Optimal. Leaf size=191 \[ -\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {(c+d x)^2}{2 a^2 d}-\frac {d x}{a^2 f g n \log (F)}+\frac {c+d x}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
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Rubi [A] time = 0.34, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2185, 2184, 2190, 2279, 2391, 2191, 2282, 266, 36, 29, 31} \[ -\frac {d \text {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}+\frac {d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {(c+d x)^2}{2 a^2 d}-\frac {d x}{a^2 f g n \log (F)}+\frac {c+d x}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rubi steps
\begin {align*} \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx &=\frac {\int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac {d \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a f g n \log (F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )} \, dx,x,F^{g (e+f x)}\right )}{a f^2 g^2 n \log ^2(F)}+\frac {d \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f g n \log (F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {d \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {d \operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {d \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {d x}{a^2 f g n \log (F)}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}\\ \end {align*}
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Mathematica [F] time = 1.13, size = 0, normalized size = 0.00 \[ \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.43, size = 400, normalized size = 2.09 \[ -\frac {2 \, {\left (a d e - a c f\right )} g n \log \relax (F) - {\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x - {\left (a d e^{2} - 2 \, a c e f\right )} g^{2} n^{2}\right )} \log \relax (F)^{2} - {\left ({\left (b d f^{2} g^{2} n^{2} x^{2} + 2 \, b c f^{2} g^{2} n^{2} x - {\left (b d e^{2} - 2 \, b c e f\right )} g^{2} n^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (b d f g n x + b d e g n\right )} \log \relax (F)\right )} F^{f g n x + e g n} + 2 \, {\left (F^{f g n x + e g n} b d + a d\right )} {\rm Li}_2\left (-\frac {F^{f g n x + e g n} b + a}{a} + 1\right ) - 2 \, {\left ({\left (a d e - a c f\right )} g n \log \relax (F) + {\left ({\left (b d e - b c f\right )} g n \log \relax (F) + b d\right )} F^{f g n x + e g n} + a d\right )} \log \left (F^{f g n x + e g n} b + a\right ) + 2 \, {\left ({\left (b d f g n x + b d e g n\right )} F^{f g n x + e g n} \log \relax (F) + {\left (a d f g n x + a d e g n\right )} \log \relax (F)\right )} \log \left (\frac {F^{f g n x + e g n} b + a}{a}\right )}{2 \, {\left (F^{f g n x + e g n} a^{2} b f^{2} g^{2} n^{2} \log \relax (F)^{2} + a^{3} f^{2} g^{2} n^{2} \log \relax (F)^{2}\right )}} \]
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 \[ \int \frac {d x + c}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 591, normalized size = 3.09 \[ \frac {d x \ln \left (F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{a^{2} f g n \ln \relax (F )}-\frac {d x \ln \left (b \,F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )}{a^{2} f g n \ln \relax (F )}+\frac {c \ln \left (F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{a^{2} f g n \ln \relax (F )}-\frac {c \ln \left (b \,F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )}{a^{2} f g n \ln \relax (F )}+\frac {d x +c}{\left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right ) a f g n \ln \relax (F )}+\frac {d \ln \left (F^{\left (f x +e \right ) g}\right )^{2}}{2 a^{2} f^{2} g^{2} \ln \relax (F )^{2}}-\frac {d \ln \left (F^{\left (f x +e \right ) g}\right ) \ln \left (F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{a^{2} f^{2} g^{2} n \ln \relax (F )^{2}}-\frac {d \ln \left (F^{\left (f x +e \right ) g}\right ) \ln \left (\frac {b \,F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}}{a}+1\right )}{a^{2} f^{2} g^{2} n \ln \relax (F )^{2}}+\frac {d \ln \left (F^{\left (f x +e \right ) g}\right ) \ln \left (b \,F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )}{a^{2} f^{2} g^{2} n \ln \relax (F )^{2}}-\frac {d \polylog \left (2, -\frac {b \,F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}}{a}\right )}{a^{2} f^{2} g^{2} n^{2} \ln \relax (F )^{2}}-\frac {d \ln \left (F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{a^{2} f^{2} g^{2} n^{2} \ln \relax (F )^{2}}+\frac {d \ln \left (b \,F^{f g n x} F^{-f g n x} \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )}{a^{2} f^{2} g^{2} n^{2} \ln \relax (F )^{2}} \]
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 \[ d {\left (\frac {x}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b f g n \log \relax (F) + a^{2} f g n \log \relax (F)} + \int \frac {f g n x \log \relax (F) - 1}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b f g n \log \relax (F) + a^{2} f g n \log \relax (F)}\,{d x}\right )} + c {\left (\frac {1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \relax (F)} + \frac {\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \relax (F)} - \frac {\log \left (\frac {{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} f g n \log \relax (F)}\right )} \]
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 \[ \int \frac {c+d\,x}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2} \,d x \]
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 \[ \frac {c + d x}{a^{2} f g n \log {\relax (F )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\relax (F )}} + \frac {\int \left (- \frac {d}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\right )\, dx + \int \frac {c f g n \log {\relax (F )}}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {d f g n x \log {\relax (F )}}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx}{a f g n \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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