Optimal. Leaf size=147 \[ -\frac {a}{2 d (c+d x)^2}+\frac {b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac {b f g n \log (F) \left (F^{e g+f g x}\right )^n}{2 d^2 (c+d x)}-\frac {b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.24, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2183, 2177, 2182, 2178} \[ -\frac {a}{2 d (c+d x)^2}+\frac {b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac {b f g n \log (F) \left (F^{e g+f g x}\right )^n}{2 d^2 (c+d x)}-\frac {b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2182
Rule 2183
Rubi steps
\begin {align*} \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^3} \, dx &=\int \left (\frac {a}{(c+d x)^3}+\frac {b \left (F^{e g+f g x}\right )^n}{(c+d x)^3}\right ) \, dx\\ &=-\frac {a}{2 d (c+d x)^2}+b \int \frac {\left (F^{e g+f g x}\right )^n}{(c+d x)^3} \, dx\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}+\frac {(b f g n \log (F)) \int \frac {\left (F^{e g+f g x}\right )^n}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac {b f \left (F^{e g+f g x}\right )^n g n \log (F)}{2 d^2 (c+d x)}+\frac {\left (b f^2 g^2 n^2 \log ^2(F)\right ) \int \frac {\left (F^{e g+f g x}\right )^n}{c+d x} \, dx}{2 d^2}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac {b f \left (F^{e g+f g x}\right )^n g n \log (F)}{2 d^2 (c+d x)}+\frac {\left (b f^2 F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n g^2 n^2 \log ^2(F)\right ) \int \frac {F^{n (e g+f g x)}}{c+d x} \, dx}{2 d^2}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac {b f \left (F^{e g+f g x}\right )^n g n \log (F)}{2 d^2 (c+d x)}+\frac {b f^2 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^2 n^2 \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right ) \log ^2(F)}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 111, normalized size = 0.76 \[ -\frac {a d^2-b f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^n F^{-\frac {f g n (c+d x)}{d}} \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right )+b d \left (F^{g (e+f x)}\right )^n (f g n \log (F) (c+d x)+d)}{2 d^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 160, normalized size = 1.09 \[ \frac {{\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x + b c^{2} f^{2} g^{2} n^{2}\right )} F^{\frac {{\left (d e - c f\right )} g n}{d}} {\rm Ei}\left (\frac {{\left (d f g n x + c f g n\right )} \log \relax (F)}{d}\right ) \log \relax (F)^{2} - a d^{2} - {\left (b d^{2} + {\left (b d^{2} f g n x + b c d f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n}}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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 \[ \int \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {b \left (F^{\left (f x +e \right ) g}\right )^{n}+a}{\left (d x +c \right )^{3}}\, dx \]
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 \[ {\left (F^{e g}\right )}^{n} b \int \frac {{\left (F^{f g x}\right )}^{n}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} - \frac {a}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
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 \[ \int \frac {a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n}{{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \frac {a + b \left (F^{e g} F^{f g x}\right )^{n}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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