Optimal. Leaf size=81 \[ -\frac{1}{18} b^3 f^a \log ^3(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^3}\right )+\frac{1}{18} b^2 x^3 \log ^2(f) f^{a+\frac{b}{x^3}}+\frac{1}{9} x^9 f^{a+\frac{b}{x^3}}+\frac{1}{18} b x^6 \log (f) f^{a+\frac{b}{x^3}} \]
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Rubi [A] time = 0.153477, antiderivative size = 81, 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 \[ -\frac{1}{18} b^3 f^a \log ^3(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^3}\right )+\frac{1}{18} b^2 x^3 \log ^2(f) f^{a+\frac{b}{x^3}}+\frac{1}{9} x^9 f^{a+\frac{b}{x^3}}+\frac{1}{18} b x^6 \log (f) f^{a+\frac{b}{x^3}} \]
Antiderivative was successfully verified.
[In] Int[f^(a + b/x^3)*x^8,x]
[Out]
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Rubi in Sympy [A] time = 12.3295, size = 76, normalized size = 0.94 \[ - \frac{b^{3} f^{a} \log{\left (f \right )}^{3} \operatorname{Ei}{\left (\frac{b \log{\left (f \right )}}{x^{3}} \right )}}{18} + \frac{b^{2} f^{a + \frac{b}{x^{3}}} x^{3} \log{\left (f \right )}^{2}}{18} + \frac{b f^{a + \frac{b}{x^{3}}} x^{6} \log{\left (f \right )}}{18} + \frac{f^{a + \frac{b}{x^{3}}} x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(a+b/x**3)*x**8,x)
[Out]
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Mathematica [A] time = 0.030855, size = 57, normalized size = 0.7 \[ \frac{1}{18} f^a \left (x^3 f^{\frac{b}{x^3}} \left (b^2 \log ^2(f)+b x^3 \log (f)+2 x^6\right )-b^3 \log ^3(f) \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^3}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[f^(a + b/x^3)*x^8,x]
[Out]
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Maple [B] time = 0.046, size = 177, normalized size = 2.2 \[{\frac{{f}^{a}{b}^{3} \left ( \ln \left ( f \right ) \right ) ^{3}}{3} \left ({\frac{{x}^{9}}{3\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}+{\frac{{x}^{6}}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{{x}^{3}}{2\,b\ln \left ( f \right ) }}+{\frac{11}{36}}+{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( -b \right ) }{6}}-{\frac{\ln \left ( \ln \left ( f \right ) \right ) }{6}}-{\frac{{x}^{9}}{72\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}} \left ( 22\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{x}^{9}}}+36\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+36\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+24 \right ) }+{\frac{{x}^{9}}{24\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}} \left ( 4\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{x}^{6}}}+4\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+8 \right ){{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}+{\frac{1}{6}\ln \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) }+{\frac{1}{6}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(a+b/x^3)*x^8,x)
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Maxima [A] time = 0.839712, size = 85, normalized size = 1.05 \[ -\frac{1}{18} \, b^{3} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right )^{3} + \frac{1}{18} \,{\left (2 \, f^{a} x^{9} + b f^{a} x^{6} \log \left (f\right ) + b^{2} f^{a} x^{3} \log \left (f\right )^{2}\right )} f^{\frac{b}{x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^3)*x^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236521, size = 81, normalized size = 1. \[ -\frac{1}{18} \, b^{3} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right )^{3} + \frac{1}{18} \,{\left (2 \, x^{9} + b x^{6} \log \left (f\right ) + b^{2} x^{3} \log \left (f\right )^{2}\right )} f^{\frac{a x^{3} + b}{x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^3)*x^8,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(a+b/x**3)*x**8,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{3}}} x^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^3)*x^8,x, algorithm="giac")
[Out]