3.141 \(\int f^{a+\frac{b}{x^2}} x^{10} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0241223, 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} \[ \frac{1}{2} x^{11} f^a \left (-\frac{b \log (f)}{x^2}\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-\frac{b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x^10,x]

[Out]

(f^a*x^11*Gamma[-11/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(11/2))/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 f^{a+\frac{b}{x^2}} x^{10} \, dx &=\frac{1}{2} f^a x^{11} \Gamma \left (-\frac{11}{2},-\frac{b \log (f)}{x^2}\right ) \left (-\frac{b \log (f)}{x^2}\right )^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0043307, size = 34, normalized size = 1. \[ \frac{1}{2} x^{11} f^a \left (-\frac{b \log (f)}{x^2}\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-\frac{b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x^10,x]

[Out]

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

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Maple [A]  time = 0.101, size = 155, normalized size = 4.6 \begin{align*}{\frac{{f}^{a}{x}^{11}}{11}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{9}}{99}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{7}}{693}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{5}}{3465}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{x}^{3}}{10395}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{32\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}x}{10395}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{32\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}\sqrt{\pi }}{10395}{\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^10,x)

[Out]

1/11*f^a*x^11*f^(b/x^2)+2/99*f^a*ln(f)*b*x^9*f^(b/x^2)+4/693*f^a*ln(f)^2*b^2*x^7*f^(b/x^2)+8/3465*f^a*ln(f)^3*
b^3*x^5*f^(b/x^2)+16/10395*f^a*ln(f)^4*b^4*x^3*f^(b/x^2)+32/10395*f^a*ln(f)^5*b^5*x*f^(b/x^2)-32/10395*f^a*ln(
f)^6*b^6*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)/x)

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Maxima [A]  time = 1.20637, size = 38, normalized size = 1.12 \begin{align*} \frac{1}{2} \, f^{a} x^{11} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{11}{2}} \Gamma \left (-\frac{11}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.85431, size = 298, normalized size = 8.76 \begin{align*} \frac{32}{10395} \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b^{5} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{5} + \frac{1}{10395} \,{\left (945 \, x^{11} + 210 \, b x^{9} \log \left (f\right ) + 60 \, b^{2} x^{7} \log \left (f\right )^{2} + 24 \, b^{3} x^{5} \log \left (f\right )^{3} + 16 \, b^{4} x^{3} \log \left (f\right )^{4} + 32 \, b^{5} x \log \left (f\right )^{5}\right )} f^{\frac{a x^{2} + b}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^10,x, algorithm="fricas")

[Out]

32/10395*sqrt(pi)*sqrt(-b*log(f))*b^5*f^a*erf(sqrt(-b*log(f))/x)*log(f)^5 + 1/10395*(945*x^11 + 210*b*x^9*log(
f) + 60*b^2*x^7*log(f)^2 + 24*b^3*x^5*log(f)^3 + 16*b^4*x^3*log(f)^4 + 32*b^5*x*log(f)^5)*f^((a*x^2 + b)/x^2)

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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**10,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{10}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)*x^10,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^10, x)