Optimal. Leaf size=34 \[ \frac {1}{3} x^5 f^a \left (-\frac {b \log (f)}{x^3}\right )^{5/3} \Gamma \left (-\frac {5}{3},-\frac {b \log (f)}{x^3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{3} x^5 f^a \left (-\frac {b \log (f)}{x^3}\right )^{5/3} \text {Gamma}\left (-\frac {5}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} x^4 \, dx &=\frac {1}{3} f^a x^5 \Gamma \left (-\frac {5}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{5/3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{3} x^5 f^a \left (-\frac {b \log (f)}{x^3}\right )^{5/3} \Gamma \left (-\frac {5}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 55, normalized size = 1.62 \[ -\frac {3}{10} \, \left (-b \log \relax (f)\right )^{\frac {2}{3}} b f^{a} \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f) + \frac {1}{10} \, {\left (2 \, x^{5} + 3 \, b x^{2} \log \relax (f)\right )} f^{\frac {a x^{3} + b}{x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 120, normalized size = 3.53 \[ -\frac {\left (-b \right )^{\frac {5}{3}} \left (-\frac {3 \left (\frac {3 b \ln \relax (f )}{2 x^{3}}+1\right ) x^{5} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{5 \left (-b \right )^{\frac {5}{3}} \ln \relax (f )^{\frac {5}{3}}}-\frac {9 b^{2} \Gamma \left (\frac {1}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {1}{3}}}{10 \left (-b \right )^{\frac {5}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} x}+\frac {3 \pi \sqrt {3}\, b^{2} \ln \relax (f )^{\frac {1}{3}}}{5 \left (-b \right )^{\frac {5}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} x}\right ) f^{a} \ln \relax (f )^{\frac {5}{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.33, size = 28, normalized size = 0.82 \[ \frac {1}{3} \, f^{a} x^{5} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {5}{3}} \Gamma \left (-\frac {5}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.57, size = 88, normalized size = 2.59 \[ \frac {f^a\,f^{\frac {b}{x^3}}\,x^5}{5}+\frac {3\,f^a\,x^5\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{5/3}}{10}+\frac {3\,b\,f^a\,f^{\frac {b}{x^3}}\,x^2\,\ln \relax (f)}{10}-\frac {\pi \,\sqrt {3}\,f^a\,x^5\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{5/3}}{5\,\Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}} x^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________