Optimal. Leaf size=1471 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.93148, antiderivative size = 2096, normalized size of antiderivative = 1.42, number of steps used = 29, number of rules used = 14, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.452, Rules used = {2497, 2500, 2394, 2393, 2391, 2433, 2375, 2317, 2374, 6589, 2440, 2437, 2435, 2315} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2497
Rule 2500
Rule 2394
Rule 2393
Rule 2391
Rule 2433
Rule 2375
Rule 2317
Rule 2374
Rule 6589
Rule 2440
Rule 2437
Rule 2435
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx &=\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac{(2 b p r) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac{(2 d q r) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{c+d x} \, dx}{h}\\ &=\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac{(2 b p r) \int \frac{\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac{(2 b p r) \int \frac{\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac{(2 d q r) \int \frac{\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x} \, dx}{h}-\frac{(2 d q r) \int \frac{\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x} \, dx}{h}-\frac{\left (2 b p r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{\log (g+h x)}{a+b x} \, dx}{h}-\frac{\left (2 d q r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{\log (g+h x)}{c+d x} \, dx}{h}\\ &=\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{2 q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac{(2 p r) \operatorname{Subst}\left (\int \frac{\log \left (x^{p r}\right ) \log \left (\frac{b g-a h}{b}+\frac{h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac{(2 p r) \operatorname{Subst}\left (\int \frac{\log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{q r}\right ) \log \left (-\frac{-b g+a h}{b}+\frac{h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac{(2 q r) \operatorname{Subst}\left (\int \frac{\log \left (x^{q r}\right ) \log \left (\frac{d g-c h}{d}+\frac{h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}-\frac{(2 q r) \operatorname{Subst}\left (\int \frac{\log \left (\left (-\frac{b c-a d}{d}+\frac{b x}{d}\right )^{p r}\right ) \log \left (-\frac{-d g+c h}{d}+\frac{h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\left (2 p r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{\log \left (\frac{h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx+\left (2 q r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{\log \left (\frac{h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx\\ &=-\frac{\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac{\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{2 q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac{\operatorname{Subst}\left (\int \frac{\log ^2\left (x^{p r}\right )}{\frac{b g-a h}{b}+\frac{h x}{b}} \, dx,x,a+b x\right )}{b}+\frac{\operatorname{Subst}\left (\int \frac{\log ^2\left (x^{q r}\right )}{\frac{d g-c h}{d}+\frac{h x}{d}} \, dx,x,c+d x\right )}{d}-\frac{\left (2 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right ) \log \left (-\frac{-b g+a h}{b}+\frac{h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac{\left (2 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d}+\frac{b x}{d}\right ) \log \left (-\frac{-d g+c h}{d}+\frac{h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\frac{\left (2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-d g+c h}{d}+\frac{h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\frac{\left (2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b g+a h}{b}+\frac{h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}+\frac{\left (2 p r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}+\frac{\left (2 q r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}\\ &=-\frac{\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac{2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x)}{h}-\frac{2 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (g+h x)}{h}+\frac{2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x)}{h}-\frac{\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{2 q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac{\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{p q r^2 \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )+\log \left (\frac{b g-a h}{b (g+h x)}\right )-\log \left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}+\frac{p q r^2 \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )-\log \left (-\frac{h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2}{h}-\frac{p q r^2 \left (\log \left (-\frac{d (a+b x)}{b c-a d}\right )+\log \left (\frac{d g-c h}{d (g+h x)}\right )-\log \left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{p q r^2 \left (\log \left (-\frac{d (a+b x)}{b c-a d}\right )-\log \left (-\frac{h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2}{h}-\frac{2 p q r^2 \left (\log (g+h x)-\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{h}-\frac{2 p q r^2 \left (\log (g+h x)-\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{h}+\frac{2 p q r^2 \log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text{Li}_2\left (\frac{h (a+b x)}{b (g+h x)}\right )}{h}-\frac{2 p q r^2 \log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text{Li}_2\left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text{Li}_2\left (\frac{h (c+d x)}{d (g+h x)}\right )}{h}-\frac{2 p q r^2 \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text{Li}_2\left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p r \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}-\frac{2 p q r^2 \left (\log (c+d x)+\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 q r \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{2 p q r^2 \left (\log (a+b x)+\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{h (a+b x)}{b (g+h x)}\right )}{h}-\frac{2 p q r^2 \text{Li}_3\left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{h (c+d x)}{d (g+h x)}\right )}{h}-\frac{2 p q r^2 \text{Li}_3\left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{(2 p r) \operatorname{Subst}\left (\int \frac{\log \left (x^{p r}\right ) \log \left (1+\frac{h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac{(2 q r) \operatorname{Subst}\left (\int \frac{\log \left (x^{q r}\right ) \log \left (1+\frac{h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h}-\frac{\left (2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{h x}{-d g+c h}\right )}{-\frac{-d g+c h}{d}+\frac{h x}{d}} \, dx,x,c+d x\right )}{d}-\frac{\left (2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{h x}{-b g+a h}\right )}{-\frac{-b g+a h}{b}+\frac{h x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac{2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x)}{h}-\frac{2 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (g+h x)}{h}+\frac{2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x)}{h}-\frac{\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{2 q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac{\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{p q r^2 \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )+\log \left (\frac{b g-a h}{b (g+h x)}\right )-\log \left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}+\frac{p q r^2 \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )-\log \left (-\frac{h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2}{h}-\frac{p q r^2 \left (\log \left (-\frac{d (a+b x)}{b c-a d}\right )+\log \left (\frac{d g-c h}{d (g+h x)}\right )-\log \left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{p q r^2 \left (\log \left (-\frac{d (a+b x)}{b c-a d}\right )-\log \left (-\frac{h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2}{h}-\frac{2 p q r^2 \left (\log (g+h x)-\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{h}+\frac{2 p r \log \left ((a+b x)^{p r}\right ) \text{Li}_2\left (-\frac{h (a+b x)}{b g-a h}\right )}{h}-\frac{2 p q r^2 \left (\log (g+h x)-\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{h}+\frac{2 q r \log \left ((c+d x)^{q r}\right ) \text{Li}_2\left (-\frac{h (c+d x)}{d g-c h}\right )}{h}+\frac{2 p q r^2 \log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text{Li}_2\left (\frac{h (a+b x)}{b (g+h x)}\right )}{h}-\frac{2 p q r^2 \log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text{Li}_2\left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text{Li}_2\left (\frac{h (c+d x)}{d (g+h x)}\right )}{h}-\frac{2 p q r^2 \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text{Li}_2\left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 p r \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}-\frac{2 p q r^2 \left (\log (c+d x)+\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{2 q r \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{2 p q r^2 \left (\log (a+b x)+\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{h (a+b x)}{b (g+h x)}\right )}{h}-\frac{2 p q r^2 \text{Li}_3\left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{h (c+d x)}{d (g+h x)}\right )}{h}-\frac{2 p q r^2 \text{Li}_3\left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{\left (2 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac{\left (2 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h}\\ &=-\frac{\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac{2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x)}{h}-\frac{2 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (g+h x)}{h}+\frac{2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x)}{h}-\frac{\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac{2 p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{2 q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac{\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{p q r^2 \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )+\log \left (\frac{b g-a h}{b (g+h x)}\right )-\log \left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}+\frac{p q r^2 \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )-\log \left (-\frac{h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2}{h}-\frac{p q r^2 \left (\log \left (-\frac{d (a+b x)}{b c-a d}\right )+\log \left (\frac{d g-c h}{d (g+h x)}\right )-\log \left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{p q r^2 \left (\log \left (-\frac{d (a+b x)}{b c-a d}\right )-\log \left (-\frac{h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2}{h}-\frac{2 p q r^2 \left (\log (g+h x)-\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{h}+\frac{2 p r \log \left ((a+b x)^{p r}\right ) \text{Li}_2\left (-\frac{h (a+b x)}{b g-a h}\right )}{h}-\frac{2 p q r^2 \left (\log (g+h x)-\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{h}+\frac{2 q r \log \left ((c+d x)^{q r}\right ) \text{Li}_2\left (-\frac{h (c+d x)}{d g-c h}\right )}{h}+\frac{2 p q r^2 \log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text{Li}_2\left (\frac{h (a+b x)}{b (g+h x)}\right )}{h}-\frac{2 p q r^2 \log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text{Li}_2\left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text{Li}_2\left (\frac{h (c+d x)}{d (g+h x)}\right )}{h}-\frac{2 p q r^2 \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text{Li}_2\left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 p r \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}-\frac{2 p q r^2 \left (\log (c+d x)+\log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{2 q r \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\frac{2 p q r^2 \left (\log (a+b x)+\log \left (-\frac{(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{h}-\frac{2 p^2 r^2 \text{Li}_3\left (-\frac{h (a+b x)}{b g-a h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{h}-\frac{2 q^2 r^2 \text{Li}_3\left (-\frac{h (c+d x)}{d g-c h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{h (a+b x)}{b (g+h x)}\right )}{h}-\frac{2 p q r^2 \text{Li}_3\left (-\frac{(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{h (c+d x)}{d (g+h x)}\right )}{h}-\frac{2 p q r^2 \text{Li}_3\left (\frac{(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{2 p q r^2 \text{Li}_3\left (\frac{d (g+h x)}{d g-c h}\right )}{h}\\ \end{align*}
Mathematica [A] time = 0.280696, size = 1370, normalized size = 0.93 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.578, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}}{hx+g}}\, 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*} \int \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g}\,{d x} \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 (\frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g}, 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 \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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