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