3.125 \(\int \frac{f^{a+\frac{b}{x}}}{x^5} \, dx\)

Optimal. Leaf size=82 \[ \frac{6 f^{a+\frac{b}{x}}}{b^4 \log ^4(f)}-\frac{6 f^{a+\frac{b}{x}}}{b^3 x \log ^3(f)}+\frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.132043, antiderivative size = 82, 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 \[ \frac{6 f^{a+\frac{b}{x}}}{b^4 \log ^4(f)}-\frac{6 f^{a+\frac{b}{x}}}{b^3 x \log ^3(f)}+\frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.2205, size = 70, normalized size = 0.85 \[ - \frac{f^{a + \frac{b}{x}}}{b x^{3} \log{\left (f \right )}} + \frac{3 f^{a + \frac{b}{x}}}{b^{2} x^{2} \log{\left (f \right )}^{2}} - \frac{6 f^{a + \frac{b}{x}}}{b^{3} x \log{\left (f \right )}^{3}} + \frac{6 f^{a + \frac{b}{x}}}{b^{4} \log{\left (f \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0162884, size = 53, normalized size = 0.65 \[ \frac{f^{a+\frac{b}{x}} \left (-b^3 \log ^3(f)+3 b^2 x \log ^2(f)-6 b x^2 \log (f)+6 x^3\right )}{b^4 x^3 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)/x^5,x]

[Out]

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

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Maple [A]  time = 0.019, size = 96, normalized size = 1.2 \[{\frac{1}{{x}^{4}} \left ( 6\,{\frac{{x}^{4}}{ \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-6\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+3\,{\frac{{x}^{2}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-{\frac{x}{b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)/x^5,x)

[Out]

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

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Maxima [A]  time = 0.838125, size = 28, normalized size = 0.34 \[ \frac{f^{a} \Gamma \left (4, -\frac{b \log \left (f\right )}{x}\right )}{b^{4} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.252607, size = 74, normalized size = 0.9 \[ -\frac{{\left (b^{3} \log \left (f\right )^{3} - 3 \, b^{2} x \log \left (f\right )^{2} + 6 \, b x^{2} \log \left (f\right ) - 6 \, x^{3}\right )} f^{\frac{a x + b}{x}}}{b^{4} x^{3} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.315081, size = 53, normalized size = 0.65 \[ \frac{f^{a + \frac{b}{x}} \left (- b^{3} \log{\left (f \right )}^{3} + 3 b^{2} x \log{\left (f \right )}^{2} - 6 b x^{2} \log{\left (f \right )} + 6 x^{3}\right )}{b^{4} x^{3} \log{\left (f \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)/x**5,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x^5,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)/x^5, x)