Optimal. Leaf size=127 \[ -\frac{2 B \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i}+\frac{2 B^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{d i}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d i} \]
[Out]
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Rubi [B] time = 3.26832, antiderivative size = 721, normalized size of antiderivative = 5.68, number of steps used = 46, number of rules used = 23, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.719, Rules used = {2524, 12, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2499, 2396, 2433, 2374, 6589, 2302, 30, 2500, 2375, 2317, 2440, 2434} \[ -\frac{2 A B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{2 B^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) \left (-\log \left (\frac{e (a+b x)}{c+d x}\right )+\log (a+b x)+\log \left (\frac{1}{c+d x}\right )\right )}{d i}-\frac{2 B^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{d i}-\frac{2 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{2 B^2 \log (a+b x) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d i}-\frac{2 B^2 \log \left (\frac{1}{c+d x}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{\log (c i+d i x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d i}-\frac{2 A B \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}+\frac{B^2 \log ^2(c i+d i x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d i}+\frac{2 B^2 \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (-\log \left (\frac{e (a+b x)}{c+d x}\right )+\log (a+b x)+\log \left (\frac{1}{c+d x}\right )\right )}{d i}-\frac{B^2 \log (a+b x) \log ^2(i (c+d x))}{d i}+\frac{B^2 \log ^2(i (c+d x)) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}-\frac{B^2 \log ^2(a+b x) \log (i (c+d x))}{d i}-\frac{B^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{d i}+\frac{B^2 \log ^2\left (\frac{1}{c+d x}\right ) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}-\frac{B^2 \log ^2(c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}+\frac{B^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d i}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (i (c+d x))}{d i}+\frac{A B \log ^2(i (c+d x))}{d i}+\frac{B^2 \log ^3(i (c+d x))}{3 d i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2524
Rule 12
Rule 2528
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 6742
Rule 2499
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2302
Rule 30
Rule 2500
Rule 2375
Rule 2317
Rule 2440
Rule 2434
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{87 c+87 d x} \, dx &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{(2 B) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{e (a+b x)} \, dx}{87 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{(2 B) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{a+b x} \, dx}{87 d e}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{(2 B) \int \frac{(b c-a d) e \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{(a+b x) (c+d x)} \, dx}{87 d e}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{(2 B (b c-a d)) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{(a+b x) (c+d x)} \, dx}{87 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{(2 B (b c-a d)) \int \left (\frac{d \left (-A-B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{(b c-a d) (c+d x)}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{87 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{1}{87} (2 B) \int \frac{\left (-A-B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{c+d x} \, dx-\frac{(2 b B) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{a+b x} \, dx}{87 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{1}{87} (2 B) \int \left (\frac{A \log (87 c+87 d x)}{-c-d x}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right ) \log (87 c+87 d x)}{-c-d x}\right ) \, dx-\frac{(2 b B) \int \left (\frac{A \log (87 c+87 d x)}{a+b x}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right ) \log (87 c+87 d x)}{a+b x}\right ) \, dx}{87 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{1}{87} (2 A B) \int \frac{\log (87 c+87 d x)}{-c-d x} \, dx-\frac{1}{87} \left (2 B^2\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (87 c+87 d x)}{-c-d x} \, dx-\frac{(2 A b B) \int \frac{\log (87 c+87 d x)}{a+b x} \, dx}{87 d}-\frac{\left (2 b B^2\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (87 c+87 d x)}{a+b x} \, dx}{87 d}\\ &=-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}+(2 A B) \int \frac{\log \left (\frac{87 d (a+b x)}{-87 b c+87 a d}\right )}{87 c+87 d x} \, dx+\frac{1}{87} B^2 \int \frac{\log ^2(87 c+87 d x)}{c+d x} \, dx-\frac{(2 A B) \operatorname{Subst}\left (\int -\frac{87 \log (x)}{x} \, dx,x,87 c+87 d x\right )}{7569 d}-\frac{\left (b B^2\right ) \int \frac{\log ^2(87 c+87 d x)}{a+b x} \, dx}{87 d}-\frac{\left (2 b B^2\right ) \int \frac{\log (a+b x) \log (87 c+87 d x)}{a+b x} \, dx}{87 d}-\frac{\left (2 b B^2\right ) \int \frac{\log \left (\frac{1}{c+d x}\right ) \log (87 c+87 d x)}{a+b x} \, dx}{87 d}-\frac{\left (2 b B^2 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log (87 c+87 d x)}{a+b x} \, dx}{87 d}\\ &=-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}+\left (2 B^2\right ) \int \frac{\log \left (\frac{87 d (a+b x)}{-87 b c+87 a d}\right ) \log (87 c+87 d x)}{87 c+87 d x} \, dx+\frac{(2 A B) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,87 c+87 d x\right )}{87 d}+\frac{(2 A B) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-87 b c+87 a d}\right )}{x} \, dx,x,87 c+87 d x\right )}{87 d}+\frac{B^2 \operatorname{Subst}\left (\int \frac{87 \log ^2(x)}{x} \, dx,x,87 c+87 d x\right )}{7569 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{87 b c-87 a d}{b}+\frac{87 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{87 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{-\frac{-b c+a d}{b}+\frac{d x}{b}}\right ) \log \left (-\frac{-87 b c+87 a d}{b}+\frac{87 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{87 d}+\left (2 B^2 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log \left (\frac{87 d (a+b x)}{-87 b c+87 a d}\right )}{87 c+87 d x} \, dx\\ &=-\frac{B^2 \log ^2(a+b x) \log (87 (c+d x))}{87 d}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (87 (c+d x))}{87 d}+\frac{A B \log ^2(87 (c+d x))}{87 d}-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}-\frac{2 A B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{1}{-\frac{-b c+a d}{b}+\frac{d x}{b}}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{87 b}+\frac{B^2 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{87 b c-87 a d}{b}+\frac{87 d x}{b}} \, dx,x,a+b x\right )}{b}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-87 b c+87 a d}{b}+\frac{87 d x}{b}\right )}{-\frac{-87 b c+87 a d}{b}+\frac{87 d x}{b}} \, dx,x,a+b x\right )}{b}+\frac{B^2 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,87 c+87 d x\right )}{87 d}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{87 d \left (\frac{-87 b c+87 a d}{87 d}+\frac{b x}{87 d}\right )}{-87 b c+87 a d}\right )}{x} \, dx,x,87 c+87 d x\right )}{87 d}+\frac{\left (2 B^2 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-87 b c+87 a d}\right )}{x} \, dx,x,87 c+87 d x\right )}{87 d}\\ &=-\frac{B^2 \log ^2(a+b x) \log (87 (c+d x))}{87 d}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (87 (c+d x))}{87 d}+\frac{A B \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}-\frac{2 A B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \log (87 (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{B^2 \operatorname{Subst}\left (\int x^2 \, dx,x,\log (87 (c+d x))\right )}{87 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-87 b c+87 a d}{87 d}+\frac{b x}{87 d}\right )}{x} \, dx,x,87 (c+d x)\right )}{87 d}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x}\right ) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{87 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{87 d x}{87 b c-87 a d}\right )}{x} \, dx,x,a+b x\right )}{87 d}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-87 b c+87 a d}\right )}{x} \, dx,x,87 c+87 d x\right )}{87 d}\\ &=-\frac{B^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}-\frac{B^2 \log ^2(a+b x) \log (87 (c+d x))}{87 d}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (87 (c+d x))}{87 d}+\frac{A B \log ^2(87 (c+d x))}{87 d}-\frac{B^2 \log (a+b x) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log ^3(87 (c+d x))}{261 d}+\frac{B^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{2 B^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}-\frac{2 A B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \log (87 (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{\left (b B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-87 b c+87 a d}{87 d}+\frac{b x}{87 d}} \, dx,x,87 (c+d x)\right )}{7569 d^2}+\frac{\left (b B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (\frac{1}{x}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{87 d^2}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{87 d x}{87 b c-87 a d}\right )}{x} \, dx,x,a+b x\right )}{87 d}\\ &=-\frac{B^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}-\frac{B^2 \log ^2(a+b x) \log (87 (c+d x))}{87 d}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (87 (c+d x))}{87 d}+\frac{A B \log ^2(87 (c+d x))}{87 d}-\frac{B^2 \log (a+b x) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log ^3(87 (c+d x))}{261 d}+\frac{B^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{2 B^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}-\frac{2 A B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \log (87 (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x}\right ) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{87 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{b x}{-87 b c+87 a d}\right )}{x} \, dx,x,87 (c+d x)\right )}{87 d}\\ &=-\frac{B^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}-\frac{B^2 \log ^2(a+b x) \log (87 (c+d x))}{87 d}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (87 (c+d x))}{87 d}+\frac{A B \log ^2(87 (c+d x))}{87 d}-\frac{B^2 \log (a+b x) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log ^3(87 (c+d x))}{261 d}+\frac{B^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{2 B^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}-\frac{2 A B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{87 d}-\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-87 b c+87 a d}\right )}{x} \, dx,x,87 (c+d x)\right )}{87 d}\\ &=-\frac{B^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{87 d}-\frac{B^2 \log ^2(a+b x) \log (87 (c+d x))}{87 d}-\frac{2 B^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (87 (c+d x))}{87 d}+\frac{A B \log ^2(87 (c+d x))}{87 d}-\frac{B^2 \log (a+b x) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 (c+d x))}{87 d}+\frac{B^2 \log ^3(87 (c+d x))}{261 d}+\frac{B^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 A B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (87 c+87 d x)}{87 d}+\frac{2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (87 c+87 d x)}{87 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (87 c+87 d x)}{87 d}-\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{B^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(87 c+87 d x)}{87 d}+\frac{2 B^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}-\frac{2 A B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}+\frac{2 B^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{87 d}-\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{87 d}\\ \end{align*}
Mathematica [A] time = 0.252336, size = 251, normalized size = 1.98 \[ \frac{2 A B \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-2 B^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )+2 B^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+2 A B \log (c+d x) \log \left (\frac{e (a+b x)}{c+d x}\right )-2 A B \log \left (\frac{a}{b}+x\right ) \log (c+d x)+2 A B \log \left (\frac{a}{b}+x\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )-B^2 \log \left (\frac{b c-a d}{b c+b d x}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )+A^2 \log (c+d x)-A B \log ^2\left (\frac{c}{d}+x\right )+2 A B \log \left (\frac{c}{d}+x\right ) \log (c+d x)}{d i} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 888, normalized size = 7. \begin{align*} \text{result too large to display} \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*} \frac{B^{2} \log \left (d x + c\right )^{3}}{3 \, d i} + \frac{A^{2} \log \left (d i x + c i\right )}{d i} - \int -\frac{B^{2} \log \left (b x + a\right )^{2} + B^{2} \log \left (e\right )^{2} + 2 \, A B \log \left (e\right ) + 2 \,{\left (B^{2} \log \left (e\right ) + A B\right )} \log \left (b x + a\right ) - 2 \,{\left (B^{2} \log \left (b x + a\right ) + B^{2} \log \left (e\right ) + A B\right )} \log \left (d x + c\right )}{d i x + c i}\,{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{B^{2} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac{b e x + a e}{d x + c}\right ) + A^{2}}{d i x + c i}, 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{{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{d i x + c i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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