Optimal. Leaf size=98 \[ -\frac{e \sqrt [3]{d+e x} \left (F^{c (a+b x)}\right )^n F^{c n \left (a-\frac{b d}{e}\right )-c n (a+b x)} \text{Gamma}\left (\frac{7}{3},-\frac{b c n \log (F) (d+e x)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac{b c n \log (F) (d+e x)}{e}}} \]
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Rubi [A] time = 0.104381, antiderivative size = 98, normalized size of antiderivative = 1., 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.
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Rule 2182
Rule 2181
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 = 0.134194, size = 78, normalized size = 0.8 \[ -\frac{(d+e x)^{7/3} \left (F^{c (a+b x)}\right )^n F^{-\frac{b c n (d+e x)}{e}} \text{Gamma}\left (\frac{7}{3},-\frac{b c n \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c n \log (F) (d+e x)}{e}\right )^{7/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int \left ({F}^{c \left ( bx+a \right ) } \right ) ^{n} \left ( ex+d \right ) ^{{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{4}{3}}{\left (F^{{\left (b x + a\right )} c}\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55899, size = 316, normalized size = 3.22 \begin{align*} \frac{\frac{4 \, \left (-\frac{b c n \log \left (F\right )}{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 \left (F\right )}{e}\right )}{F^{\frac{{\left (b c d - a c e\right )} n}{e}}} - 3 \,{\left (4 \, b c e n \log \left (F\right ) - 3 \,{\left (b^{2} c^{2} e n^{2} x + b^{2} c^{2} d n^{2}\right )} \log \left (F\right )^{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 \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{4}{3}}{\left (F^{{\left (b x + a\right )} c}\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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