Optimal. Leaf size=2240 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.43569, antiderivative size = 2220, normalized size of antiderivative = 0.99, number of steps used = 49, number of rules used = 14, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.452, Rules used = {2498, 2513, 2411, 43, 2334, 12, 2301, 2418, 2389, 2295, 2394, 2393, 2391, 2395} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2498
Rule 2513
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 2301
Rule 2418
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rubi steps
\begin{align*} \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{(b p r) \int \frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx}{2 h}-\frac{(d q r) \int \frac{(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{2 h}\\ &=\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{\left (b p^2 r^2\right ) \int \frac{(g+h x)^4 \log (a+b x)}{a+b x} \, dx}{2 h}-\frac{\left (b p q r^2\right ) \int \frac{(g+h x)^4 \log (c+d x)}{a+b x} \, dx}{2 h}-\frac{\left (d p q r^2\right ) \int \frac{(g+h x)^4 \log (a+b x)}{c+d x} \, dx}{2 h}-\frac{\left (d q^2 r^2\right ) \int \frac{(g+h x)^4 \log (c+d x)}{c+d x} \, dx}{2 h}+\frac{\left (b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{(g+h x)^4}{a+b x} \, dx}{2 h}+\frac{\left (d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{(g+h x)^4}{c+d x} \, dx}{2 h}\\ &=\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{\left (p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{b g-a h}{b}+\frac{h x}{b}\right )^4 \log (x)}{x} \, dx,x,a+b x\right )}{2 h}-\frac{\left (b p q r^2\right ) \int \left (\frac{h (b g-a h)^3 \log (c+d x)}{b^4}+\frac{(b g-a h)^4 \log (c+d x)}{b^4 (a+b x)}+\frac{h (b g-a h)^2 (g+h x) \log (c+d x)}{b^3}+\frac{h (b g-a h) (g+h x)^2 \log (c+d x)}{b^2}+\frac{h (g+h x)^3 \log (c+d x)}{b}\right ) \, dx}{2 h}-\frac{\left (d p q r^2\right ) \int \left (\frac{h (d g-c h)^3 \log (a+b x)}{d^4}+\frac{(d g-c h)^4 \log (a+b x)}{d^4 (c+d x)}+\frac{h (d g-c h)^2 (g+h x) \log (a+b x)}{d^3}+\frac{h (d g-c h) (g+h x)^2 \log (a+b x)}{d^2}+\frac{h (g+h x)^3 \log (a+b x)}{d}\right ) \, dx}{2 h}-\frac{\left (q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{d g-c h}{d}+\frac{h x}{d}\right )^4 \log (x)}{x} \, dx,x,c+d x\right )}{2 h}+\frac{\left (b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \left (\frac{h (b g-a h)^3}{b^4}+\frac{(b g-a h)^4}{b^4 (a+b x)}+\frac{h (b g-a h)^2 (g+h x)}{b^3}+\frac{h (b g-a h) (g+h x)^2}{b^2}+\frac{h (g+h x)^3}{b}\right ) \, dx}{2 h}+\frac{\left (d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \left (\frac{h (d g-c h)^3}{d^4}+\frac{(d g-c h)^4}{d^4 (c+d x)}+\frac{h (d g-c h)^2 (g+h x)}{d^3}+\frac{h (d g-c h) (g+h x)^2}{d^2}+\frac{h (g+h x)^3}{d}\right ) \, dx}{2 h}\\ &=-\frac{p^2 r^2 \log (a+b x) \left (\frac{48 h (b g-a h)^3 (a+b x)}{b^4}+\frac{36 h^2 (b g-a h)^2 (a+b x)^2}{b^4}+\frac{16 h^3 (b g-a h) (a+b x)^3}{b^4}+\frac{3 h^4 (a+b x)^4}{b^4}+\frac{12 (b g-a h)^4 \log (a+b x)}{b^4}\right )}{24 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{48 h (d g-c h)^3 (c+d x)}{d^4}+\frac{36 h^2 (d g-c h)^2 (c+d x)^2}{d^4}+\frac{16 h^3 (d g-c h) (c+d x)^3}{d^4}+\frac{3 h^4 (c+d x)^4}{d^4}+\frac{12 (d g-c h)^4 \log (c+d x)}{d^4}\right )}{24 h}+\frac{(b g-a h)^3 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^3}+\frac{(d g-c h)^3 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^3}+\frac{(b g-a h)^2 p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 b^2 h}+\frac{(d g-c h)^2 q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 d^2 h}+\frac{(b g-a h) p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 b h}+\frac{(d g-c h) q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 d h}+\frac{p r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{q r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{(b g-a h)^4 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^4 h}+\frac{(d g-c h)^4 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^4 h}+\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}+\frac{\left (p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{48 h (b g-a h)^3+36 h^2 (b g-a h)^2 x+16 h^3 (b g-a h) x^2+3 h^4 x^3+\frac{12 (b g-a h)^4 \log (x)}{x}}{12 b^4} \, dx,x,a+b x\right )}{2 h}-\frac{1}{2} \left (p q r^2\right ) \int (g+h x)^3 \log (a+b x) \, dx-\frac{1}{2} \left (p q r^2\right ) \int (g+h x)^3 \log (c+d x) \, dx-\frac{\left ((b g-a h) p q r^2\right ) \int (g+h x)^2 \log (c+d x) \, dx}{2 b}-\frac{\left ((b g-a h)^2 p q r^2\right ) \int (g+h x) \log (c+d x) \, dx}{2 b^2}-\frac{\left ((b g-a h)^3 p q r^2\right ) \int \log (c+d x) \, dx}{2 b^3}-\frac{\left ((b g-a h)^4 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 b^3 h}-\frac{\left ((d g-c h) p q r^2\right ) \int (g+h x)^2 \log (a+b x) \, dx}{2 d}-\frac{\left ((d g-c h)^2 p q r^2\right ) \int (g+h x) \log (a+b x) \, dx}{2 d^2}-\frac{\left ((d g-c h)^3 p q r^2\right ) \int \log (a+b x) \, dx}{2 d^3}-\frac{\left ((d g-c h)^4 p q r^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 d^3 h}+\frac{\left (q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{48 h (d g-c h)^3+36 h^2 (d g-c h)^2 x+16 h^3 (d g-c h) x^2+3 h^4 x^3+\frac{12 (d g-c h)^4 \log (x)}{x}}{12 d^4} \, dx,x,c+d x\right )}{2 h}\\ &=-\frac{(d g-c h)^2 p q r^2 (g+h x)^2 \log (a+b x)}{4 d^2 h}-\frac{(d g-c h) p q r^2 (g+h x)^3 \log (a+b x)}{6 d h}-\frac{p q r^2 (g+h x)^4 \log (a+b x)}{8 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{48 h (b g-a h)^3 (a+b x)}{b^4}+\frac{36 h^2 (b g-a h)^2 (a+b x)^2}{b^4}+\frac{16 h^3 (b g-a h) (a+b x)^3}{b^4}+\frac{3 h^4 (a+b x)^4}{b^4}+\frac{12 (b g-a h)^4 \log (a+b x)}{b^4}\right )}{24 h}-\frac{(b g-a h)^2 p q r^2 (g+h x)^2 \log (c+d x)}{4 b^2 h}-\frac{(b g-a h) p q r^2 (g+h x)^3 \log (c+d x)}{6 b h}-\frac{p q r^2 (g+h x)^4 \log (c+d x)}{8 h}-\frac{(b g-a h)^4 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{48 h (d g-c h)^3 (c+d x)}{d^4}+\frac{36 h^2 (d g-c h)^2 (c+d x)^2}{d^4}+\frac{16 h^3 (d g-c h) (c+d x)^3}{d^4}+\frac{3 h^4 (c+d x)^4}{d^4}+\frac{12 (d g-c h)^4 \log (c+d x)}{d^4}\right )}{24 h}-\frac{(d g-c h)^4 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 d^4 h}+\frac{(b g-a h)^3 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^3}+\frac{(d g-c h)^3 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^3}+\frac{(b g-a h)^2 p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 b^2 h}+\frac{(d g-c h)^2 q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 d^2 h}+\frac{(b g-a h) p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 b h}+\frac{(d g-c h) q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 d h}+\frac{p r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{q r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{(b g-a h)^4 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^4 h}+\frac{(d g-c h)^4 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^4 h}+\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}+\frac{\left (p^2 r^2\right ) \operatorname{Subst}\left (\int \left (48 h (b g-a h)^3+36 h^2 (b g-a h)^2 x+16 h^3 (b g-a h) x^2+3 h^4 x^3+\frac{12 (b g-a h)^4 \log (x)}{x}\right ) \, dx,x,a+b x\right )}{24 b^4 h}+\frac{\left (b p q r^2\right ) \int \frac{(g+h x)^4}{a+b x} \, dx}{8 h}+\frac{\left (d p q r^2\right ) \int \frac{(g+h x)^4}{c+d x} \, dx}{8 h}+\frac{\left (d (b g-a h) p q r^2\right ) \int \frac{(g+h x)^3}{c+d x} \, dx}{6 b h}+\frac{\left (d (b g-a h)^2 p q r^2\right ) \int \frac{(g+h x)^2}{c+d x} \, dx}{4 b^2 h}-\frac{\left ((b g-a h)^3 p q r^2\right ) \operatorname{Subst}(\int \log (x) \, dx,x,c+d x)}{2 b^3 d}+\frac{\left (d (b g-a h)^4 p q r^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b^4 h}+\frac{\left (b (d g-c h) p q r^2\right ) \int \frac{(g+h x)^3}{a+b x} \, dx}{6 d h}+\frac{\left (b (d g-c h)^2 p q r^2\right ) \int \frac{(g+h x)^2}{a+b x} \, dx}{4 d^2 h}-\frac{\left ((d g-c h)^3 p q r^2\right ) \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{2 b d^3}+\frac{\left (b (d g-c h)^4 p q r^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 d^4 h}+\frac{\left (q^2 r^2\right ) \operatorname{Subst}\left (\int \left (48 h (d g-c h)^3+36 h^2 (d g-c h)^2 x+16 h^3 (d g-c h) x^2+3 h^4 x^3+\frac{12 (d g-c h)^4 \log (x)}{x}\right ) \, dx,x,c+d x\right )}{24 d^4 h}\\ &=\frac{2 (b g-a h)^3 p^2 r^2 x}{b^3}+\frac{(b g-a h)^3 p q r^2 x}{2 b^3}+\frac{(d g-c h)^3 p q r^2 x}{2 d^3}+\frac{2 (d g-c h)^3 q^2 r^2 x}{d^3}+\frac{3 h (b g-a h)^2 p^2 r^2 (a+b x)^2}{4 b^4}+\frac{2 h^2 (b g-a h) p^2 r^2 (a+b x)^3}{9 b^4}+\frac{h^3 p^2 r^2 (a+b x)^4}{32 b^4}+\frac{3 h (d g-c h)^2 q^2 r^2 (c+d x)^2}{4 d^4}+\frac{2 h^2 (d g-c h) q^2 r^2 (c+d x)^3}{9 d^4}+\frac{h^3 q^2 r^2 (c+d x)^4}{32 d^4}-\frac{(d g-c h)^3 p q r^2 (a+b x) \log (a+b x)}{2 b d^3}-\frac{(d g-c h)^2 p q r^2 (g+h x)^2 \log (a+b x)}{4 d^2 h}-\frac{(d g-c h) p q r^2 (g+h x)^3 \log (a+b x)}{6 d h}-\frac{p q r^2 (g+h x)^4 \log (a+b x)}{8 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{48 h (b g-a h)^3 (a+b x)}{b^4}+\frac{36 h^2 (b g-a h)^2 (a+b x)^2}{b^4}+\frac{16 h^3 (b g-a h) (a+b x)^3}{b^4}+\frac{3 h^4 (a+b x)^4}{b^4}+\frac{12 (b g-a h)^4 \log (a+b x)}{b^4}\right )}{24 h}-\frac{(b g-a h)^3 p q r^2 (c+d x) \log (c+d x)}{2 b^3 d}-\frac{(b g-a h)^2 p q r^2 (g+h x)^2 \log (c+d x)}{4 b^2 h}-\frac{(b g-a h) p q r^2 (g+h x)^3 \log (c+d x)}{6 b h}-\frac{p q r^2 (g+h x)^4 \log (c+d x)}{8 h}-\frac{(b g-a h)^4 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{48 h (d g-c h)^3 (c+d x)}{d^4}+\frac{36 h^2 (d g-c h)^2 (c+d x)^2}{d^4}+\frac{16 h^3 (d g-c h) (c+d x)^3}{d^4}+\frac{3 h^4 (c+d x)^4}{d^4}+\frac{12 (d g-c h)^4 \log (c+d x)}{d^4}\right )}{24 h}-\frac{(d g-c h)^4 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 d^4 h}+\frac{(b g-a h)^3 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^3}+\frac{(d g-c h)^3 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^3}+\frac{(b g-a h)^2 p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 b^2 h}+\frac{(d g-c h)^2 q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 d^2 h}+\frac{(b g-a h) p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 b h}+\frac{(d g-c h) q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 d h}+\frac{p r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{q r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{(b g-a h)^4 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^4 h}+\frac{(d g-c h)^4 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^4 h}+\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}+\frac{\left ((b g-a h)^4 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 h}+\frac{\left (b p q r^2\right ) \int \left (\frac{h (b g-a h)^3}{b^4}+\frac{(b g-a h)^4}{b^4 (a+b x)}+\frac{h (b g-a h)^2 (g+h x)}{b^3}+\frac{h (b g-a h) (g+h x)^2}{b^2}+\frac{h (g+h x)^3}{b}\right ) \, dx}{8 h}+\frac{\left (d p q r^2\right ) \int \left (\frac{h (d g-c h)^3}{d^4}+\frac{(d g-c h)^4}{d^4 (c+d x)}+\frac{h (d g-c h)^2 (g+h x)}{d^3}+\frac{h (d g-c h) (g+h x)^2}{d^2}+\frac{h (g+h x)^3}{d}\right ) \, dx}{8 h}+\frac{\left (d (b g-a h) p q r^2\right ) \int \left (\frac{h (d g-c h)^2}{d^3}+\frac{(d g-c h)^3}{d^3 (c+d x)}+\frac{h (d g-c h) (g+h x)}{d^2}+\frac{h (g+h x)^2}{d}\right ) \, dx}{6 b h}+\frac{\left (d (b g-a h)^2 p q r^2\right ) \int \left (\frac{h (d g-c h)}{d^2}+\frac{(d g-c h)^2}{d^2 (c+d x)}+\frac{h (g+h x)}{d}\right ) \, dx}{4 b^2 h}+\frac{\left ((b g-a h)^4 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 b^4 h}+\frac{\left (b (d g-c h) p q r^2\right ) \int \left (\frac{h (b g-a h)^2}{b^3}+\frac{(b g-a h)^3}{b^3 (a+b x)}+\frac{h (b g-a h) (g+h x)}{b^2}+\frac{h (g+h x)^2}{b}\right ) \, dx}{6 d h}+\frac{\left (b (d g-c h)^2 p q r^2\right ) \int \left (\frac{h (b g-a h)}{b^2}+\frac{(b g-a h)^2}{b^2 (a+b x)}+\frac{h (g+h x)}{b}\right ) \, dx}{4 d^2 h}+\frac{\left ((d g-c h)^4 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2 d^4 h}+\frac{\left ((d g-c h)^4 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 d^4 h}\\ &=\frac{2 (b g-a h)^3 p^2 r^2 x}{b^3}+\frac{5 (b g-a h)^3 p q r^2 x}{8 b^3}+\frac{5 (b g-a h)^2 (d g-c h) p q r^2 x}{12 b^2 d}+\frac{5 (b g-a h) (d g-c h)^2 p q r^2 x}{12 b d^2}+\frac{5 (d g-c h)^3 p q r^2 x}{8 d^3}+\frac{2 (d g-c h)^3 q^2 r^2 x}{d^3}+\frac{3 h (b g-a h)^2 p^2 r^2 (a+b x)^2}{4 b^4}+\frac{2 h^2 (b g-a h) p^2 r^2 (a+b x)^3}{9 b^4}+\frac{h^3 p^2 r^2 (a+b x)^4}{32 b^4}+\frac{3 h (d g-c h)^2 q^2 r^2 (c+d x)^2}{4 d^4}+\frac{2 h^2 (d g-c h) q^2 r^2 (c+d x)^3}{9 d^4}+\frac{h^3 q^2 r^2 (c+d x)^4}{32 d^4}+\frac{3 (b g-a h)^2 p q r^2 (g+h x)^2}{16 b^2 h}+\frac{(b g-a h) (d g-c h) p q r^2 (g+h x)^2}{6 b d h}+\frac{3 (d g-c h)^2 p q r^2 (g+h x)^2}{16 d^2 h}+\frac{7 (b g-a h) p q r^2 (g+h x)^3}{72 b h}+\frac{7 (d g-c h) p q r^2 (g+h x)^3}{72 d h}+\frac{p q r^2 (g+h x)^4}{16 h}+\frac{(b g-a h)^4 p q r^2 \log (a+b x)}{8 b^4 h}+\frac{(b g-a h)^3 (d g-c h) p q r^2 \log (a+b x)}{6 b^3 d h}+\frac{(b g-a h)^2 (d g-c h)^2 p q r^2 \log (a+b x)}{4 b^2 d^2 h}-\frac{(d g-c h)^3 p q r^2 (a+b x) \log (a+b x)}{2 b d^3}-\frac{(d g-c h)^2 p q r^2 (g+h x)^2 \log (a+b x)}{4 d^2 h}-\frac{(d g-c h) p q r^2 (g+h x)^3 \log (a+b x)}{6 d h}-\frac{p q r^2 (g+h x)^4 \log (a+b x)}{8 h}+\frac{(b g-a h)^4 p^2 r^2 \log ^2(a+b x)}{4 b^4 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{48 h (b g-a h)^3 (a+b x)}{b^4}+\frac{36 h^2 (b g-a h)^2 (a+b x)^2}{b^4}+\frac{16 h^3 (b g-a h) (a+b x)^3}{b^4}+\frac{3 h^4 (a+b x)^4}{b^4}+\frac{12 (b g-a h)^4 \log (a+b x)}{b^4}\right )}{24 h}+\frac{(b g-a h)^2 (d g-c h)^2 p q r^2 \log (c+d x)}{4 b^2 d^2 h}+\frac{(b g-a h) (d g-c h)^3 p q r^2 \log (c+d x)}{6 b d^3 h}+\frac{(d g-c h)^4 p q r^2 \log (c+d x)}{8 d^4 h}-\frac{(b g-a h)^3 p q r^2 (c+d x) \log (c+d x)}{2 b^3 d}-\frac{(b g-a h)^2 p q r^2 (g+h x)^2 \log (c+d x)}{4 b^2 h}-\frac{(b g-a h) p q r^2 (g+h x)^3 \log (c+d x)}{6 b h}-\frac{p q r^2 (g+h x)^4 \log (c+d x)}{8 h}-\frac{(b g-a h)^4 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 h}+\frac{(d g-c h)^4 q^2 r^2 \log ^2(c+d x)}{4 d^4 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{48 h (d g-c h)^3 (c+d x)}{d^4}+\frac{36 h^2 (d g-c h)^2 (c+d x)^2}{d^4}+\frac{16 h^3 (d g-c h) (c+d x)^3}{d^4}+\frac{3 h^4 (c+d x)^4}{d^4}+\frac{12 (d g-c h)^4 \log (c+d x)}{d^4}\right )}{24 h}-\frac{(d g-c h)^4 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 d^4 h}+\frac{(b g-a h)^3 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^3}+\frac{(d g-c h)^3 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^3}+\frac{(b g-a h)^2 p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 b^2 h}+\frac{(d g-c h)^2 q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 d^2 h}+\frac{(b g-a h) p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 b h}+\frac{(d g-c h) q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{6 d h}+\frac{p r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{q r (g+h x)^4 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{8 h}+\frac{(b g-a h)^4 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 b^4 h}+\frac{(d g-c h)^4 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 d^4 h}+\frac{(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac{(d g-c h)^4 p q r^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 d^4 h}-\frac{(b g-a h)^4 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 h}\\ \end{align*}
Mathematica [A] time = 3.01291, size = 1386, normalized size = 0.62 \[ \frac{72 a \left (-4 b^3 g^3+6 a b^2 h g^2-4 a^2 b h^2 g+a^3 h^3\right ) p^2 r^2 \log ^2(a+b x) d^4+12 p r \log (a+b x) \left (12 c \left (-4 d^3 g^3+6 c d^2 h g^2-4 c^2 d h^2 g+c^3 h^3\right ) q r \log (c+d x) b^4-12 \left (c \left (-4 d^3 g^3+6 c d^2 h g^2-4 c^2 d h^2 g+c^3 h^3\right ) b^4+4 a d^4 g^3 b^3-6 a^2 d^4 g^2 h b^2+4 a^3 d^4 g h^2 b-a^4 d^4 h^3\right ) q r \log \left (\frac{b (c+d x)}{b c-a d}\right )+a d \left (12 \left (4 b^3 g^3-6 a b^2 h g^2+4 a^2 b h^2 g-a^3 h^3\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3+\left (12 \left (-4 d^3 g^3+6 c d^2 h g^2-4 c^2 d h^2 g+c^3 h^3\right ) q b^3+6 a d h \left (6 d^2 (3 p+q) g^2-4 c d h q g+c^2 h^2 q\right ) b^2-4 a^2 d^2 h^2 (22 d g p+4 d g q-c h q) b+a^3 d^3 h^3 (25 p+3 q)\right ) r\right )\right )+b \left (72 c \left (-4 d^3 g^3+6 c d^2 h g^2-4 c^2 d h^2 g+c^3 h^3\right ) q^2 r^2 \log ^2(c+d x) b^3+12 q r \log (c+d x) \left (\left (c \left (-48 d^3 (p+q) g^3+36 c d^2 h (p+3 q) g^2-8 c^2 d h^2 (2 p+11 q) g+c^3 h^3 (3 p+25 q)\right ) b^3+4 a d \left (12 d^3 g^3+18 c d^2 h g^2-6 c^2 d h^2 g+c^3 h^3\right ) p b^2+6 a^2 c d^2 h^2 (c h-8 d g) p b+12 a^3 c d^3 h^3 p\right ) r-12 b^3 c \left (-4 d^3 g^3+6 c d^2 h g^2-4 c^2 d h^2 g+c^3 h^3\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+d \left (72 b^3 x \left (4 g^3+6 h x g^2+4 h^2 x^2 g+h^3 x^3\right ) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3+r^2 \left (x \left ((p+q)^2 \left (576 g^3+216 h x g^2+64 h^2 x^2 g+9 h^3 x^3\right ) d^3-4 c h q (p+q) \left (324 g^2+60 h x g+7 h^2 x^2\right ) d^2+6 c^2 h^2 q (16 g (8 p+11 q)+h (9 p+13 q) x) d-60 c^3 h^3 q (3 p+5 q)\right ) b^3-4 a p \left (\left (-144 q g^3+324 h (p+q) x g^2+60 h^2 (p+q) x^2 g+7 h^3 (p+q) x^3\right ) d^3-12 c h q \left (-18 g^2+12 h x g+h^2 x^2\right ) d^2+6 c^2 h^2 q (5 h x-24 g) d+36 c^3 h^3 q\right ) b^2+6 a^2 d^2 h^2 p x (-20 c h q+16 d g (11 p+8 q)+d h (13 p+9 q) x) b-60 a^3 d^3 h^3 p (5 p+3 q) x\right )+12 r \left (-x \left ((p+q) \left (48 g^3+36 h x g^2+16 h^2 x^2 g+3 h^3 x^3\right ) d^3-4 c h q \left (18 g^2+6 h x g+h^2 x^2\right ) d^2+6 c^2 h^2 q (8 g+h x) d-12 c^3 h^3 q\right ) b^3+4 a d^3 p \left (-12 g^3+18 h x g^2+6 h^2 x^2 g+h^3 x^3\right ) b^2-6 a^2 d^3 h^2 p x (8 g+h x) b+12 a^3 d^3 h^3 p x\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-144 \left (c \left (-4 d^3 g^3+6 c d^2 h g^2-4 c^2 d h^2 g+c^3 h^3\right ) b^4+4 a d^4 g^3 b^3-6 a^2 d^4 g^2 h b^2+4 a^3 d^4 g h^2 b-a^4 d^4 h^3\right ) p q r^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{288 b^4 d^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.392, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{3} \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45213, size = 2429, normalized size = 1.08 \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 ({\left (h^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}, 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 (h x + g\right )}^{3} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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