Optimal. Leaf size=711 \[ -\frac{B^2 g^3 i^2 (b c-a d)^6 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4}-\frac{B g^3 i^2 (b c-a d)^6 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac{e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{180 b^3 d^4}-\frac{B g^3 i^2 (a+b x) (b c-a d)^5 \left (6 B \log \left (\frac{e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{180 b^3 d^3}+\frac{B g^3 i^2 (a+b x)^2 (b c-a d)^4 \left (3 B \log \left (\frac{e (a+b x)}{c+d x}\right )+3 A+B\right )}{180 b^3 d^2}-\frac{B g^3 i^2 (a+b x)^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{90 b^3 d}+\frac{g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{60 b^3}-\frac{B g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{20 b^3}+\frac{g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{15 b^2}-\frac{B g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{15 b^2}+\frac{g^3 i^2 (a+b x)^4 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{6 b}+\frac{3 B^2 g^3 i^2 x (b c-a d)^5}{20 b^2 d^3}-\frac{B^2 g^3 i^2 (b c-a d)^6 \log (c+d x)}{20 b^3 d^4}+\frac{B^2 g^3 i^2 (a+b x)^4 (b c-a d)^2}{60 b^3}-\frac{3 B^2 g^3 i^2 (c+d x)^2 (b c-a d)^4}{40 b d^4}+\frac{B^2 g^3 i^2 (c+d x)^3 (b c-a d)^3}{60 d^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.00598, antiderivative size = 790, normalized size of antiderivative = 1.11, number of steps used = 86, number of rules used = 13, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B^2 g^3 i^2 (b c-a d)^6 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{30 b^3 d^4}+\frac{B g^3 i^2 (b c-a d)^6 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{30 b^3 d^4}+\frac{B g^3 i^2 (a+b x)^2 (b c-a d)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{60 b^3 d^2}+\frac{d^2 g^3 i^2 (a+b x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{6 b^3}-\frac{A B g^3 i^2 x (b c-a d)^5}{30 b^2 d^3}-\frac{B g^3 i^2 (a+b x)^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{90 b^3 d}+\frac{g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 b^3}-\frac{7 B g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{60 b^3}+\frac{2 d g^3 i^2 (a+b x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{5 b^3}-\frac{B d g^3 i^2 (a+b x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{15 b^3}-\frac{B^2 g^3 i^2 (a+b x) (b c-a d)^5 \log \left (\frac{e (a+b x)}{c+d x}\right )}{30 b^3 d^3}+\frac{B^2 g^3 i^2 x (b c-a d)^5}{45 b^2 d^3}-\frac{7 B^2 g^3 i^2 (a+b x)^2 (b c-a d)^4}{360 b^3 d^2}+\frac{B^2 g^3 i^2 (b c-a d)^6 \log ^2(c+d x)}{60 b^3 d^4}+\frac{B^2 g^3 i^2 (b c-a d)^6 \log (c+d x)}{90 b^3 d^4}-\frac{B^2 g^3 i^2 (b c-a d)^6 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{30 b^3 d^4}+\frac{B^2 g^3 i^2 (a+b x)^3 (b c-a d)^3}{60 b^3 d}+\frac{B^2 g^3 i^2 (a+b x)^4 (b c-a d)^2}{60 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 2486
Rule 31
Rule 43
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int (64 c+64 d x)^2 (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac{4096 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2}+\frac{8192 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac{4096 d^2 (a g+b g x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}\right ) \, dx\\ &=\frac{\left (4096 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2}+\frac{\left (4096 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g^2}+\frac{(8192 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g}\\ &=\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac{\left (4096 B d^2\right ) \int \frac{(b c-a d) g^6 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b^3 g^3}-\frac{(16384 B d (b c-a d)) \int \frac{(b c-a d) g^5 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{5 b^3 g^2}-\frac{\left (2048 B (b c-a d)^2\right ) \int \frac{(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{b^3 g}\\ &=\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac{\left (4096 B d^2 (b c-a d) g^3\right ) \int \frac{(a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b^3}-\frac{\left (16384 B d (b c-a d)^2 g^3\right ) \int \frac{(a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{5 b^3}-\frac{\left (2048 B (b c-a d)^3 g^3\right ) \int \frac{(a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{b^3}\\ &=\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac{\left (4096 B d^2 (b c-a d) g^3\right ) \int \left (\frac{b (b c-a d)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^5}-\frac{b (b c-a d)^3 (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{b (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{(-b c+a d)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^5 (c+d x)}\right ) \, dx}{3 b^3}-\frac{\left (16384 B d (b c-a d)^2 g^3\right ) \int \left (-\frac{b (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{b (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{(-b c+a d)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac{\left (2048 B (b c-a d)^3 g^3\right ) \int \left (\frac{b (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{b (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{(-b c+a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^3}\\ &=\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac{\left (4096 B d (b c-a d) g^3\right ) \int (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2}+\frac{\left (4096 B (b c-a d)^2 g^3\right ) \int (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2}-\frac{\left (16384 B (b c-a d)^2 g^3\right ) \int (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2}-\frac{\left (4096 B (b c-a d)^3 g^3\right ) \int (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d}-\frac{\left (2048 B (b c-a d)^3 g^3\right ) \int (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d}+\frac{\left (16384 B (b c-a d)^3 g^3\right ) \int (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d}+\frac{\left (4096 B (b c-a d)^4 g^3\right ) \int (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d^2}+\frac{\left (2048 B (b c-a d)^4 g^3\right ) \int (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^2}-\frac{\left (16384 B (b c-a d)^4 g^3\right ) \int (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d^2}-\frac{\left (4096 B (b c-a d)^5 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d^3}-\frac{\left (2048 B (b c-a d)^5 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^3}+\frac{\left (16384 B (b c-a d)^5 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d^3}+\frac{\left (4096 B (b c-a d)^6 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^3 d^3}+\frac{\left (2048 B (b c-a d)^6 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 d^3}-\frac{\left (16384 B (b c-a d)^6 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac{\left (4096 B^2 d (b c-a d) g^3\right ) \int \frac{(b c-a d) (a+b x)^4}{c+d x} \, dx}{15 b^3}-\frac{\left (1024 B^2 (b c-a d)^2 g^3\right ) \int \frac{(b c-a d) (a+b x)^3}{c+d x} \, dx}{3 b^3}+\frac{\left (4096 B^2 (b c-a d)^2 g^3\right ) \int \frac{(b c-a d) (a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac{\left (4096 B^2 (b c-a d)^3 g^3\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{9 b^3 d}+\frac{\left (2048 B^2 (b c-a d)^3 g^3\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{3 b^3 d}-\frac{\left (16384 B^2 (b c-a d)^3 g^3\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{15 b^3 d}-\frac{\left (2048 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{3 b^3 d^2}-\frac{\left (1024 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{b^3 d^2}+\frac{\left (8192 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{5 b^3 d^2}-\frac{\left (4096 B^2 (b c-a d)^5 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{3 b^2 d^3}-\frac{\left (2048 B^2 (b c-a d)^5 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b^2 d^3}+\frac{\left (16384 B^2 (b c-a d)^5 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{5 b^2 d^3}-\frac{\left (4096 B^2 (b c-a d)^6 g^3\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}{3 b^3 d^4}-\frac{\left (2048 B^2 (b c-a d)^6 g^3\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 d^4}+\frac{\left (16384 B^2 (b c-a d)^6 g^3\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}{5 b^3 d^4}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}-\frac{2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac{\left (4096 B^2 d (b c-a d)^2 g^3\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{15 b^3}-\frac{\left (1024 B^2 (b c-a d)^3 g^3\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{3 b^3}+\frac{\left (4096 B^2 (b c-a d)^3 g^3\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac{\left (4096 B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{9 b^3 d}+\frac{\left (2048 B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b^3 d}-\frac{\left (16384 B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{15 b^3 d}-\frac{\left (2048 B^2 (b c-a d)^5 g^3\right ) \int \frac{a+b x}{c+d x} \, dx}{3 b^3 d^2}-\frac{\left (1024 B^2 (b c-a d)^5 g^3\right ) \int \frac{a+b x}{c+d x} \, dx}{b^3 d^2}+\frac{\left (8192 B^2 (b c-a d)^5 g^3\right ) \int \frac{a+b x}{c+d x} \, dx}{5 b^3 d^2}+\frac{\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac{1}{c+d x} \, dx}{3 b^3 d^3}+\frac{\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac{1}{c+d x} \, dx}{b^3 d^3}-\frac{\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac{1}{c+d x} \, dx}{5 b^3 d^3}-\frac{\left (4096 B^2 (b c-a d)^6 g^3\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}{3 b^3 d^4 e}-\frac{\left (2048 B^2 (b c-a d)^6 g^3\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 d^4 e}+\frac{\left (16384 B^2 (b c-a d)^6 g^3\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}{5 b^3 d^4 e}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}-\frac{2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{15 b^3 d^4}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac{\left (4096 B^2 d (b c-a d)^2 g^3\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{15 b^3}-\frac{\left (1024 B^2 (b c-a d)^3 g^3\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{3 b^3}+\frac{\left (4096 B^2 (b c-a d)^3 g^3\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{5 b^3}+\frac{\left (4096 B^2 (b c-a d)^4 g^3\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{9 b^3 d}+\frac{\left (2048 B^2 (b c-a d)^4 g^3\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^3 d}-\frac{\left (16384 B^2 (b c-a d)^4 g^3\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{15 b^3 d}-\frac{\left (2048 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{3 b^3 d^2}-\frac{\left (1024 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{b^3 d^2}+\frac{\left (8192 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{5 b^3 d^2}-\frac{\left (4096 B^2 (b c-a d)^6 g^3\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}{3 b^3 d^4 e}-\frac{\left (2048 B^2 (b c-a d)^6 g^3\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 d^4 e}+\frac{\left (16384 B^2 (b c-a d)^6 g^3\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}{5 b^3 d^4 e}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac{4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac{3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac{1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac{1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac{2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}-\frac{\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b^2 d^4}-\frac{\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac{\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{5 b^2 d^4}+\frac{\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b^3 d^3}+\frac{\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 d^3}-\frac{\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac{4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac{3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac{1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac{1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac{2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}-\frac{2048 B^2 (b c-a d)^6 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 d^4}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac{\left (4096 B^2 (b c-a d)^6 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 d^4}+\frac{\left (2048 B^2 (b c-a d)^6 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 d^4}-\frac{\left (16384 B^2 (b c-a d)^6 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 d^4}+\frac{\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 d^3}+\frac{\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 d^3}-\frac{\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac{4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac{3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac{1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac{1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac{2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}-\frac{2048 B^2 (b c-a d)^6 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 d^4}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac{1024 B^2 (b c-a d)^6 g^3 \log ^2(c+d x)}{15 b^3 d^4}+\frac{\left (4096 B^2 (b c-a d)^6 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 d^4}+\frac{\left (2048 B^2 (b c-a d)^6 g^3\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 d^4}-\frac{\left (16384 B^2 (b c-a d)^6 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b^3 d^4}\\ &=-\frac{2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac{4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac{3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac{1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac{1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac{2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac{1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac{2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac{7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac{4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac{1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac{8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac{2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac{2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}-\frac{2048 B^2 (b c-a d)^6 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 d^4}+\frac{2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac{1024 B^2 (b c-a d)^6 g^3 \log ^2(c+d x)}{15 b^3 d^4}-\frac{2048 B^2 (b c-a d)^6 g^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{15 b^3 d^4}\\ \end{align*}
Mathematica [B] time = 1.3484, size = 1559, normalized size = 2.19 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.027, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{2} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.13798, size = 6990, normalized size = 9.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{3} d^{2} g^{3} i^{2} x^{5} + A^{2} a^{3} c^{2} g^{3} i^{2} +{\left (2 \, A^{2} b^{3} c d + 3 \, A^{2} a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} +{\left (A^{2} b^{3} c^{2} + 6 \, A^{2} a b^{2} c d + 3 \, A^{2} a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} +{\left (3 \, A^{2} a b^{2} c^{2} + 6 \, A^{2} a^{2} b c d + A^{2} a^{3} d^{2}\right )} g^{3} i^{2} x^{2} +{\left (3 \, A^{2} a^{2} b c^{2} + 2 \, A^{2} a^{3} c d\right )} g^{3} i^{2} x +{\left (B^{2} b^{3} d^{2} g^{3} i^{2} x^{5} + B^{2} a^{3} c^{2} g^{3} i^{2} +{\left (2 \, B^{2} b^{3} c d + 3 \, B^{2} a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} +{\left (B^{2} b^{3} c^{2} + 6 \, B^{2} a b^{2} c d + 3 \, B^{2} a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} +{\left (3 \, B^{2} a b^{2} c^{2} + 6 \, B^{2} a^{2} b c d + B^{2} a^{3} d^{2}\right )} g^{3} i^{2} x^{2} +{\left (3 \, B^{2} a^{2} b c^{2} + 2 \, B^{2} a^{3} c d\right )} g^{3} i^{2} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B b^{3} d^{2} g^{3} i^{2} x^{5} + A B a^{3} c^{2} g^{3} i^{2} +{\left (2 \, A B b^{3} c d + 3 \, A B a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} +{\left (A B b^{3} c^{2} + 6 \, A B a b^{2} c d + 3 \, A B a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} +{\left (3 \, A B a b^{2} c^{2} + 6 \, A B a^{2} b c d + A B a^{3} d^{2}\right )} g^{3} i^{2} x^{2} +{\left (3 \, A B a^{2} b c^{2} + 2 \, A B a^{3} c d\right )} g^{3} i^{2} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}^{3}{\left (d i x + c i\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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