3.163 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^{10}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 f^{a+\frac{b}{x^3}}}{3 b^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^6 \log (f)} \]

[Out]

(-2*f^(a + b/x^3))/(3*b^3*Log[f]^3) + (2*f^(a + b/x^3))/(3*b^2*x^3*Log[f]^2) - f
^(a + b/x^3)/(3*b*x^6*Log[f])

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Rubi [A]  time = 0.112556, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 f^{a+\frac{b}{x^3}}}{3 b^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^6 \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^3)/x^10,x]

[Out]

(-2*f^(a + b/x^3))/(3*b^3*Log[f]^3) + (2*f^(a + b/x^3))/(3*b^2*x^3*Log[f]^2) - f
^(a + b/x^3)/(3*b*x^6*Log[f])

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Rubi in Sympy [A]  time = 11.3278, size = 61, normalized size = 0.91 \[ - \frac{f^{a + \frac{b}{x^{3}}}}{3 b x^{6} \log{\left (f \right )}} + \frac{2 f^{a + \frac{b}{x^{3}}}}{3 b^{2} x^{3} \log{\left (f \right )}^{2}} - \frac{2 f^{a + \frac{b}{x^{3}}}}{3 b^{3} \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**3)/x**10,x)

[Out]

-f**(a + b/x**3)/(3*b*x**6*log(f)) + 2*f**(a + b/x**3)/(3*b**2*x**3*log(f)**2) -
2*f**(a + b/x**3)/(3*b**3*log(f)**3)

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Mathematica [A]  time = 0.0148043, size = 45, normalized size = 0.67 \[ -\frac{f^{a+\frac{b}{x^3}} \left (b^2 \log ^2(f)-2 b x^3 \log (f)+2 x^6\right )}{3 b^3 x^6 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^3)/x^10,x]

[Out]

-(f^(a + b/x^3)*(2*x^6 - 2*b*x^3*Log[f] + b^2*Log[f]^2))/(3*b^3*x^6*Log[f]^3)

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Maple [A]  time = 0.028, size = 75, normalized size = 1.1 \[{\frac{1}{{x}^{9}} \left ( -{\frac{2\,{x}^{9}}{3\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}+{\frac{2\,{x}^{6}}{3\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{3}}{3\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^3)/x^10,x)

[Out]

(-2/3/b^3/ln(f)^3*x^9*exp((a+b/x^3)*ln(f))+2/3/b^2/ln(f)^2*x^6*exp((a+b/x^3)*ln(
f))-1/3/b/ln(f)*x^3*exp((a+b/x^3)*ln(f)))/x^9

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Maxima [A]  time = 0.819117, size = 30, normalized size = 0.45 \[ -\frac{f^{a} \Gamma \left (3, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{3} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^10,x, algorithm="maxima")

[Out]

-1/3*f^a*gamma(3, -b*log(f)/x^3)/(b^3*log(f)^3)

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Fricas [A]  time = 0.256633, size = 63, normalized size = 0.94 \[ -\frac{{\left (2 \, x^{6} - 2 \, b x^{3} \log \left (f\right ) + b^{2} \log \left (f\right )^{2}\right )} f^{\frac{a x^{3} + b}{x^{3}}}}{3 \, b^{3} x^{6} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^10,x, algorithm="fricas")

[Out]

-1/3*(2*x^6 - 2*b*x^3*log(f) + b^2*log(f)^2)*f^((a*x^3 + b)/x^3)/(b^3*x^6*log(f)
^3)

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Sympy [A]  time = 0.290538, size = 44, normalized size = 0.66 \[ \frac{f^{a + \frac{b}{x^{3}}} \left (- b^{2} \log{\left (f \right )}^{2} + 2 b x^{3} \log{\left (f \right )} - 2 x^{6}\right )}{3 b^{3} x^{6} \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**3)/x**10,x)

[Out]

f**(a + b/x**3)*(-b**2*log(f)**2 + 2*b*x**3*log(f) - 2*x**6)/(3*b**3*x**6*log(f)
**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^10,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^10, x)