Optimal. Leaf size=96 \[ -\frac{4}{15} \sqrt{\pi } b^{5/2} f^a \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{4}{15} b^2 x \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{5} x^5 f^{a+\frac{b}{x^2}}+\frac{2}{15} b x^3 \log (f) f^{a+\frac{b}{x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.146236, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{4}{15} \sqrt{\pi } b^{5/2} f^a \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{4}{15} b^2 x \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{5} x^5 f^{a+\frac{b}{x^2}}+\frac{2}{15} b x^3 \log (f) f^{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
[In] Int[f^(a + b/x^2)*x^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.9026, size = 94, normalized size = 0.98 \[ - \frac{4 \sqrt{\pi } b^{\frac{5}{2}} f^{a} \log{\left (f \right )}^{\frac{5}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{15} + \frac{4 b^{2} f^{a + \frac{b}{x^{2}}} x \log{\left (f \right )}^{2}}{15} + \frac{2 b f^{a + \frac{b}{x^{2}}} x^{3} \log{\left (f \right )}}{15} + \frac{f^{a + \frac{b}{x^{2}}} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(a+b/x**2)*x**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0443343, size = 74, normalized size = 0.77 \[ \frac{1}{15} f^a \left (x f^{\frac{b}{x^2}} \left (4 b^2 \log ^2(f)+2 b x^2 \log (f)+3 x^4\right )-4 \sqrt{\pi } b^{5/2} \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[f^(a + b/x^2)*x^4,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.032, size = 89, normalized size = 0.9 \[{\frac{{f}^{a}{x}^{5}}{5}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{3}}{15}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}x}{15}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}\sqrt{\pi }}{15}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(a+b/x^2)*x^4,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.927673, size = 113, normalized size = 1.18 \[ -\frac{4 \, \sqrt{\pi } b^{3} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{3}}{15 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{15} \,{\left (3 \, f^{a} x^{5} + 2 \, b f^{a} x^{3} \log \left (f\right ) + 4 \, b^{2} f^{a} x \log \left (f\right )^{2}\right )} f^{\frac{b}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^2)*x^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.319589, size = 112, normalized size = 1.17 \[ -\frac{4 \, \sqrt{\pi } b^{3} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{3} -{\left (3 \, x^{5} + 2 \, b x^{3} \log \left (f\right ) + 4 \, b^{2} x \log \left (f\right )^{2}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{15 \, \sqrt{-b \log \left (f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^2)*x^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(a+b/x**2)*x**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^2)*x^4,x, algorithm="giac")
[Out]