Optimal. Leaf size=32 \[ \frac {1}{3} x f^a \sqrt [3]{-\frac {b \log (f)}{x^3}} \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2208} \[ \frac {1}{3} x f^a \sqrt [3]{-\frac {b \log (f)}{x^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2208
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} \, dx &=\frac {1}{3} f^a x \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \sqrt [3]{-\frac {b \log (f)}{x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 32, normalized size = 1.00 \[ \frac {1}{3} x f^a \sqrt [3]{-\frac {b \log (f)}{x^3}} \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 38, normalized size = 1.19 \[ -\left (-b \log \relax (f)\right )^{\frac {1}{3}} f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) + f^{\frac {a x^{3} + b}{x^{3}}} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 98, normalized size = 3.06 \[ -\frac {\left (-b \right )^{\frac {1}{3}} \left (-\frac {3 x \,{\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{\left (-b \right )^{\frac {1}{3}} \ln \relax (f )^{\frac {1}{3}}}-\frac {3 b \Gamma \left (\frac {2}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {2}{3}}}{\left (-b \right )^{\frac {1}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}+\frac {3 \Gamma \left (\frac {2}{3}\right ) b \ln \relax (f )^{\frac {2}{3}}}{\left (-b \right )^{\frac {1}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}\right ) f^{a} \ln \relax (f )^{\frac {1}{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.24, size = 26, normalized size = 0.81 \[ \frac {1}{3} \, f^{a} x \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {1}{3}} \Gamma \left (-\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.59, size = 48, normalized size = 1.50 \[ f^a\,x\,\left (f^{\frac {b}{x^3}}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{1/3}-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{1/3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________