Integrand size = 25, antiderivative size = 439 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)} \]
[Out]
Time = 0.87 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2216, 2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438, 272, 36, 29, 31} \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {3 d (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {(c+d x)^2}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {(c+d x)^2}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 272
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx}{a} \\ & = \frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {\int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a^2}-\frac {d \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a f g n \log (F)} \\ & = \frac {(c+d x)^3}{3 a^3 d}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 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)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^3}-\frac {d \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f g n \log (F)}-\frac {(2 d) \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f g n \log (F)}+\frac {(b d) \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^2 f g n \log (F)} \\ & = \frac {(c+d x)^3}{3 a^3 d}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {d^2 \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f^2 g^2 n^2 \log ^2(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^3 f g n \log (F)}+\frac {(b d) \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^3 f g n \log (F)}+\frac {(2 b d) \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^3 f g n \log (F)} \\ & = \frac {(c+d x)^3}{3 a^3 d}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 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^3 f^2 g^2 n^2 \log ^2(F)}+\frac {d^2 \text {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )} \, dx,x,F^{g (e+f x)}\right )}{a^2 f^3 g^3 n^2 \log ^3(F)}-\frac {d^2 \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {\left (2 d^2\right ) \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 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^3 f^2 g^2 n^2 \log ^2(F)} \\ & = \frac {(c+d x)^3}{3 a^3 d}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 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^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d^2 \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^3 f^3 g^3 n^3 \log ^3(F)}-\frac {\left (2 d^2\right ) \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^3 f^3 g^3 n^3 \log ^3(F)}+\frac {d^2 \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^3 g^3 n^3 \log ^3(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^3 f^3 g^3 n^2 \log ^3(F)} \\ & = \frac {(c+d x)^3}{3 a^3 d}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 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^3 f^3 g^3 n^3 \log ^3(F)}+\frac {d^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)} \\ & = \frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 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^3 f^3 g^3 n^3 \log ^3(F)} \\ \end{align*}
\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1886\) vs. \(2(433)=866\).
Time = 0.43 (sec) , antiderivative size = 1887, normalized size of antiderivative = 4.30
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (431) = 862\).
Time = 0.30 (sec) , antiderivative size = 1518, normalized size of antiderivative = 3.46 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (a + b \left (F^{e g + f g x}\right )^{n}\right )^{3}}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {1}{2} \, c^{2} {\left (\frac {2 \, F^{f g n x + e g n} b + 3 \, a}{{\left (2 \, F^{f g n x + e g n} a^{3} b + F^{2 \, f g n x + 2 \, e g n} a^{2} b^{2} + a^{4}\right )} f g n \log \left (F\right )} + \frac {2 \, {\left (f g n x + e g n\right )}}{a^{3} f g n} - \frac {2 \, \log \left (F^{f g n x + e g n} b + a\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac {3 \, a d^{2} f g n x^{2} \log \left (F\right ) - 2 \, a c d + 2 \, {\left (F^{e g n} b d^{2} f g n x^{2} \log \left (F\right ) - F^{e g n} b c d + {\left (2 \, F^{e g n} b c d f g n \log \left (F\right ) - F^{e g n} b d^{2}\right )} x\right )} F^{f g n x} + 2 \, {\left (3 \, a c d f g n \log \left (F\right ) - a d^{2}\right )} x}{2 \, {\left (2 \, F^{f g n x} F^{e g n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x} F^{2 \, e g n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} - \frac {{\left (3 \, c d f g n \log \left (F\right ) - d^{2}\right )} x}{a^{3} 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^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (2 \, c d f g n \log \left (F\right ) - 3 \, d^{2}\right )} {\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 )}}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (3 \, c d f g n \log \left (F\right ) - d^{2}\right )} \log \left (F^{f g n x} F^{e g n} b + a\right )}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, {\left (2 \, c d f g n \log \left (F\right ) - 3 \, d^{2}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{6 \, a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
[In]
[Out]
\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \]
[In]
[Out]