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 = 0.0393777, 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 \[ \frac{1}{2} x^{11} f^a \left (-\frac{b \log (f)}{x^2}\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-\frac{b \log (f)}{x^2}\right ) \]

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

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Rubi in Sympy [A]  time = 3.19013, size = 36, normalized size = 1.06 \[ \frac{f^{a} x^{11} \left (- \frac{b \log{\left (f \right )}}{x^{2}}\right )^{\frac{11}{2}} \Gamma{\left (- \frac{11}{2},- \frac{b \log{\left (f \right )}}{x^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)*x**10,x)

[Out]

f**a*x**11*(-b*log(f)/x**2)**(11/2)*Gamma(-11/2, -b*log(f)/x**2)/2

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Mathematica [B]  time = 0.0782941, size = 110, normalized size = 3.24 \[ \frac{f^a \left (x f^{\frac{b}{x^2}} \left (32 b^5 \log ^5(f)+16 b^4 x^2 \log ^4(f)+24 b^3 x^4 \log ^3(f)+60 b^2 x^6 \log ^2(f)+210 b x^8 \log (f)+945 x^{10}\right )-32 \sqrt{\pi } b^{11/2} \log ^{\frac{11}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right )}{10395} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)*x^10,x]

[Out]

(f^a*(-32*b^(11/2)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(11/2) + f^(b/
x^2)*x*(945*x^10 + 210*b*x^8*Log[f] + 60*b^2*x^6*Log[f]^2 + 24*b^3*x^4*Log[f]^3
+ 16*b^4*x^2*Log[f]^4 + 32*b^5*Log[f]^5)))/10395

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Maple [A]  time = 0.072, size = 155, normalized size = 4.6 \[{\frac{{f}^{a}{x}^{11}}{11}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{9}}{99}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{7}}{693}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{5}}{3465}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{x}^{3}}{10395}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{32\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}x}{10395}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{32\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}\sqrt{\pi }}{10395}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

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*f^(b/x^2)*x^11+2/99*f^a*ln(f)*b*f^(b/x^2)*x^9+4/693*f^a*ln(f)^2*b^2*f^(
b/x^2)*x^7+8/3465*f^a*ln(f)^3*b^3*f^(b/x^2)*x^5+16/10395*f^a*ln(f)^4*b^4*f^(b/x^
2)*x^3+32/10395*f^a*ln(f)^5*b^5*f^(b/x^2)*x-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 = 0.986898, size = 174, normalized size = 5.12 \[ -\frac{32 \, \sqrt{\pi } b^{6} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{6}}{10395 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{10395} \,{\left (945 \, f^{a} x^{11} + 210 \, b f^{a} x^{9} \log \left (f\right ) + 60 \, b^{2} f^{a} x^{7} \log \left (f\right )^{2} + 24 \, b^{3} f^{a} x^{5} \log \left (f\right )^{3} + 16 \, b^{4} f^{a} x^{3} \log \left (f\right )^{4} + 32 \, b^{5} f^{a} x \log \left (f\right )^{5}\right )} f^{\frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/10395*sqrt(pi)*b^6*f^a*(erf(sqrt(-b*log(f)/x^2)) - 1)*log(f)^6/(x*sqrt(-b*lo
g(f)/x^2)) + 1/10395*(945*f^a*x^11 + 210*b*f^a*x^9*log(f) + 60*b^2*f^a*x^7*log(f
)^2 + 24*b^3*f^a*x^5*log(f)^3 + 16*b^4*f^a*x^3*log(f)^4 + 32*b^5*f^a*x*log(f)^5)
*f^(b/x^2)

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Fricas [A]  time = 0.29887, size = 161, normalized size = 4.74 \[ -\frac{32 \, \sqrt{\pi } b^{6} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{6} -{\left (945 \, x^{11} + 210 \, b x^{9} \log \left (f\right ) + 60 \, b^{2} x^{7} \log \left (f\right )^{2} + 24 \, b^{3} x^{5} \log \left (f\right )^{3} + 16 \, b^{4} x^{3} \log \left (f\right )^{4} + 32 \, b^{5} x \log \left (f\right )^{5}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{10395 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10395*(32*sqrt(pi)*b^6*f^a*erf(sqrt(-b*log(f))/x)*log(f)^6 - (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)*sqrt(-b*log(f))*f^((a*x^2 + b)/x^2))/sqrt(-b*log(f))

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{10}\,{d x} \]

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)