Optimal. Leaf size=49 \[ \frac {F^a (c+d x)^2 \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2218} \[ \frac {F^a (c+d x)^2 \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2218
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^3}} (c+d x) \, dx &=\frac {F^a (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (F)}{(c+d x)^3}\right ) \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{2/3}}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {F^a (c+d x)^2 \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.47, size = 142, normalized size = 2.90 \[ -\frac {F^{a} d^{2} \left (-\frac {b \log \relax (F)}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (F)}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) F^{a +\frac {b}{\left (d x +c \right )^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (F^{a} d x^{2} + 2 \, F^{a} c x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac {3 \, {\left (F^{a} b d^{2} x^{2} \log \relax (F) + 2 \, F^{a} b c d x \log \relax (F)\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{2 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.99, size = 107, normalized size = 2.18 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,{\left (c+d\,x\right )}^2}{2\,d}-\frac {F^a\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )\,{\left (c+d\,x\right )}^2\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{2/3}}{2\,d}+\frac {\pi \,\sqrt {3}\,F^a\,{\left (c+d\,x\right )}^2\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{2/3}}{3\,d\,\Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________