3.129 \(\int f^{a+\frac{b}{x^2}} x^9 \, dx\)

Optimal. Leaf size=24 \[ -\frac{1}{2} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x^2}\right ) \]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x^2)]*Log[f]^5)/2

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Rubi [A]  time = 0.039484, antiderivative size = 24, 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 \[ -\frac{1}{2} b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)*x^9,x]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x^2)]*Log[f]^5)/2

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Rubi in Sympy [A]  time = 3.62792, size = 27, normalized size = 1.12 \[ - \frac{b^{5} f^{a} \Gamma{\left (-5,- \frac{b \log{\left (f \right )}}{x^{2}} \right )} \log{\left (f \right )}^{5}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)*x**9,x)

[Out]

-b**5*f**a*Gamma(-5, -b*log(f)/x**2)*log(f)**5/2

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Mathematica [B]  time = 0.0439682, size = 81, normalized size = 3.38 \[ \frac{1}{240} f^a \left (x^2 f^{\frac{b}{x^2}} \left (b^4 \log ^4(f)+b^3 x^2 \log ^3(f)+2 b^2 x^4 \log ^2(f)+6 b x^6 \log (f)+24 x^8\right )-b^5 \log ^5(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)*x^9,x]

[Out]

(f^a*(-(b^5*ExpIntegralEi[(b*Log[f])/x^2]*Log[f]^5) + f^(b/x^2)*x^2*(24*x^8 + 6*
b*x^6*Log[f] + 2*b^2*x^4*Log[f]^2 + b^3*x^2*Log[f]^3 + b^4*Log[f]^4)))/240

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Maple [B]  time = 0.039, size = 123, normalized size = 5.1 \[{\frac{{f}^{a}{x}^{10}}{10}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a}\ln \left ( f \right ) b{x}^{8}}{40}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{6}}{120}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{4}}{240}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{x}^{2}}{240}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}}{240}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)*x^9,x)

[Out]

1/10*f^a*x^10*f^(b/x^2)+1/40*f^a*ln(f)*b*x^8*f^(b/x^2)+1/120*f^a*ln(f)^2*b^2*x^6
*f^(b/x^2)+1/240*f^a*ln(f)^3*b^3*x^4*f^(b/x^2)+1/240*f^a*ln(f)^4*b^4*x^2*f^(b/x^
2)+1/240*f^a*ln(f)^5*b^5*Ei(1,-b*ln(f)/x^2)

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Maxima [A]  time = 0.802582, size = 126, normalized size = 5.25 \[ -\frac{1}{240} \, b^{5} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{5} + \frac{1}{240} \,{\left (24 \, f^{a} x^{10} + 6 \, b f^{a} x^{8} \log \left (f\right ) + 2 \, b^{2} f^{a} x^{6} \log \left (f\right )^{2} + b^{3} f^{a} x^{4} \log \left (f\right )^{3} + b^{4} f^{a} x^{2} \log \left (f\right )^{4}\right )} f^{\frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^9,x, algorithm="maxima")

[Out]

-1/240*b^5*f^a*Ei(b*log(f)/x^2)*log(f)^5 + 1/240*(24*f^a*x^10 + 6*b*f^a*x^8*log(
f) + 2*b^2*f^a*x^6*log(f)^2 + b^3*f^a*x^4*log(f)^3 + b^4*f^a*x^2*log(f)^4)*f^(b/
x^2)

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Fricas [A]  time = 0.263752, size = 113, normalized size = 4.71 \[ -\frac{1}{240} \, b^{5} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{5} + \frac{1}{240} \,{\left (24 \, x^{10} + 6 \, b x^{8} \log \left (f\right ) + 2 \, b^{2} x^{6} \log \left (f\right )^{2} + b^{3} x^{4} \log \left (f\right )^{3} + b^{4} x^{2} \log \left (f\right )^{4}\right )} f^{\frac{a x^{2} + b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^9,x, algorithm="fricas")

[Out]

-1/240*b^5*f^a*Ei(b*log(f)/x^2)*log(f)^5 + 1/240*(24*x^10 + 6*b*x^8*log(f) + 2*b
^2*x^6*log(f)^2 + b^3*x^4*log(f)^3 + b^4*x^2*log(f)^4)*f^((a*x^2 + b)/x^2)

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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**9,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^9,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^9, x)