3.152 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^{12}} \, dx\)

Optimal. Leaf size=34 \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]

[Out]

(f^a*Gamma[11/2, -((b*Log[f])/x^2)])/(2*x^11*(-((b*Log[f])/x^2))^(11/2))

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Rubi [A]  time = 0.0256291, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^12,x]

[Out]

(f^a*Gamma[11/2, -((b*Log[f])/x^2)])/(2*x^11*(-((b*Log[f])/x^2))^(11/2))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{f^{a+\frac{b}{x^2}}}{x^{12}} \, dx &=\frac{f^a \Gamma \left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0052913, size = 34, normalized size = 1. \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^12,x]

[Out]

(f^a*Gamma[11/2, -((b*Log[f])/x^2)])/(2*x^11*(-((b*Log[f])/x^2))^(11/2))

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Maple [A]  time = 0.08, size = 146, normalized size = 4.3 \begin{align*} -{\frac{{f}^{a}}{2\,b{x}^{9}\ln \left ( f \right ) }{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{9\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{7}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{63\,{f}^{a}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{5}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{315\,{f}^{a}}{16\,{b}^{4}{x}^{3} \left ( \ln \left ( f \right ) \right ) ^{4}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{945\,{f}^{a}}{32\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}x}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{945\,{f}^{a}\sqrt{\pi }}{64\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^12,x)

[Out]

-1/2*f^a*f^(b/x^2)/x^9/b/ln(f)+9/4*f^a/ln(f)^2/b^2*f^(b/x^2)/x^7-63/8*f^a/ln(f)^3/b^3*f^(b/x^2)/x^5+315/16*f^a
/ln(f)^4/b^4*f^(b/x^2)/x^3-945/32*f^a/ln(f)^5/b^5*f^(b/x^2)/x+945/64*f^a/ln(f)^5/b^5*Pi^(1/2)/(-b*ln(f))^(1/2)
*erf((-b*ln(f))^(1/2)/x)

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Maxima [A]  time = 1.25357, size = 38, normalized size = 1.12 \begin{align*} \frac{f^{a} \Gamma \left (\frac{11}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{11} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^12,x, algorithm="maxima")

[Out]

1/2*f^a*gamma(11/2, -b*log(f)/x^2)/(x^11*(-b*log(f)/x^2)^(11/2))

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Fricas [A]  time = 1.78804, size = 293, normalized size = 8.62 \begin{align*} -\frac{945 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} x^{9} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) + 2 \,{\left (945 \, b x^{8} \log \left (f\right ) - 630 \, b^{2} x^{6} \log \left (f\right )^{2} + 252 \, b^{3} x^{4} \log \left (f\right )^{3} - 72 \, b^{4} x^{2} \log \left (f\right )^{4} + 16 \, b^{5} \log \left (f\right )^{5}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{64 \, b^{6} x^{9} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^12,x, algorithm="fricas")

[Out]

-1/64*(945*sqrt(pi)*sqrt(-b*log(f))*f^a*x^9*erf(sqrt(-b*log(f))/x) + 2*(945*b*x^8*log(f) - 630*b^2*x^6*log(f)^
2 + 252*b^3*x^4*log(f)^3 - 72*b^4*x^2*log(f)^4 + 16*b^5*log(f)^5)*f^((a*x^2 + b)/x^2))/(b^6*x^9*log(f)^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)/x**12,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^12,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^12, x)