Optimal. Leaf size=98 \[ -\frac {e \sqrt [3]{d+e x} \left (F^{c (a+b x)}\right )^n F^{c n \left (a-\frac {b d}{e}\right )-c n (a+b x)} \Gamma \left (\frac {7}{3},-\frac {b c n (d+e x) \log (F)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac {b c n \log (F) (d+e x)}{e}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2182, 2181} \[ -\frac {e \sqrt [3]{d+e x} \left (F^{c (a+b x)}\right )^n F^{c n \left (a-\frac {b d}{e}\right )-c n (a+b x)} \text {Gamma}\left (\frac {7}{3},-\frac {b c n \log (F) (d+e x)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac {b c n \log (F) (d+e x)}{e}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 2182
Rubi steps
\begin {align*} \int \left (F^{c (a+b x)}\right )^n (d+e x)^{4/3} \, dx &=\left (F^{-c n (a+b x)} \left (F^{c (a+b x)}\right )^n\right ) \int F^{c n (a+b x)} (d+e x)^{4/3} \, dx\\ &=-\frac {e F^{c \left (a-\frac {b d}{e}\right ) n-c n (a+b x)} \left (F^{c (a+b x)}\right )^n \sqrt [3]{d+e x} \Gamma \left (\frac {7}{3},-\frac {b c n (d+e x) \log (F)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac {b c n (d+e x) \log (F)}{e}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 78, normalized size = 0.80 \[ -\frac {(d+e x)^{7/3} \left (F^{c (a+b x)}\right )^n F^{-\frac {b c n (d+e x)}{e}} \Gamma \left (\frac {7}{3},-\frac {b c n (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c n \log (F) (d+e x)}{e}\right )^{7/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 133, normalized size = 1.36 \[ \frac {\frac {4 \, \left (-\frac {b c n \log \relax (F)}{e}\right )^{\frac {2}{3}} e^{2} \Gamma \left (\frac {1}{3}, -\frac {{\left (b c e n x + b c d n\right )} \log \relax (F)}{e}\right )}{F^{\frac {{\left (b c d - a c e\right )} n}{e}}} - 3 \, {\left (4 \, b c e n \log \relax (F) - 3 \, {\left (b^{2} c^{2} e n^{2} x + b^{2} c^{2} d n^{2}\right )} \log \relax (F)^{2}\right )} {\left (e x + d\right )}^{\frac {1}{3}} F^{b c n x + a c n}}{9 \, b^{3} c^{3} n^{3} \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{\frac {4}{3}} {\left (F^{{\left (b x + a\right )} c}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{\frac {4}{3}} \left (F^{\left (b x +a \right ) c}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{\frac {4}{3}} {\left (F^{{\left (b x + a\right )} c}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (F^{c\,\left (a+b\,x\right )}\right )}^n\,{\left (d+e\,x\right )}^{4/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________