Optimal. Leaf size=127 \[ -\frac {2 B \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d i}+\frac {2 B^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d i} \]
[Out]
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Rubi [B] time = 3.27, 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 12
Rule 30
Rule 2301
Rule 2302
Rule 2317
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2418
Rule 2433
Rule 2434
Rule 2440
Rule 2499
Rule 2500
Rule 2524
Rule 2528
Rule 6589
Rule 6688
Rule 6742
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*}
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Mathematica [A] time = 0.26, size = 251, normalized size = 1.98 \[ \frac {2 A B \log (c+d x) \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A B \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )-2 A B \log \left (\frac {a}{b}+x\right ) \log (c+d x)+2 A B \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-2 B^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-B^2 \log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+2 B^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+A^2 \log (c+d x)-A B \log ^2\left (\frac {c}{d}+x\right )+2 A B \log \left (\frac {c}{d}+x\right ) \log (c+d x)}{d i} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac {b e x + a e}{d x + c}\right ) + A^{2}}{d i x + c i}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 888, normalized size = 6.99 \[ -\frac {B^{2} a \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}+1\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right ) i}+\frac {B^{2} b c \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}+1\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right ) d i}-\frac {2 A B a \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) i}+\frac {2 A B b c \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) d i}-\frac {2 B^{2} a \polylog \left (2, \frac {\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) i}+\frac {2 B^{2} b c \polylog \left (2, \frac {\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) d i}-\frac {A^{2} a \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{\left (a d -b c \right ) i}+\frac {A^{2} b c \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{\left (a d -b c \right ) d i}-\frac {2 A B a \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) i}+\frac {2 A B b c \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) d i}+\frac {2 B^{2} a \polylog \left (3, \frac {\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) i}-\frac {2 B^{2} b c \polylog \left (3, \frac {\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) d i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B^{2} \log \left (d x + c\right )^{3}}{3 \, d i} + \frac {A^{2} \log \left (d i x + c i\right )}{d i} - \int -\frac {B^{2} \log \left (b x + a\right )^{2} + B^{2} \log \relax (e)^{2} + 2 \, A B \log \relax (e) + 2 \, {\left (B^{2} \log \relax (e) + A B\right )} \log \left (b x + a\right ) - 2 \, {\left (B^{2} \log \left (b x + a\right ) + B^{2} \log \relax (e) + A B\right )} \log \left (d x + c\right )}{d i x + c i}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{c\,i+d\,i\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A^{2}}{c + d x}\, dx + \int \frac {B^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
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