Integrand size = 23, antiderivative size = 98 \[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)} \]
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Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2215, 2221, 2317, 2438} \[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=-\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}-\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {(c+d x)^2}{2 a d} \]
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Rule 2215
Rule 2221
Rule 2317
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2}{2 a d}-\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} \\ & = \frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}+\frac {d \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a f g n \log (F)} \\ & = \frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}+\frac {d \text {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 f^2 g^2 n^2 \log ^2(F)} \\ & = \frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {-f g n (c+d x) \log (F) \log \left (1+\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+d \operatorname {PolyLog}\left (2,-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )}{a f^2 g^2 n^2 \log ^2(F)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs. \(2(96)=192\).
Time = 0.06 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.65
method | result | size |
risch | \(-\frac {c \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right )}{\ln \left (F \right ) f g n a}+\frac {c \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g n a}+\frac {d \ln \left (F^{g \left (f x +e \right )}\right )^{2}}{2 \ln \left (F \right )^{2} f^{2} g^{2} a}-\frac {d \ln \left (F^{g \left (f x +e \right )}\right ) \ln \left (1+\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n a}-\frac {d \,\operatorname {Li}_{2}\left (-\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2} a}-\frac {d \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right ) x}{\ln \left (F \right ) f g n a}+\frac {d \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n} F^{-n g f x} F^{n g f x} b +a \right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n a}+\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) x}{\ln \left (F \right ) f g n a}-\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n a}\) | \(456\) |
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Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.50 \[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {2 \, {\left (d e - c f\right )} g n \log \left (F^{f g n x + e g n} b + a\right ) \log \left (F\right ) + {\left (d f^{2} g^{2} n^{2} x^{2} + 2 \, c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} - 2 \, {\left (d f g n x + d e g n\right )} \log \left (F\right ) \log \left (\frac {F^{f g n x + e g n} b + a}{a}\right ) - 2 \, d {\rm Li}_2\left (-\frac {F^{f g n x + e g n} b + a}{a} + 1\right )}{2 \, a f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \]
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\[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {c + d x}{a + b \left (F^{e g + f g x}\right )^{n}}\, dx \]
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\[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int { \frac {d x + c}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a} \,d x } \]
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\[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int { \frac {d x + c}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a} \,d x } \]
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Timed out. \[ \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {c+d\,x}{a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n} \,d x \]
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