Optimal. Leaf size=34 \[ \frac{f^a \text{Gamma}\left (\frac{2}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac{b \log (f)}{x^3}\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0222683, antiderivative size = 34, 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, Rules used = {2218} \[ \frac{f^a \text{Gamma}\left (\frac{2}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac{b \log (f)}{x^3}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2218
Rubi steps
\begin{align*} \int \frac{f^{a+\frac{b}{x^3}}}{x^3} \, dx &=\frac{f^a \Gamma \left (\frac{2}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac{b \log (f)}{x^3}\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0046829, size = 34, normalized size = 1. \[ \frac{f^a \text{Gamma}\left (\frac{2}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac{b \log (f)}{x^3}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.03, size = 78, normalized size = 2.3 \begin{align*}{\frac{{f}^{a}}{3\,b}\sqrt [3]{-b} \left ({\frac{\Gamma \left ({\frac{2}{3}} \right ) }{{x}^{2}} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{{x}^{2}} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}}\Gamma \left ({\frac{2}{3}},-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-{\frac{2}{3}}}} \right ) \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.2261, size = 38, normalized size = 1.12 \begin{align*} \frac{f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, x^{2} \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59512, size = 88, normalized size = 2.59 \begin{align*} -\frac{\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]