}{2}} \left (-\frac {2 \left (-\frac {8}{15} b^{3} x^{3}+\frac {4}{15} b^{2} x^{2}-\frac {2}{5} b x +1\right ) {\mathrm e}^{-b x}}{7 x^{\frac {7}{2}} b^{\frac {7}{2}}}+\frac {16 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{105}\right )+b^{\frac {7}{2}} {\mathrm e}^{-a} \left (-\frac {2 \left (-\frac {8}{15} b^{3} x^{3}+\frac {4}{15} b^{2} x^{2}-\frac {2}{5} b x +1\right ) {\mathrm e}^{-b x}}{7 x^{\frac {7}{2}} b^{\frac {7}{2}}}+\frac {16 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{105}\right )\) | \(166\) |
| derivativedivides | \(2 \,{\mathrm e}^{-a} \left (-\frac {{\mathrm e}^{-b x}}{7 x^{\frac {7}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )}{7}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {{\mathrm e}^{-b x}}{7 x^{\frac {7}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )}{7}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )\) | \(215\) |
| default | \(2 \,{\mathrm e}^{-a} \left (-\frac {{\mathrm e}^{-b x}}{7 x^{\frac {7}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )}{7}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {{\mathrm e}^{-b x}}{7 x^{\frac {7}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )}{7}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {{\mathrm e}^{-b x}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )}{3}\right )}{5}\right )\) | \(215\) |
Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(9/2),x,method=_RETURNVERBOSE)
Output:
b^(7/2)*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)))+exp(-a)*a*b^(7/2)*(-2/7/x^(7/2)/b^(7/2)* (-8/15*b^3*x^3+4/15*b^2*x^2-2/5*b*x+1)*exp(-b*x)+16/105*Pi^(1/2)*erf(b^(1/ 2)*x^(1/2)))+b^(7/2)*exp(-a)*(-2/7/x^(7/2)/b^(7/2)*(-8/15*b^3*x^3+4/15*b^2 *x^2-2/5*b*x+1)*exp(-b*x)+16/105*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14 \[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\frac {2 \, {\left (4 \, \sqrt {\pi } {\left (2 \, a - 5\right )} b^{\frac {7}{2}} x^{4} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} + {\left (2 \, {\left (2 \, {\left (2 \, a - 5\right )} b^{3} x^{3} - {\left (2 \, a - 5\right )} b^{2} x^{2} + 3 \, a b x\right )} e^{\left (-b x - a\right )} - 15 \, \Gamma \left (2, b x + a\right )\right )} \sqrt {x}\right )}}{105 \, x^{4}} \] Input:
integrate(gamma(2,b*x+a)/x^(9/2),x, algorithm="fricas")
Output:
2/105*(4*sqrt(pi)*(2*a - 5)*b^(7/2)*x^4*erf(sqrt(b)*sqrt(x))*e^(-a) + (2*( 2*(2*a - 5)*b^3*x^3 - (2*a - 5)*b^2*x^2 + 3*a*b*x)*e^(-b*x - a) - 15*gamma (2, b*x + a))*sqrt(x))/x^4
Timed out. \[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(uppergamma(2,b*x+a)/x**(9/2),x)
Output:
Timed out
\[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{x^{\frac {9}{2}}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(9/2),x, algorithm="maxima")
Output:
integrate(gamma(2, b*x + a)/x^(9/2), x)
\[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{x^{\frac {9}{2}}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(9/2),x, algorithm="giac")
Output:
integrate(gamma(2, b*x + a)/x^(9/2), x)
Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.42 \[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=-\frac {30\,b^2\,x^2\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}-20\,b^3\,x^3\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}+40\,b^4\,x^4\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}+30\,b\,x\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}-12\,a\,b^2\,x^2\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}+8\,a\,b^3\,x^3\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}-16\,a\,b^4\,x^4\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}+30\,a\,b\,x\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}-40\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {b\,x}+16\,a\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {b\,x}}{105\,b\,x^{9/2}} \] Input:
int((exp(- a - b*x)*(a + b*x + 1))/x^(9/2),x)
Output:
-(30*b^2*x^2*exp(-a)*exp(-b*x) - 20*b^3*x^3*exp(-a)*exp(-b*x) + 40*b^4*x^4 *exp(-a)*exp(-b*x) + 30*b*x*exp(-a)*exp(-b*x) - 12*a*b^2*x^2*exp(-a)*exp(- b*x) + 8*a*b^3*x^3*exp(-a)*exp(-b*x) - 16*a*b^4*x^4*exp(-a)*exp(-b*x) + 30 *a*b*x*exp(-a)*exp(-b*x) - 40*b^4*x^4*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2))*( b*x)^(1/2) + 16*a*b^4*x^4*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2))*(b*x)^(1/2))/ (105*b*x^(9/2))
\[ \int \frac {\Gamma (2,a+b x)}{x^{9/2}} \, dx=\frac {-\frac {8 \sqrt {x}\, e^{b x} \left (\int \frac {1}{\sqrt {x}\, e^{b x} x}d x \right ) a \,b^{3} x^{3}}{105}+\frac {4 \sqrt {x}\, e^{b x} \left (\int \frac {1}{\sqrt {x}\, e^{b x} x}d x \right ) b^{3} x^{3}}{21}-\frac {8 a \,b^{2} x^{2}}{105}+\frac {4 a b x}{35}-\frac {2 a}{7}+\frac {4 b^{2} x^{2}}{21}-\frac {2 b x}{7}-\frac {2}{7}}{\sqrt {x}\, e^{b x +a} x^{3}} \] Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(9/2),x)
Output:
(2*( - 4*sqrt(x)*e**(b*x)*int(1/(sqrt(x)*e**(b*x)*x),x)*a*b**3*x**3 + 10*s qrt(x)*e**(b*x)*int(1/(sqrt(x)*e**(b*x)*x),x)*b**3*x**3 - 4*a*b**2*x**2 + 6*a*b*x - 15*a + 10*b**2*x**2 - 15*b*x - 15))/(105*sqrt(x)*e**(a + b*x)*x* *3)