Optimal. Leaf size=305 \[ -\frac {a^3}{d (c+d x)}-\frac {3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}-\frac {b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)}+\frac {3 a^2 b f F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n g n \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}+\frac {6 a b^2 f F^{2 \left (e-\frac {c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} g n \text {Ei}\left (\frac {2 f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}+\frac {3 b^3 f F^{3 \left (e-\frac {c f}{d}\right ) g n-3 g n (e+f x)} \left (F^{e g+f g x}\right )^{3 n} g n \text {Ei}\left (\frac {3 f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2} \]
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Rubi [A]
time = 0.35, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2214, 2208,
2213, 2209} \begin {gather*} -\frac {a^3}{d (c+d x)}+\frac {3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac {3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac {6 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac {c f}{d}\right )-2 g n (e+f x)} \text {Ei}\left (\frac {2 f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac {3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac {c f}{d}\right )-3 g n (e+f x)} \text {Ei}\left (\frac {3 f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac {b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2209
Rule 2213
Rule 2214
Rubi steps
\begin {align*} \int \frac {\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3}{(c+d x)^2} \, dx &=\int \left (\frac {a^3}{(c+d x)^2}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n}{(c+d x)^2}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^2}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n}}{(c+d x)^2}\right ) \, dx\\ &=-\frac {a^3}{d (c+d x)}+\left (3 a^2 b\right ) \int \frac {\left (F^{e g+f g x}\right )^n}{(c+d x)^2} \, dx+\left (3 a b^2\right ) \int \frac {\left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^2} \, dx+b^3 \int \frac {\left (F^{e g+f g x}\right )^{3 n}}{(c+d x)^2} \, dx\\ &=-\frac {a^3}{d (c+d x)}-\frac {3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}-\frac {b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)}+\frac {\left (3 a^2 b f g n \log (F)\right ) \int \frac {\left (F^{e g+f g x}\right )^n}{c+d x} \, dx}{d}+\frac {\left (6 a b^2 f g n \log (F)\right ) \int \frac {\left (F^{e g+f g x}\right )^{2 n}}{c+d x} \, dx}{d}+\frac {\left (3 b^3 f g n \log (F)\right ) \int \frac {\left (F^{e g+f g x}\right )^{3 n}}{c+d x} \, dx}{d}\\ &=-\frac {a^3}{d (c+d x)}-\frac {3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}-\frac {b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)}+\frac {\left (3 a^2 b f F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n g n \log (F)\right ) \int \frac {F^{n (e g+f g x)}}{c+d x} \, dx}{d}+\frac {\left (6 a b^2 f F^{-2 n (e g+f g x)} \left (F^{e g+f g x}\right )^{2 n} g n \log (F)\right ) \int \frac {F^{2 n (e g+f g x)}}{c+d x} \, dx}{d}+\frac {\left (3 b^3 f F^{-3 n (e g+f g x)} \left (F^{e g+f g x}\right )^{3 n} g n \log (F)\right ) \int \frac {F^{3 n (e g+f g x)}}{c+d x} \, dx}{d}\\ &=-\frac {a^3}{d (c+d x)}-\frac {3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}-\frac {b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)}+\frac {3 a^2 b f F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n g n \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}+\frac {6 a b^2 f F^{2 \left (e-\frac {c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} g n \text {Ei}\left (\frac {2 f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}+\frac {3 b^3 f F^{3 \left (e-\frac {c f}{d}\right ) g n-3 g n (e+f x)} \left (F^{e g+f g x}\right )^{3 n} g n \text {Ei}\left (\frac {3 f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 250, normalized size = 0.82 \begin {gather*} -\frac {a^3 d+3 a^2 b d \left (F^{g (e+f x)}\right )^n+3 a b^2 d \left (F^{g (e+f x)}\right )^{2 n}+b^3 d \left (F^{g (e+f x)}\right )^{3 n}-3 a^2 b f F^{-\frac {f g n (c+d x)}{d}} \left (F^{g (e+f x)}\right )^n g n (c+d x) \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right ) \log (F)-6 a b^2 f F^{-\frac {2 f g n (c+d x)}{d}} \left (F^{g (e+f x)}\right )^{2 n} g n (c+d x) \text {Ei}\left (\frac {2 f g n (c+d x) \log (F)}{d}\right ) \log (F)-3 b^3 f F^{-\frac {3 f g n (c+d x)}{d}} \left (F^{g (e+f x)}\right )^{3 n} g n (c+d x) \text {Ei}\left (\frac {3 f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{3}}{\left (d x +c \right )^{2}}\, dx\]
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 = 0.40, size = 277, normalized size = 0.91 \begin {gather*} -\frac {3 \, F^{f g n x + g n e} a^{2} b d + 3 \, F^{2 \, f g n x + 2 \, g n e} a b^{2} d + F^{3 \, f g n x + 3 \, g n e} b^{3} d + a^{3} d - \frac {3 \, {\left (b^{3} d f g n x + b^{3} c f g n\right )} {\rm Ei}\left (\frac {3 \, {\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )}{F^{\frac {3 \, {\left (c f g n - d g n e\right )}}{d}}} - \frac {6 \, {\left (a b^{2} d f g n x + a b^{2} c f g n\right )} {\rm Ei}\left (\frac {2 \, {\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )}{F^{\frac {2 \, {\left (c f g n - d g n e\right )}}{d}}} - \frac {3 \, {\left (a^{2} b d f g n x + a^{2} b c f g n\right )} {\rm Ei}\left (\frac {{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )}{F^{\frac {c f g n - d g n e}{d}}}}{d^{3} x + c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \left (F^{e g} F^{f g x}\right )^{n}\right )^{3}}{\left (c + d x\right )^{2}}\, dx \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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