Optimal. Leaf size=1210 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.79024, antiderivative size = 1179, normalized size of antiderivative = 0.97, number of steps used = 73, number of rules used = 27, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.844, Rules used = {2439, 2416, 2389, 2296, 2295, 2401, 2390, 2305, 2304, 2396, 2433, 2374, 6589, 6742, 2411, 43, 2334, 12, 14, 2301, 2430, 2394, 2393, 2391, 2395, 2375, 2317} \[ -\frac{1}{4} g m n^2 x^2 b^2+\frac{f n^2 (d+e x)^2 b^2}{4 e^2}-\frac{g m n^2 (d+e x)^2 b^2}{8 e^2}+\frac{d^2 f n^2 \log ^2(d+e x) b^2}{2 e^2}-\frac{2 d f n^2 x b^2}{e}+\frac{15 d g m n^2 x b^2}{4 e}+\frac{7 g i m n^2 x b^2}{4 j}-\frac{d^2 g m n^2 \log (d+e x) b^2}{4 e^2}-\frac{2 d g m n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2}-\frac{3 g i m n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{2 e j}-\frac{g i^2 m n^2 \log (i+j x) b^2}{4 j^2}+\frac{1}{4} g n^2 x^2 \log \left (h (i+j x)^m\right ) b^2-\frac{3 d g n^2 (i+j x) \log \left (h (i+j x)^m\right ) b^2}{2 e j}+\frac{3 d^2 g n^2 \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right ) b^2}{2 e^2}+\frac{d g i m n^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{e j}+\frac{g i^2 m n^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{2 j^2}+\frac{3 d^2 g m n^2 \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right ) b^2}{2 e^2}-\frac{d^2 g m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{e^2}+\frac{g i^2 m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{j^2}-\frac{2 a d g m n x b}{e}-\frac{3 a g i m n x b}{2 j}+\frac{1}{4} g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b+\frac{g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{4 e^2}+\frac{1}{2} f n \left (-\frac{2 \log (d+e x) d^2}{e^2}+\frac{4 (d+e x) d}{e^2}-\frac{(d+e x)^2}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) b+\frac{d g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right ) b}{e j}+\frac{g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right ) b}{2 j^2}-\frac{1}{2} g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b+\frac{d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b}{e}+\frac{d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b}{e^2}-\frac{g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b}{j^2}-\frac{g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}+\frac{d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac{g i^2 m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (i+j x)}{e i-d j}\right )}{2 j^2}-\frac{d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac{1}{2} x^2 \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 ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2439
Rule 2416
Rule 2389
Rule 2296
Rule 2295
Rule 2401
Rule 2390
Rule 2305
Rule 2304
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 6742
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rule 2430
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rule 2375
Rule 2317
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{1}{2} (g j m) \int \frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{393+j x} \, dx-(b e n) \int \frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (393+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{1}{2} (g j m) \int \left (-\frac{393 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2}+\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}+\frac{154449 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2 (393+j x)}\right ) \, dx-(b e n) \int \left (\frac{f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}+\frac{g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{d+e x}\right ) \, dx\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{1}{2} (g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx+\frac{(393 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 j}-\frac{(154449 g m) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{393+j x} \, dx}{2 j}-(b e f n) \int \frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(b e g n) \int \frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{d+e x} \, dx\\ &=-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{1}{2} (g m) \int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx+\frac{(393 g m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e j}-(b f n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-(b e g n) \int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e^2}+\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}+\frac{d^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e^2 (d+e x)}\right ) \, dx+\frac{(154449 b e g m n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{d+e x} \, dx}{j^2}\\ &=\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{(g m) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 e}+\frac{(d g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 e}-(b g n) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right ) \, dx+\frac{(b d g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right ) \, dx}{e}-\frac{\left (b d^2 g n\right ) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{d+e x} \, dx}{e}+\frac{(154449 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{393 e-d j}{e}+\frac{j x}{e}\right )}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2}-\frac{(393 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 j}+\left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{x (-4 d+x)+2 d^2 \log (x)}{2 e^2 x} \, dx,x,d+e x\right )\\ &=-\frac{393 a b g m n x}{j}+\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac{1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{154449 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)}{393 e-d j}\right )}{j^2}-\frac{(g m) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}+\frac{(d g m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}-\frac{\left (b d^2 g n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac{393 e-d j}{e}+\frac{j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac{\left (393 b^2 g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j}+\frac{1}{2} (b g j m n) \int \frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{393+j x} \, dx-\frac{(b d g j m n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{393+j x} \, dx}{e}+\frac{\left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{x (-4 d+x)+2 d^2 \log (x)}{x} \, dx,x,d+e x\right )}{2 e^2}-\left (b^2 d g n^2\right ) \int \frac{x \log \left (h (393+j x)^m\right )}{d+e x} \, dx+\frac{1}{2} \left (b^2 e g n^2\right ) \int \frac{x^2 \log \left (h (393+j x)^m\right )}{d+e x} \, dx+\frac{\left (154449 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2}\\ &=-\frac{393 a b g m n x}{j}+\frac{393 b^2 g m n^2 x}{j}-\frac{393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac{1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac{d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac{154449 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)}{393 e-d j}\right )}{j^2}+\frac{154449 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{j^2}+\frac{\left (d^2 g j m\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\frac{393 e-d j}{e}+\frac{j x}{e}} \, dx,x,d+e x\right )}{2 e^3}+\frac{(b g m n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}-\frac{(b d g m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac{1}{2} (b g j m n) \int \left (-\frac{393 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac{154449 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2 (393+j x)}\right ) \, dx-\frac{(b d g j m n) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{j}-\frac{393 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (393+j x)}\right ) \, dx}{e}+\frac{\left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \left (-4 d+x+\frac{2 d^2 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{2 e^2}-\left (b^2 d g n^2\right ) \int \left (\frac{\log \left (h (393+j x)^m\right )}{e}-\frac{d \log \left (h (393+j x)^m\right )}{e (d+e x)}\right ) \, dx+\frac{1}{2} \left (b^2 e g n^2\right ) \int \left (-\frac{d \log \left (h (393+j x)^m\right )}{e^2}+\frac{x \log \left (h (393+j x)^m\right )}{e}+\frac{d^2 \log \left (h (393+j x)^m\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{a b d g m n x}{e}-\frac{393 a b g m n x}{j}-\frac{2 b^2 d f n^2 x}{e}+\frac{393 b^2 g m n^2 x}{j}+\frac{b^2 f n^2 (d+e x)^2}{4 e^2}-\frac{b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac{393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac{b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac{1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac{d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac{d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{393 e-d j}\right )}{2 e^2}-\frac{154449 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)}{393 e-d j}\right )}{j^2}+\frac{154449 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{j^2}+\frac{1}{2} (b g m n) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-\frac{\left (b^2 d g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac{\left (b d^2 g m n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac{(b d g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{e}+\frac{(393 b d g m n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{393+j x} \, dx}{e}-\frac{(393 b g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{2 j}+\frac{(154449 b g m n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{393+j x} \, dx}{2 j}+\frac{\left (b^2 d^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e x\right )}{e^2}+\frac{1}{2} \left (b^2 g n^2\right ) \int x \log \left (h (393+j x)^m\right ) \, dx-\frac{\left (b^2 d g n^2\right ) \int \log \left (h (393+j x)^m\right ) \, dx}{2 e}-\frac{\left (b^2 d g n^2\right ) \int \log \left (h (393+j x)^m\right ) \, dx}{e}+\frac{\left (b^2 d^2 g n^2\right ) \int \frac{\log \left (h (393+j x)^m\right )}{d+e x} \, dx}{2 e}+\frac{\left (b^2 d^2 g n^2\right ) \int \frac{\log \left (h (393+j x)^m\right )}{d+e x} \, dx}{e}\\ &=-\frac{2 a b d g m n x}{e}-\frac{1179 a b g m n x}{2 j}-\frac{2 b^2 d f n^2 x}{e}+\frac{b^2 d g m n^2 x}{e}+\frac{393 b^2 g m n^2 x}{j}+\frac{b^2 f n^2 (d+e x)^2}{4 e^2}-\frac{b^2 g m n^2 (d+e x)^2}{8 e^2}+\frac{b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac{b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac{393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac{1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{393 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{e j}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{1}{4} b^2 g n^2 x^2 \log \left (h (393+j x)^m\right )+\frac{3 b^2 d^2 g n^2 \log \left (-\frac{j (d+e x)}{393 e-d j}\right ) \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac{1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac{d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac{d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{393 e-d j}\right )}{2 e^2}+\frac{b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{393 e-d j}\right )}{e^2}-\frac{154449 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)}{393 e-d j}\right )}{j^2}+\frac{154449 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{j^2}-\frac{\left (b^2 d g m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{e}-\frac{\left (393 b^2 g m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{2 j}-\frac{\left (b^2 d g n^2\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,393+j x\right )}{2 e j}-\frac{\left (b^2 d g n^2\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,393+j x\right )}{e j}-\frac{\left (b^2 d^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac{1}{4} \left (b^2 e g m n^2\right ) \int \frac{x^2}{d+e x} \, dx-\frac{\left (154449 b^2 e g m n^2\right ) \int \frac{\log \left (\frac{e (393+j x)}{393 e-d j}\right )}{d+e x} \, dx}{2 j^2}-\frac{\left (393 b^2 d g m n^2\right ) \int \frac{\log \left (\frac{e (393+j x)}{393 e-d j}\right )}{d+e x} \, dx}{j}-\frac{1}{4} \left (b^2 g j m n^2\right ) \int \frac{x^2}{393+j x} \, dx-\frac{\left (b^2 d^2 g j m n^2\right ) \int \frac{\log \left (\frac{j (d+e x)}{-393 e+d j}\right )}{393+j x} \, dx}{2 e^2}-\frac{\left (b^2 d^2 g j m n^2\right ) \int \frac{\log \left (\frac{j (d+e x)}{-393 e+d j}\right )}{393+j x} \, dx}{e^2}\\ &=-\frac{2 a b d g m n x}{e}-\frac{1179 a b g m n x}{2 j}-\frac{2 b^2 d f n^2 x}{e}+\frac{5 b^2 d g m n^2 x}{2 e}+\frac{393 b^2 g m n^2 x}{j}+\frac{b^2 f n^2 (d+e x)^2}{4 e^2}-\frac{b^2 g m n^2 (d+e x)^2}{8 e^2}+\frac{b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac{b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac{393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac{1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{393 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{e j}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{1}{4} b^2 g n^2 x^2 \log \left (h (393+j x)^m\right )-\frac{3 b^2 d g n^2 (393+j x) \log \left (h (393+j x)^m\right )}{2 e j}+\frac{3 b^2 d^2 g n^2 \log \left (-\frac{j (d+e x)}{393 e-d j}\right ) \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac{1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac{d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac{d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{393 e-d j}\right )}{2 e^2}+\frac{b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{393 e-d j}\right )}{e^2}-\frac{154449 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)}{393 e-d j}\right )}{j^2}-\frac{b^2 d^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{e^2}+\frac{154449 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{j^2}-\frac{\left (b^2 d g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac{\left (393 b^2 g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{2 e j}-\frac{\left (b^2 d^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-393 e+d j}\right )}{x} \, dx,x,393+j x\right )}{2 e^2}-\frac{\left (b^2 d^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-393 e+d j}\right )}{x} \, dx,x,393+j x\right )}{e^2}-\frac{1}{4} \left (b^2 e g m n^2\right ) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx-\frac{\left (154449 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{2 j^2}-\frac{\left (393 b^2 d g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e j}-\frac{1}{4} \left (b^2 g j m n^2\right ) \int \left (-\frac{393}{j^2}+\frac{x}{j}+\frac{154449}{j^2 (393+j x)}\right ) \, dx\\ &=-\frac{2 a b d g m n x}{e}-\frac{1179 a b g m n x}{2 j}-\frac{2 b^2 d f n^2 x}{e}+\frac{15 b^2 d g m n^2 x}{4 e}+\frac{2751 b^2 g m n^2 x}{4 j}-\frac{1}{4} b^2 g m n^2 x^2+\frac{b^2 f n^2 (d+e x)^2}{4 e^2}-\frac{b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac{b^2 d^2 g m n^2 \log (d+e x)}{4 e^2}+\frac{b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac{2 b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac{1179 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}+\frac{1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac{1}{2} b f n \left (\frac{4 d (d+e x)}{e^2}-\frac{(d+e x)^2}{e^2}-\frac{2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac{g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac{154449 b^2 g m n^2 \log (393+j x)}{4 j^2}+\frac{154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{393 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{e j}-\frac{154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac{1}{4} b^2 g n^2 x^2 \log \left (h (393+j x)^m\right )-\frac{3 b^2 d g n^2 (393+j x) \log \left (h (393+j x)^m\right )}{2 e j}+\frac{3 b^2 d^2 g n^2 \log \left (-\frac{j (d+e x)}{393 e-d j}\right ) \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac{1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac{d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac{d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{393 e-d j}\right )}{2 e^2}+\frac{154449 b^2 g m n^2 \text{Li}_2\left (-\frac{j (d+e x)}{393 e-d j}\right )}{2 j^2}+\frac{393 b^2 d g m n^2 \text{Li}_2\left (-\frac{j (d+e x)}{393 e-d j}\right )}{e j}+\frac{b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{393 e-d j}\right )}{e^2}-\frac{154449 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)}{393 e-d j}\right )}{j^2}+\frac{3 b^2 d^2 g m n^2 \text{Li}_2\left (\frac{e (393+j x)}{393 e-d j}\right )}{2 e^2}-\frac{b^2 d^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{e^2}+\frac{154449 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{393 e-d j}\right )}{j^2}\\ \end{align*}
Mathematica [A] time = 0.942629, size = 2067, normalized size = 1.71 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.207, size = 0, normalized size = 0. \begin{align*} \int x \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 x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b f x \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} f x +{\left (b^{2} g x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b g x \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} g x\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 )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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