Optimal. Leaf size=98 \[ -\frac {e F^{c \left (a-\frac {b d}{e}\right ) n-c n (a+b x)} \left (F^{c (a+b x)}\right )^n \sqrt [3]{d+e x} \Gamma \left (\frac {7}{3},-\frac {b c n (d+e x) \log (F)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac {b c n (d+e x) \log (F)}{e}}} \]
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Rubi [A]
time = 0.07, 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 = {2213, 2212}
\begin {gather*} -\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 2213
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*}
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Mathematica [A]
time = 0.20, size = 78, normalized size = 0.80 \begin {gather*} -\frac {F^{-\frac {b c n (d+e x)}{e}} \left (F^{c (a+b x)}\right )^n (d+e x)^{7/3} \Gamma \left (\frac {7}{3},-\frac {b c n (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c n (d+e x) \log (F)}{e}\right )^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (F^{c \left (b x +a \right )}\right )^{n} \left (e x +d \right )^{\frac {4}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 135, normalized size = 1.38 \begin {gather*} -\frac {3 \, {\left (4 \, b c n e \log \left (F\right ) - 3 \, {\left (b^{2} c^{2} n^{2} x e + b^{2} c^{2} d n^{2}\right )} \log \left (F\right )^{2}\right )} {\left (x e + d\right )}^{\frac {1}{3}} F^{b c n x + a c n} - \frac {4 \, \left (-b c n e^{\left (-1\right )} \log \left (F\right )\right )^{\frac {2}{3}} e^{2} \Gamma \left (\frac {1}{3}, -{\left (b c n x e + b c d n\right )} e^{\left (-1\right )} \log \left (F\right )\right )}{F^{{\left (b c d n - a c n e\right )} e^{\left (-1\right )}}}}{9 \, b^{3} c^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (F^{c\,\left (a+b\,x\right )}\right )}^n\,{\left (d+e\,x\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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