Optimal. Leaf size=649 \[ -\frac{2 b d g m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b g i m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{j}-\frac{2 b^2 d g m n^2 \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+\frac{2 b^2 d g m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{e}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b g n x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{2 b g i m n \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{d g m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{g i m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b f n x+4 a b g m n x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}+2 b^2 f n^2 x+\frac{2 b^2 g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{j}-6 b^2 g m n^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.48128, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 19, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.613, Rules used = {2430, 2416, 2389, 2296, 2295, 2396, 2433, 2374, 6589, 6742, 2411, 2346, 2301, 43, 2394, 2393, 2391, 2375, 2317} \[ -\frac{2 b d g m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b g i m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{j}-\frac{2 b^2 d g m n^2 \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+\frac{2 b^2 d g m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{e}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b g n x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{2 b g i m n \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{d g m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{g i m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b f n x+4 a b g m n x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}+2 b^2 f n^2 x+\frac{2 b^2 g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{j}-6 b^2 g m n^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2430
Rule 2416
Rule 2389
Rule 2296
Rule 2295
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 6742
Rule 2411
Rule 2346
Rule 2301
Rule 43
Rule 2394
Rule 2393
Rule 2391
Rule 2375
Rule 2317
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right ) \, dx &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-(g j m) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{394+j x} \, dx-(2 b e n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (394+j x)^m\right )\right )}{d+e x} \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-(g j m) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{394 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j (394+j x)}\right ) \, dx-(2 b e n) \int \left (\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}+\frac{g x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{d+e x}\right ) \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-(g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx+(394 g m) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{394+j x} \, dx-(2 b e f n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(2 b e g n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{d+e x} \, dx\\ &=\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{(g m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-(2 b f n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-(2 b e g n) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{e}-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac{(788 b e g m n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{d+e x} \, dx}{j}\\ &=-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{(2 b f n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac{(2 b d f n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{e}-(2 b g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right ) \, dx+(2 b d g n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{d+e x} \, dx+\frac{(2 b g m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac{(788 b g m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{394 e-d j}{e}+\frac{j x}{e}\right )}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-2 a b f n x+2 a b g m n x+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{\left (2 b^2 f n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{(2 b d g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac{394 e-d j}{e}+\frac{j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (2 b^2 g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+(2 b g j m n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{394+j x} \, dx+\left (2 b^2 e g n^2\right ) \int \frac{x \log \left (h (394+j x)^m\right )}{d+e x} \, dx-\frac{\left (788 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-2 a b f n x+2 a b g m n x+2 b^2 f n^2 x-2 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{(d g j m) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\frac{394 e-d j}{e}+\frac{j x}{e}} \, dx,x,d+e x\right )}{e^2}+(2 b g j m n) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{j}-\frac{394 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (394+j x)}\right ) \, dx+\left (2 b^2 e g n^2\right ) \int \left (\frac{\log \left (h (394+j x)^m\right )}{e}-\frac{d \log \left (h (394+j x)^m\right )}{e (d+e x)}\right ) \, dx\\ &=-2 a b f n x+2 a b g m n x+2 b^2 f n^2 x-2 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+(2 b g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-(788 b g m n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{394+j x} \, dx+\frac{(2 b d g m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\left (2 b^2 g n^2\right ) \int \log \left (h (394+j x)^m\right ) \, dx-\left (2 b^2 d g n^2\right ) \int \frac{\log \left (h (394+j x)^m\right )}{d+e x} \, dx\\ &=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-2 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{394 e-d j}\right ) \log \left (h (394+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+\left (2 b^2 g m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx+\frac{\left (2 b^2 g n^2\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,394+j x\right )}{j}+\frac{\left (2 b^2 d g m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (788 b^2 e g m n^2\right ) \int \frac{\log \left (\frac{e (394+j x)}{394 e-d j}\right )}{d+e x} \, dx}{j}+\frac{\left (2 b^2 d g j m n^2\right ) \int \frac{\log \left (\frac{j (d+e x)}{-394 e+d j}\right )}{394+j x} \, dx}{e}\\ &=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-4 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{2 b^2 g n^2 (394+j x) \log \left (h (394+j x)^m\right )}{j}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{394 e-d j}\right ) \log \left (h (394+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+\frac{2 b^2 d g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+\frac{\left (2 b^2 g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{\left (2 b^2 d g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-394 e+d j}\right )}{x} \, dx,x,394+j x\right )}{e}+\frac{\left (788 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-6 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{2 b^2 g n^2 (394+j x) \log \left (h (394+j x)^m\right )}{j}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{394 e-d j}\right ) \log \left (h (394+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{788 b^2 g m n^2 \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{2 b^2 d g m n^2 \text{Li}_2\left (\frac{e (394+j x)}{394 e-d j}\right )}{e}+\frac{2 b^2 d g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}\\ \end{align*}
Mathematica [B] time = 0.533498, size = 1355, normalized size = 2.09 \[ \frac{e f j x a^2-e g j m x a^2+e g i m \log (i+j x) a^2+e g j x \log \left (h (i+j x)^m\right ) a^2-2 b d f j n a+2 b d g j m n a-2 b e f j n x a+4 b e g j m n x a+2 b d f j n \log (d+e x) a-2 b d g j m n \log (d+e x) a+2 b e f j x \log \left (c (d+e x)^n\right ) a-2 b e g j m x \log \left (c (d+e x)^n\right ) a-2 b e g i m n \log (i+j x) a+2 b d g j m n \log (i+j x) a-2 b e g i m n \log (d+e x) \log (i+j x) a+2 b e g i m \log \left (c (d+e x)^n\right ) \log (i+j x) a+2 b e g i m n \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right ) a-2 b d g j m n \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right ) a-2 b d g j n \log \left (h (i+j x)^m\right ) a-2 b e g j n x \log \left (h (i+j x)^m\right ) a+2 b d g j n \log (d+e x) \log \left (h (i+j x)^m\right ) a+2 b e g j x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right ) a-2 b^2 d g j m n^2-b^2 d f j n^2 \log ^2(d+e x)+b^2 d g j m n^2 \log ^2(d+e x)+b^2 e f j x \log ^2\left (c (d+e x)^n\right )-b^2 e g j m x \log ^2\left (c (d+e x)^n\right )+2 b^2 e f j n^2 x-6 b^2 e g j m n^2 x+2 b^2 d g j m n^2 \log (d+e x)-2 b^2 d f j n \log \left (c (d+e x)^n\right )+2 b^2 d g j m n \log \left (c (d+e x)^n\right )-2 b^2 e f j n x \log \left (c (d+e x)^n\right )+4 b^2 e g j m n x \log \left (c (d+e x)^n\right )+2 b^2 d f j n \log (d+e x) \log \left (c (d+e x)^n\right )-2 b^2 d g j m n \log (d+e x) \log \left (c (d+e x)^n\right )+2 b^2 e g i m n^2 \log (i+j x)+b^2 e g i m n^2 \log ^2(d+e x) \log (i+j x)+b^2 e g i m \log ^2\left (c (d+e x)^n\right ) \log (i+j x)+2 b^2 e g i m n^2 \log (d+e x) \log (i+j x)-2 b^2 d g j m n^2 \log (d+e x) \log (i+j x)-2 b^2 e g i m n \log \left (c (d+e x)^n\right ) \log (i+j x)+2 b^2 d g j m n \log \left (c (d+e x)^n\right ) \log (i+j x)-2 b^2 e g i m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log (i+j x)-b^2 e g i m n^2 \log ^2(d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )+b^2 d g j m n^2 \log ^2(d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )-2 b^2 e g i m n^2 \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )+2 b^2 d g j m n^2 \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )+2 b^2 e g i m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )-2 b^2 d g j m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )-b^2 d g j n^2 \log ^2(d+e x) \log \left (h (i+j x)^m\right )+b^2 e g j x \log ^2\left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b^2 e g j n^2 x \log \left (h (i+j x)^m\right )-2 b^2 d g j n \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )-2 b^2 e g j n x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b^2 d g j n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b g (e i-d j) m n \left (a-b n+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )+2 b^2 g (d j-e i) m n^2 \text{PolyLog}\left (3,\frac{j (d+e x)}{d j-e i}\right )}{e j} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.82, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \left ( f+g\ln \left ( h \left ( jx+i \right ) ^{m} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} f \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b f \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} f +{\left (b^{2} g \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b g \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} g\right )} \log \left ({\left (j x + i\right )}^{m} h\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}{\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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