Optimal. Leaf size=44 \[ \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g i (b c-a d)} \]
[Out]
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Rubi [C] time = 5.53, antiderivative size = 1163, normalized size of antiderivative = 26.43, number of steps used = 61, number of rules used = 29, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ -\frac {B^2 \log ^3(c+d x)}{3 (b c-a d) g i}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{(b c-a d) g i}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{(b c-a d) g i}-\frac {A B \log ^2(c+d x)}{(b c-a d) g i}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{(b c-a d) g i}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{(b c-a d) g i}+\frac {2 A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{(b c-a d) g i}+\frac {2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{(b c-a d) g i}-\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 (c+d x)}{(b c-a d) g i}-\frac {A B \log ^2(a+b x)}{(b c-a d) g i}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{(b c-a d) g i}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{(b c-a d) g i}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) g i}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) g i}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) g i}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}-\frac {2 B^2 \log (a+b x) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 A B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) g 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 )}{(b c-a d) g i}-\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 {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 A B \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{(b c-a d) g i}+\frac {2 B^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 B^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 B^2 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{(b c-a d) g i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2301
Rule 2302
Rule 2317
Rule 2344
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2411
Rule 2418
Rule 2433
Rule 2434
Rule 2440
Rule 2488
Rule 2499
Rule 2500
Rule 2506
Rule 2507
Rule 2524
Rule 2528
Rule 6589
Rule 6610
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}{(88 c+88 d x) (a g+b g x)} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac {b \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{88 (b c-a d) g}-\frac {d \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{88 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{44 (b c-a d) g}+\frac {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 (c+d x)}{e (a+b x)} \, dx}{44 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{44 (b c-a d) e g}+\frac {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 (c+d x)}{a+b x} \, dx}{44 (b c-a d) e g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{44 (b c-a d) e g}+\frac {B \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{44 (b c-a d) e g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{44 g}+\frac {B \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{44 g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{44 g}+\frac {B \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{44 g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {(A B) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{44 g}-\frac {B^2 \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{44 g}+\frac {(b B) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}-\frac {(B d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}\\ &=-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{44 b g}+\frac {(b B) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{88 (b c-a d) g}-\frac {(B d) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{44 (b c-a d) g}\\ &=-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}+\frac {B^2 \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{44 g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}+\frac {(A b B) \int \frac {\log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}-\frac {(A B d) \int \frac {\log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}+\frac {(A B d) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{44 b (b c-a d) g}-\frac {\left (B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{44 g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{88 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}-\frac {(A B d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{44 (b c-a d) g}-\frac {\left (B^2 d\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{88 (b c-a d) g}+\frac {\left (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 (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{88 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}-\frac {\left (B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}-\frac {\left (B^2 d \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 {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{88 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}-\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{88 b (b c-a d) g}-\frac {\left (B^2 d\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 )}{44 b (b c-a d) g}+\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\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 )}{44 b (b c-a d) g}-\frac {\left (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}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 )}{44 (b c-a d) g}+\frac {B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \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 )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 )}{44 (b c-a d) g}+\frac {B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}-\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 )}{88 d (b c-a d) g}-\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{88 d (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 )}{44 (b c-a d) g}+\frac {B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \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 )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+2 \frac {B^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 79, normalized size = 1.80 \[ \frac {3 A^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A B \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{3 b c g i-3 a d g i} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 87, normalized size = 1.98 \[ \frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, A B \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, A^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left (b c - a d\right )} g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 145, normalized size = 3.30 \[ -\frac {{\left (B^{2} i e \log \left (\frac {b x e + a e}{d x + c}\right )^{3} + 3 \, A B i e \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 3 \, A^{2} i e \log \left (\frac {b x e + a e}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{3 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 312, normalized size = 7.09 \[ -\frac {B^{2} a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{3}}{3 \left (a d -b c \right )^{2} g i}+\frac {B^{2} b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{3}}{3 \left (a d -b c \right )^{2} g i}-\frac {A B a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right )^{2} g i}+\frac {A B b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right )^{2} g i}-\frac {A^{2} a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} g i}+\frac {A^{2} b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 397, normalized size = 9.02 \[ B^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} + 2 \, A B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b c g i - a d g i} - \frac {\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - \log \left (d x + c\right )^{3}}{b c g i - a d g i}\right )} + A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} A B}{b c g i - a d g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.76, size = 96, normalized size = 2.18 \[ -\frac {-6{}\mathrm {i}\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,A^2+3\,A\,B\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2+B^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.63, size = 206, normalized size = 4.68 \[ A^{2} \left (\frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )}\right ) - \frac {A B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a d g i - b c g i} - \frac {B^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a d g i - 3 b c g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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