Optimal. Leaf size=387 \[ \frac {2 B d^2 i^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac {d^2 i^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^3 g^3}-\frac {d i^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^3 (a+b x)}-\frac {2 B d i^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b g^3 (a+b x)^2}-\frac {B i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g^3 (a+b x)^2}+\frac {2 B^2 d^2 i^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac {2 B^2 d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac {B^2 i^2 (c+d x)^2}{4 b g^3 (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 4.06, antiderivative size = 932, normalized size of antiderivative = 2.41, number of steps used = 73, number of rules used = 20, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac {3 B^2 d^2 \log ^2(a+b x) i^2}{2 b^3 g^3}-\frac {A B d^2 \log ^2(a+b x) i^2}{b^3 g^3}-\frac {B^2 d^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 ) i^2}{b^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) i^2}{b^3 g^3}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 i^2}{b^3 g^3}-\frac {2 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 i^2}{b^3 g^3 (a+b x)}-\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 i^2}{2 b^3 g^3 (a+b x)^2}+\frac {3 B^2 d^2 \log ^2(c+d x) i^2}{2 b^3 g^3}-\frac {5 B^2 d^2 \log (a+b x) i^2}{2 b^3 g^3}-\frac {3 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) i^2}{b^3 g^3}-\frac {3 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) i^2}{b^3 g^3 (a+b x)}-\frac {B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) i^2}{2 b^3 g^3 (a+b x)^2}-\frac {3 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^3 g^3}+\frac {3 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x) i^2}{b^3 g^3}+\frac {5 B^2 d^2 \log (c+d x) i^2}{2 b^3 g^3}-\frac {3 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^3}+\frac {2 A B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^3}-\frac {3 B^2 d^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^3}+\frac {2 A B d^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^3}-\frac {3 B^2 d^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^3}+\frac {2 B^2 d^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 ) i^2}{b^3 g^3}+\frac {2 B^2 d^2 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right ) i^2}{b^3 g^3}-\frac {5 B^2 d (b c-a d) i^2}{2 b^3 g^3 (a+b x)}-\frac {B^2 (b c-a d)^2 i^2}{4 b^3 g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2301
Rule 2317
Rule 2344
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2418
Rule 2488
Rule 2506
Rule 2507
Rule 2524
Rule 2525
Rule 2528
Rule 6610
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {(70 c+70 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx &=\int \left (\frac {4900 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)^3}+\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)^2}+\frac {4900 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (4900 d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{b^2 g^3}+\frac {(9800 d (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^2 g^3}+\frac {\left (4900 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}-\frac {\left (9800 B d^2\right ) \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}{b^3 g^3}+\frac {(19600 B d (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (4900 B (b c-a d)^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}\\ &=-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {\left (19600 B d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (4900 B (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}-\frac {\left (9800 B d^2\right ) \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}{b^3 e g^3}\\ &=-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {\left (19600 B d (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^3}+\frac {\left (4900 B (b c-a d)^3\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^3}-\frac {\left (9800 B d^2\right ) \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}{b^3 e g^3}\\ &=-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {\left (4900 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^3}-\frac {\left (19600 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^3}-\frac {\left (4900 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 g^3}+\frac {\left (19600 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 g^3}-\frac {(4900 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^3}+\frac {(19600 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^3}-\frac {\left (9800 B d^2 (b c-a d)\right ) \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}{b^3 g^3}+\frac {\left (4900 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \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)}{e (a+b x)} \, dx}{b^3 g^3}+\frac {\left (4900 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 g^3}+\frac {\left (19600 B^2 d^2\right ) \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)}{e (a+b x)} \, dx}{b^3 g^3}-\frac {\left (19600 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 g^3}-\frac {\left (4900 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (19600 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}-\frac {\left (9800 B d^2 (b c-a d)\right ) \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}{b^3 g^3}+\frac {\left (2450 B^2 (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}\\ &=-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}-\frac {\left (9800 A B d^2 (b c-a d)\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^3 g^3}-\frac {\left (9800 B^2 d^2 (b c-a d)\right ) \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^3}-\frac {\left (4900 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (19600 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (2450 B^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \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)}{a+b x} \, dx}{b^3 e g^3}+\frac {\left (4900 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 e g^3}+\frac {\left (19600 B^2 d^2\right ) \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)}{a+b x} \, dx}{b^3 e g^3}-\frac {\left (19600 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 e g^3}\\ &=-\frac {4900 B^2 d^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^3}-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}+\frac {\left (4900 B^2 d^2\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^3}-\frac {\left (9800 A B d^2 (b c-a d)\right ) \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 )}{b^4 g^3}-\frac {\left (4900 B^2 d (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^3}+\frac {\left (19600 B^2 d (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^3}+\frac {\left (2450 B^2 (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 e g^3}+\frac {\left (4900 B^2 d^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 e g^3}+\frac {\left (19600 B^2 d^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 e g^3}-\frac {\left (19600 B^2 d^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 e g^3}\\ &=-\frac {1225 B^2 (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {12250 B^2 d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {12250 B^2 d^2 \log (a+b x)}{b^3 g^3}-\frac {4900 B^2 d^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^3 g^3}-\frac {4900 B^2 d^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^3}-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {12250 B^2 d^2 \log (c+d x)}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}-\frac {\left (9800 A B d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^3}+\frac {\left (4900 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 g^3}+\frac {\left (19600 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^3}-\frac {\left (19600 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 g^3}+\frac {\left (9800 A B d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^4 g^3}+\frac {\left (4900 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^3}-\frac {\left (4900 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 g^3}-\frac {\left (19600 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^3}+\frac {\left (19600 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 g^3}+\frac {\left (9800 B^2 d^2 (b c-a d)\right ) \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}{b^3 g^3}\\ &=-\frac {1225 B^2 (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {12250 B^2 d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {12250 B^2 d^2 \log (a+b x)}{b^3 g^3}-\frac {4900 A B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {4900 B^2 d^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^3 g^3}-\frac {4900 B^2 d^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^3}-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {12250 B^2 d^2 \log (c+d x)}{b^3 g^3}-\frac {14700 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}+\frac {9800 A B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {14700 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {9800 B^2 d^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 )}{b^3 g^3}-\frac {\left (9800 A B d^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 )}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^3}+\frac {\left (19600 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}+\frac {\left (19600 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^3}-\frac {\left (4900 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}+\frac {\left (19600 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}-\frac {\left (4900 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^3}+\frac {\left (19600 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^3}-\frac {\left (9800 B^2 d^2 (b c-a d)\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^3}\\ &=-\frac {1225 B^2 (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {12250 B^2 d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {12250 B^2 d^2 \log (a+b x)}{b^3 g^3}-\frac {4900 A B d^2 \log ^2(a+b x)}{b^3 g^3}+\frac {7350 B^2 d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {4900 B^2 d^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^3 g^3}-\frac {4900 B^2 d^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^3}-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {12250 B^2 d^2 \log (c+d x)}{b^3 g^3}-\frac {14700 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}+\frac {7350 B^2 d^2 \log ^2(c+d x)}{b^3 g^3}+\frac {9800 A B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {14700 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {9800 A B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}+\frac {9800 B^2 d^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 )}{b^3 g^3}+\frac {9800 B^2 d^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^3}-\frac {\left (4900 B^2 d^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 )}{b^3 g^3}-\frac {\left (4900 B^2 d^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 )}{b^3 g^3}+\frac {\left (19600 B^2 d^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 )}{b^3 g^3}+\frac {\left (19600 B^2 d^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 )}{b^3 g^3}\\ &=-\frac {1225 B^2 (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {12250 B^2 d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {12250 B^2 d^2 \log (a+b x)}{b^3 g^3}-\frac {4900 A B d^2 \log ^2(a+b x)}{b^3 g^3}+\frac {7350 B^2 d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {4900 B^2 d^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^3 g^3}-\frac {4900 B^2 d^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^3}-\frac {2450 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {14700 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {14700 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {2450 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)^2}-\frac {9800 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3 (a+b x)}+\frac {4900 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^3}+\frac {12250 B^2 d^2 \log (c+d x)}{b^3 g^3}-\frac {14700 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^3}+\frac {14700 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^3}+\frac {7350 B^2 d^2 \log ^2(c+d x)}{b^3 g^3}+\frac {9800 A B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {14700 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {9800 A B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}-\frac {14700 B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}-\frac {14700 B^2 d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {9800 B^2 d^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 )}{b^3 g^3}+\frac {9800 B^2 d^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^3}\\ \end {align*}
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Mathematica [B] time = 7.13, size = 3582, normalized size = 9.26 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} d^{2} i^{2} x^{2} + 2 \, A^{2} c d i^{2} x + A^{2} c^{2} i^{2} + {\left (B^{2} d^{2} i^{2} x^{2} + 2 \, B^{2} c d i^{2} x + B^{2} c^{2} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B d^{2} i^{2} x^{2} + 2 \, A B c d i^{2} x + A B c^{2} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{3} g^{3} x^{3} + 3 \, a b^{2} g^{3} x^{2} + 3 \, a^{2} b g^{3} x + a^{3} g^{3}}, x\right ) \]
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 [F] time = 2.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}{\left (b g x +a g \right )^{3}}\, dx \]
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 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^3} \,d x \]
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 \[ \frac {i^{2} \left (\int \frac {A^{2} c^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {A^{2} d^{2} x^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B^{2} c^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A B c^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A^{2} c d x}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B^{2} d^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A B d^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 B^{2} c d x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {4 A B c d x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx\right )}{g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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