3.164 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^{13}} \, dx\)

Optimal. Leaf size=83 \[ \frac{2 f^{a+\frac{b}{x^3}}}{b^4 \log ^4(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)} \]

[Out]

(2*f^(a + b/x^3))/(b^4*Log[f]^4) - (2*f^(a + b/x^3))/(b^3*x^3*Log[f]^3) + f^(a +
b/x^3)/(b^2*x^6*Log[f]^2) - f^(a + b/x^3)/(3*b*x^9*Log[f])

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Rubi [A]  time = 0.153882, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 f^{a+\frac{b}{x^3}}}{b^4 \log ^4(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^3)/x^13,x]

[Out]

(2*f^(a + b/x^3))/(b^4*Log[f]^4) - (2*f^(a + b/x^3))/(b^3*x^3*Log[f]^3) + f^(a +
b/x^3)/(b^2*x^6*Log[f]^2) - f^(a + b/x^3)/(3*b*x^9*Log[f])

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Rubi in Sympy [A]  time = 17.1628, size = 78, normalized size = 0.94 \[ - \frac{f^{a + \frac{b}{x^{3}}}}{3 b x^{9} \log{\left (f \right )}} + \frac{f^{a + \frac{b}{x^{3}}}}{b^{2} x^{6} \log{\left (f \right )}^{2}} - \frac{2 f^{a + \frac{b}{x^{3}}}}{b^{3} x^{3} \log{\left (f \right )}^{3}} + \frac{2 f^{a + \frac{b}{x^{3}}}}{b^{4} \log{\left (f \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**3)/x**13,x)

[Out]

-f**(a + b/x**3)/(3*b*x**9*log(f)) + f**(a + b/x**3)/(b**2*x**6*log(f)**2) - 2*f
**(a + b/x**3)/(b**3*x**3*log(f)**3) + 2*f**(a + b/x**3)/(b**4*log(f)**4)

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Mathematica [A]  time = 0.0176563, size = 58, normalized size = 0.7 \[ \frac{f^{a+\frac{b}{x^3}} \left (-b^3 \log ^3(f)+3 b^2 x^3 \log ^2(f)-6 b x^6 \log (f)+6 x^9\right )}{3 b^4 x^9 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^3)/x^13,x]

[Out]

(f^(a + b/x^3)*(6*x^9 - 6*b*x^6*Log[f] + 3*b^2*x^3*Log[f]^2 - b^3*Log[f]^3))/(3*
b^4*x^9*Log[f]^4)

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Maple [A]  time = 0.035, size = 97, normalized size = 1.2 \[{\frac{1}{{x}^{12}} \left ({\frac{{x}^{6}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}+2\,{\frac{{x}^{12}}{ \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-2\,{\frac{{x}^{9}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{3}}{3\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^3)/x^13,x)

[Out]

(1/b^2/ln(f)^2*x^6*exp((a+b/x^3)*ln(f))+2/b^4/ln(f)^4*x^12*exp((a+b/x^3)*ln(f))-
2/b^3/ln(f)^3*x^9*exp((a+b/x^3)*ln(f))-1/3/b/ln(f)*x^3*exp((a+b/x^3)*ln(f)))/x^1
2

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Maxima [A]  time = 0.816781, size = 30, normalized size = 0.36 \[ \frac{f^{a} \Gamma \left (4, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{4} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^13,x, algorithm="maxima")

[Out]

1/3*f^a*gamma(4, -b*log(f)/x^3)/(b^4*log(f)^4)

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Fricas [A]  time = 0.236218, size = 81, normalized size = 0.98 \[ \frac{{\left (6 \, x^{9} - 6 \, b x^{6} \log \left (f\right ) + 3 \, b^{2} x^{3} \log \left (f\right )^{2} - b^{3} \log \left (f\right )^{3}\right )} f^{\frac{a x^{3} + b}{x^{3}}}}{3 \, b^{4} x^{9} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^13,x, algorithm="fricas")

[Out]

1/3*(6*x^9 - 6*b*x^6*log(f) + 3*b^2*x^3*log(f)^2 - b^3*log(f)^3)*f^((a*x^3 + b)/
x^3)/(b^4*x^9*log(f)^4)

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Sympy [A]  time = 0.329948, size = 58, normalized size = 0.7 \[ \frac{f^{a + \frac{b}{x^{3}}} \left (- b^{3} \log{\left (f \right )}^{3} + 3 b^{2} x^{3} \log{\left (f \right )}^{2} - 6 b x^{6} \log{\left (f \right )} + 6 x^{9}\right )}{3 b^{4} x^{9} \log{\left (f \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**3)/x**13,x)

[Out]

f**(a + b/x**3)*(-b**3*log(f)**3 + 3*b**2*x**3*log(f)**2 - 6*b*x**6*log(f) + 6*x
**9)/(3*b**4*x**9*log(f)**4)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{13}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^13,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^13, x)