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

Optimal. Leaf size=56 \[ -\frac{1}{2} b^2 f^a \log ^2(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right )+\frac{1}{2} x^2 f^{a+\frac{b}{x}}+\frac{1}{2} b x \log (f) f^{a+\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.0607181, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{2} b^2 f^a \log ^2(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right )+\frac{1}{2} x^2 f^{a+\frac{b}{x}}+\frac{1}{2} b x \log (f) f^{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.05451, size = 48, normalized size = 0.86 \[ - \frac{b^{2} f^{a} \log{\left (f \right )}^{2} \operatorname{Ei}{\left (\frac{b \log{\left (f \right )}}{x} \right )}}{2} + \frac{b f^{a + \frac{b}{x}} x \log{\left (f \right )}}{2} + \frac{f^{a + \frac{b}{x}} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x)*x,x)

[Out]

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

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Mathematica [A]  time = 0.0226263, size = 40, normalized size = 0.71 \[ \frac{1}{2} f^a \left (x f^{b/x} (b \log (f)+x)-b^2 \log ^2(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)*x,x]

[Out]

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

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Maple [A]  time = 0.023, size = 57, normalized size = 1. \[{\frac{{x}^{2}}{2}{f}^{{\frac{ax+b}{x}}}}+{\frac{\ln \left ( f \right ) bx}{2}{f}^{{\frac{ax+b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{a}}{2}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)*x,x)

[Out]

1/2*f^((a*x+b)/x)*x^2+1/2*b*ln(f)*f^((a*x+b)/x)*x+1/2*b^2*ln(f)^2*f^a*Ei(1,-b*ln
(f)/x)

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Maxima [A]  time = 0.823798, size = 62, normalized size = 1.11 \[ -\frac{1}{2} \, b^{2} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{2} + \frac{1}{2} \,{\left (b f^{a} x \log \left (f\right ) + f^{a} x^{2}\right )} f^{\frac{b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255763, size = 58, normalized size = 1.04 \[ -\frac{1}{2} \, b^{2} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{2} + \frac{1}{2} \,{\left (b x \log \left (f\right ) + x^{2}\right )} f^{\frac{a x + b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)*x,x)

[Out]

Integral(f**(a + b/x)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(a + b/x)*x, x)