Optimal. Leaf size=111 \[ -\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f g n \log (F)}+\frac {x}{a^3}+\frac {1}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {1}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]
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Rubi [A] time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2282, 266, 44} \[ -\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f g n \log (F)}+\frac {1}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {x}{a^3}+\frac {1}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )^3} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {x}{a^3}+\frac {1}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {1}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f g n \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 84, normalized size = 0.76 \[ \frac {\frac {a \left (3 a+2 b \left (F^{g (e+f x)}\right )^n\right )}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}-2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )+2 f g n x \log (F)}{2 a^3 f g n \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 190, normalized size = 1.71 \[ \frac {2 \, F^{2 \, f g n x + 2 \, e g n} b^{2} f g n x \log \relax (F) + 2 \, a^{2} f g n x \log \relax (F) + 2 \, {\left (2 \, a b f g n x \log \relax (F) + a b\right )} F^{f g n x + e g n} + 3 \, a^{2} - 2 \, {\left (2 \, F^{f g n x + e g n} a b + F^{2 \, f g n x + 2 \, e g n} b^{2} + a^{2}\right )} \log \left (F^{f g n x + e g n} b + a\right )}{2 \, {\left (2 \, F^{f g n x + e g n} a^{4} b f g n \log \relax (F) + F^{2 \, f g n x + 2 \, e g n} a^{3} b^{2} f g n \log \relax (F) + a^{5} f g n \log \relax (F)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 136, normalized size = 1.23 \[ \frac {\log \left ({\left | F \right |}^{f g n x} {\left | F \right |}^{g n e}\right )}{a^{3} f g n \log \relax (F)} - \frac {\log \left ({\left | F^{f g n x} F^{g n e} b + a \right |}\right )}{a^{3} f g n \log \relax (F)} + \frac {2 \, F^{f g n x} F^{g n e} a b + 3 \, a^{2}}{2 \, {\left (F^{f g n x} F^{g n e} b + a\right )}^{2} a^{3} f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 134, normalized size = 1.21 \[ \frac {1}{2 \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{2} a f g n \ln \relax (F )}+\frac {1}{\left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right ) a^{2} f g n \ln \relax (F )}+\frac {\ln \left (\left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{a^{3} f g n \ln \relax (F )}-\frac {\ln \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )}{a^{3} f g n \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 145, normalized size = 1.31 \[ \frac {2 \, {\left (F^{f g x + e g}\right )}^{n} b + 3 \, a}{2 \, {\left (2 \, {\left (F^{f g x + e g}\right )}^{n} a^{3} b n + {\left (F^{f g x + e g}\right )}^{2 \, n} a^{2} b^{2} n + a^{4} n\right )} f g \log \relax (F)} + \frac {\log \left (F^{f g x + e g}\right )}{a^{3} f g \log \relax (F)} - \frac {\log \left (\frac {{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{3} f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 142, normalized size = 1.28 \[ \frac {x}{a^3}+\frac {1}{2\,a\,f\,g\,n\,\ln \relax (F)\,\left (a^2+b^2\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}+2\,a\,b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\right )}+\frac {1}{a^2\,f\,g\,n\,\ln \relax (F)\,\left (a+b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\right )}-\frac {\ln \left (a+b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\right )}{a^3\,f\,g\,n\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 116, normalized size = 1.05 \[ \frac {3 a + 2 b \left (F^{g \left (e + f x\right )}\right )^{n}}{2 a^{4} f g n \log {\relax (F )} + 4 a^{3} b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\relax (F )} + 2 a^{2} b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log {\relax (F )}} + \frac {x}{a^{3}} - \frac {\log {\left (\frac {a}{b} + \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{3} f g n \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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