Optimal. Leaf size=730 \[ \frac {B^2 (b c-a d)^4 g i^3 x}{60 b^3 d}+\frac {B^2 (b c-a d)^3 g i^3 (c+d x)^2}{30 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 (c+d x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {B (b c-a d)^4 g i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {B (b c-a d)^5 g i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d^2}-\frac {11 B^2 (b c-a d)^5 g i^3 \log (c+d x)}{60 b^4 d^2}-\frac {B^2 (b c-a d)^5 g i^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.56, antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2562, 2383,
2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 78} \begin {gather*} -\frac {B^2 g i^3 (b c-a d)^5 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2}-\frac {B g i^3 (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{10 b^4 d^2}-\frac {B g i^3 (a+b x) (b c-a d)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^4 d}+\frac {g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{20 b^4}-\frac {B g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^4}+\frac {g i^3 (a+b x)^2 (c+d x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{10 b^3}+\frac {3 B g i^3 (c+d x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{20 b^2 d^2}+\frac {3 g i^3 (a+b x)^2 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{20 b^2}+\frac {B g i^3 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{30 b d^2}-\frac {B g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 d^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b}-\frac {B^2 g i^3 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {11 B^2 g i^3 (b c-a d)^5 \log (c+d x)}{60 b^4 d^2}+\frac {B^2 g i^3 x (b c-a d)^4}{60 b^3 d}+\frac {B^2 g i^3 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}+\frac {B^2 g i^3 (c+d x)^3 (b c-a d)^2}{30 b d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 78
Rule 2354
Rule 2373
Rule 2381
Rule 2382
Rule 2383
Rule 2384
Rule 2438
Rule 2562
Rubi steps
\begin {align*} \int (76 c+76 d x)^3 (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac {(-b c+a d) g (76 c+76 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac {b g (76 c+76 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{76 d}\right ) \, dx\\ &=\frac {(b g) \int (76 c+76 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{76 d}+\frac {((-b c+a d) g) \int (76 c+76 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{d}\\ &=-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(b B g) \int \frac {2535525376 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{14440 d^2}+\frac {(B (b c-a d) g) \int \frac {33362176 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{152 d^2}\\ &=-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(877952 b B (b c-a d) g) \int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{5 d^2}+\frac {\left (219488 B (b c-a d)^2 g\right ) \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^2}\\ &=-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(877952 b B (b c-a d) g) \int \left (\frac {d (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac {(b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{5 d^2}+\frac {\left (219488 B (b c-a d)^2 g\right ) \int \left (\frac {d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {(b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^2}\\ &=-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(877952 B (b c-a d) g) \int (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 d}-\frac {\left (877952 B (b c-a d)^2 g\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b d}+\frac {\left (219488 B (b c-a d)^2 g\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d}-\frac {\left (877952 B (b c-a d)^3 g\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d}+\frac {\left (219488 B (b c-a d)^3 g\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d}-\frac {\left (877952 B (b c-a d)^4 g\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^3 d}+\frac {\left (219488 B (b c-a d)^4 g\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 d}-\frac {\left (877952 B (b c-a d)^5 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{5 b^3 d^2}+\frac {\left (219488 B (b c-a d)^5 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}+\frac {\left (219488 B^2 (b c-a d) g\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{5 d^2}+\frac {\left (877952 B^2 (b c-a d)^2 g\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{15 b d^2}-\frac {\left (219488 B^2 (b c-a d)^2 g\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^2}+\frac {\left (438976 B^2 (b c-a d)^3 g\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^2 d^2}-\frac {\left (109744 B^2 (b c-a d)^3 g\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^2}-\frac {\left (877952 B^2 (b c-a d)^4 g\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{5 b^3 d}+\frac {\left (219488 B^2 (b c-a d)^4 g\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 d}+\frac {\left (877952 B^2 (b c-a d)^5 g\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}{5 b^4 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\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^4 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}+\frac {\left (219488 B^2 (b c-a d)^2 g\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{5 d^2}+\frac {\left (877952 B^2 (b c-a d)^3 g\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{15 b d^2}-\frac {\left (219488 B^2 (b c-a d)^3 g\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b d^2}+\frac {\left (438976 B^2 (b c-a d)^4 g\right ) \int \frac {c+d x}{a+b x} \, dx}{5 b^2 d^2}-\frac {\left (109744 B^2 (b c-a d)^4 g\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {1}{c+d x} \, dx}{5 b^4 d}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {1}{c+d x} \, dx}{b^4 d}+\frac {\left (877952 B^2 (b c-a d)^5 g\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}{5 b^4 d^2 e}-\frac {\left (219488 B^2 (b c-a d)^5 g\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^4 d^2 e}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {\left (219488 B^2 (b c-a d)^2 g\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{5 d^2}+\frac {\left (877952 B^2 (b c-a d)^3 g\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{15 b d^2}-\frac {\left (219488 B^2 (b c-a d)^3 g\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{3 b d^2}+\frac {\left (438976 B^2 (b c-a d)^4 g\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{5 b^2 d^2}-\frac {\left (109744 B^2 (b c-a d)^4 g\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\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}{5 b^4 d^2 e}-\frac {\left (219488 B^2 (b c-a d)^5 g\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^4 d^2 e}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^3 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^2}-\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^4 d}+\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^4 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^3 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}-\frac {109744 B^2 (b c-a d)^5 g \log ^2(a+b x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{5 b^4 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}-\frac {109744 B^2 (b c-a d)^5 g \log ^2(a+b x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{5 b^4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 901, normalized size = 1.23 \begin {gather*} \frac {g i^3 \left (-5 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 b (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {5 B (b c-a d)^2 \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 (b d x+(b c-a d) \log (a+b x))-B (b c-a d) \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-3 B (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{3 b^4}-\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x-12 B (b c-a d)^3 (b d x+(b c-a d) \log (a+b x))-4 B (b c-a d)^2 \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+12 b^2 (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 b^3 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 b^4 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-12 B (b c-a d)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{3 b^4}\right )}{20 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2508 vs. \(2 (659) = 1318\).
time = 0.46, size = 2508, normalized size = 3.44 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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