3.116 \(\int f^{a+\frac{b}{x}} x^4 \, dx\)

Optimal. Leaf size=22 \[ -b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0346772, antiderivative size = 22, 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 \[ -b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)*x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.65573, size = 24, normalized size = 1.09 \[ - b^{5} f^{a} \Gamma{\left (-5,- \frac{b \log{\left (f \right )}}{x} \right )} \log{\left (f \right )}^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x)*x**4,x)

[Out]

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

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Mathematica [B]  time = 0.047434, size = 77, normalized size = 3.5 \[ \frac{1}{120} f^a \left (x f^{b/x} \left (b^4 \log ^4(f)+b^3 x \log ^3(f)+2 b^2 x^2 \log ^2(f)+6 b x^3 \log (f)+24 x^4\right )-b^5 \log ^5(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)*x^4,x]

[Out]

(f^a*(-(b^5*ExpIntegralEi[(b*Log[f])/x]*Log[f]^5) + f^(b/x)*x*(24*x^4 + 6*b*x^3*
Log[f] + 2*b^2*x^2*Log[f]^2 + b^3*x*Log[f]^3 + b^4*Log[f]^4)))/120

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Maple [B]  time = 0.039, size = 126, normalized size = 5.7 \[{\frac{{x}^{5}}{5}{f}^{{\frac{ax+b}{x}}}}+{\frac{b\ln \left ( f \right ){x}^{4}}{20}{f}^{{\frac{ax+b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{3}}{60}{f}^{{\frac{ax+b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{2}}{120}{f}^{{\frac{ax+b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}x}{120}{f}^{{\frac{ax+b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}{f}^{a}}{120}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)*x^4,x)

[Out]

1/5*f^((a*x+b)/x)*x^5+1/20*b*ln(f)*f^((a*x+b)/x)*x^4+1/60*b^2*ln(f)^2*f^((a*x+b)
/x)*x^3+1/120*b^3*ln(f)^3*f^((a*x+b)/x)*x^2+1/120*b^4*ln(f)^4*f^((a*x+b)/x)*x+1/
120*b^5*ln(f)^5*f^a*Ei(1,-b*ln(f)/x)

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Maxima [A]  time = 0.818673, size = 123, normalized size = 5.59 \[ -\frac{1}{120} \, b^{5} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{5} + \frac{1}{120} \,{\left (b^{4} f^{a} x \log \left (f\right )^{4} + b^{3} f^{a} x^{2} \log \left (f\right )^{3} + 2 \, b^{2} f^{a} x^{3} \log \left (f\right )^{2} + 6 \, b f^{a} x^{4} \log \left (f\right ) + 24 \, f^{a} x^{5}\right )} f^{\frac{b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)*x^4,x, algorithm="maxima")

[Out]

-1/120*b^5*f^a*Ei(b*log(f)/x)*log(f)^5 + 1/120*(b^4*f^a*x*log(f)^4 + b^3*f^a*x^2
*log(f)^3 + 2*b^2*f^a*x^3*log(f)^2 + 6*b*f^a*x^4*log(f) + 24*f^a*x^5)*f^(b/x)

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Fricas [A]  time = 0.261188, size = 108, normalized size = 4.91 \[ -\frac{1}{120} \, b^{5} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{5} + \frac{1}{120} \,{\left (b^{4} x \log \left (f\right )^{4} + b^{3} x^{2} \log \left (f\right )^{3} + 2 \, b^{2} x^{3} \log \left (f\right )^{2} + 6 \, b x^{4} \log \left (f\right ) + 24 \, x^{5}\right )} f^{\frac{a x + b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)*x^4,x, algorithm="fricas")

[Out]

-1/120*b^5*f^a*Ei(b*log(f)/x)*log(f)^5 + 1/120*(b^4*x*log(f)^4 + b^3*x^2*log(f)^
3 + 2*b^2*x^3*log(f)^2 + 6*b*x^4*log(f) + 24*x^5)*f^((a*x + b)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x}} x^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)*x**4,x)

[Out]

Integral(f**(a + b/x)*x**4, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)*x^4,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)*x^4, x)