Integrand size = 25, antiderivative size = 145 \[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)} \]
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Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2215, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}+\frac {(c+d x)^3}{3 a d} \]
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Rule 2215
Rule 2221
Rule 2320
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a} \\ & = \frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}+\frac {(2 d) \int (c+d x) \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)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {2 d (c+d x) \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)}+\frac {\left (2 d^2\right ) \int \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a f^2 g^2 n^2 \log ^2(F)} \\ & = \frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {2 d (c+d x) \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)}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a f^3 g^3 n^2 \log ^3(F)} \\ & = \frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {2 d (c+d x) \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)}+\frac {2 d^2 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {-f^2 g^2 n^2 (c+d x)^2 \log ^2(F) \log \left (1+\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+2 d f g n (c+d x) \log (F) \operatorname {PolyLog}\left (2,-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+2 d^2 \operatorname {PolyLog}\left (3,-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )}{a f^3 g^3 n^3 \log ^3(F)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1123\) vs. \(2(143)=286\).
Time = 0.30 (sec) , antiderivative size = 1124, normalized size of antiderivative = 7.75
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Time = 0.26 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=-\frac {3 \, {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} g^{2} n^{2} \log \left (F^{f g n x + e g n} b + a\right ) \log \left (F\right )^{2} - {\left (d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, c d f^{3} g^{3} n^{3} x^{2} + 3 \, c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \, {\left (d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, c d f^{2} g^{2} n^{2} x - {\left (d^{2} e^{2} - 2 \, c d e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} \log \left (\frac {F^{f g n x + e g n} b + a}{a}\right ) + 6 \, {\left (d^{2} f g n x + c d f g n\right )} {\rm Li}_2\left (-\frac {F^{f g n x + e g n} b + a}{a} + 1\right ) \log \left (F\right ) - 6 \, d^{2} {\rm polylog}\left (3, -\frac {F^{f g n x + e g n} b}{a}\right )}{3 \, a f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
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\[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {\left (c + d x\right )^{2}}{a + b \left (F^{e g + f g x}\right )^{n}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (142) = 284\).
Time = 0.25 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.09 \[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=c^{2} {\left (\frac {f g n x + e g n}{a f g n} - \frac {\log \left (F^{f g n x + e g n} b + a\right )}{a f g n \log \left (F\right )}\right )} - \frac {2 \, {\left (f g n x \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right )\right )} c d}{a f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{e g n} b}{a})\right )} d^{2}}{a f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, c d f^{3} g^{3} n^{3} x^{2} \log \left (F\right )^{3}}{3 \, a f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
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\[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n} \,d x \]
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