Optimal. Leaf size=442 \[ -\frac {2 B^2 (b c-a d) i^2 (c+d x)}{b^2 g^2 (a+b x)}-\frac {2 B (b c-a d) i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {2 B d (b c-a d) i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {d^2 i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {(b c-a d) i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}-\frac {2 d (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {2 B^2 d (b c-a d) i^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g^2}+\frac {4 B d (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {4 B^2 d (b c-a d) i^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.37, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2562, 2395,
2342, 2341, 2355, 2354, 2438, 2379, 2421, 6724} \begin {gather*} \frac {4 B d i^2 (b c-a d) \text {PolyLog}\left (2,\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^2}+\frac {2 B^2 d i^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g^2}+\frac {4 B^2 d i^2 (b c-a d) \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {d^2 i^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^3 g^2}+\frac {2 B d i^2 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac {2 d i^2 (b c-a d) \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^2}-\frac {i^2 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2 (a+b x)}-\frac {2 B i^2 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2 (a+b x)}-\frac {2 B^2 i^2 (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2342
Rule 2354
Rule 2355
Rule 2379
Rule 2395
Rule 2421
Rule 2438
Rule 2562
Rule 6724
Rubi steps
\begin {align*} \int \frac {(69 c+69 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac {4761 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)^2}+\frac {9522 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}\right ) \, dx\\ &=\frac {\left (4761 d^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g^2}+\frac {(9522 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} \, dx}{b^2 g^2}+\frac {\left (4761 (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)^2} \, dx}{b^2 g^2}\\ &=\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {\left (9522 B d^2\right ) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {(19044 B d (b c-a d)) \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^2}+\frac {\left (9522 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)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {\left (9522 B d^2 (b c-a d)\right ) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac {\left (9522 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)^2 (c+d x)} \, dx}{b^3 g^2}-\frac {(19044 B d (b c-a d)) \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^2}\\ &=\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {\left (9522 B d^2 (b c-a d)\right ) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b^2 g^2}+\frac {\left (9522 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)^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^2}-\frac {(19044 B d (b c-a d)) \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^2}\\ &=\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {\left (9522 a 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^2}-\frac {\left (9522 B c d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 g^2}-\frac {(9522 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^2}+\frac {\left (9522 B d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 g^2}+\frac {\left (9522 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)^2} \, dx}{b^2 g^2}-\frac {\left (19044 B d (b c-a d)^2\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^2}\\ &=-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {\left (9522 B^2 c d\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^2 g^2}-\frac {\left (9522 a 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^2}+\frac {\left (9522 B^2 d (b c-a d)\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^2}-\frac {\left (9522 B^2 d (b c-a d)\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^2}+\frac {\left (9522 B^2 (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}-\frac {\left (19044 B d (b c-a d)^2\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^2}\\ &=-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {\left (19044 A B d (b c-a d)^2\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^3 g^2}-\frac {\left (19044 B^2 d (b c-a d)^2\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^2}+\frac {\left (9522 B^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}+\frac {\left (9522 B^2 c d\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^2 e g^2}-\frac {\left (9522 a 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^2}+\frac {\left (9522 B^2 d (b c-a d)\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^2}-\frac {\left (9522 B^2 d (b c-a d)\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^2}\\ &=-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {\left (9522 B^2 d (b c-a d)\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^2}-\frac {\left (19044 A B d (b c-a d)^2\right ) \text {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^2}+\frac {\left (9522 B^2 (b c-a d)^3\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^2}+\frac {\left (9522 B^2 c d\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^2 e g^2}-\frac {\left (9522 a 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^2}+\frac {\left (9522 B^2 d (b c-a d)\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^2}-\frac {\left (9522 B^2 d (b c-a d)\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^2}\\ &=-\frac {9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \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^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {\left (9522 B^2 c d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b g^2}-\frac {\left (9522 a B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (9522 B^2 c d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac {\left (9522 a B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^2}-\frac {(19044 A B d (b c-a d)) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (9522 B^2 d (b c-a d)\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (9522 B^2 d (b c-a d)\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 g^2}+\frac {\left (19044 A B d^2 (b c-a d)\right ) \text {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^2}-\frac {\left (9522 B^2 d^2 (b c-a d)\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (9522 B^2 d^2 (b c-a d)\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (19044 B^2 d (b c-a d)^2\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^2}\\ &=-\frac {9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac {9522 A B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \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^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac {9522 B^2 c d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac {9522 B^2 d (b c-a d) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {9522 a B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 A B d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 B^2 d (b c-a d) \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^2}-\frac {\left (9522 B^2 c d\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}-\frac {\left (9522 a B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac {\left (9522 a 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^2}-\frac {\left (9522 B^2 c d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}-\frac {(19044 A B d (b c-a d)) \text {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^2}+\frac {\left (9522 B^2 d (b c-a d)\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (9522 B^2 d (b c-a d)\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^2}+\frac {\left (9522 B^2 d (b c-a d)\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}+\frac {\left (9522 B^2 d^2 (b c-a d)\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^2}-\frac {\left (19044 B^2 d (b c-a d)^2\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^2}\\ &=-\frac {9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac {4761 a B^2 d^2 \log ^2(a+b x)}{b^3 g^2}-\frac {9522 A B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}+\frac {4761 B^2 d (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \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^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac {9522 B^2 c d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac {9522 B^2 d (b c-a d) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {4761 B^2 c d \log ^2(c+d x)}{b^2 g^2}+\frac {4761 B^2 d (b c-a d) \log ^2(c+d x)}{b^3 g^2}+\frac {9522 a B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 A B d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 A B d (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 B^2 d (b c-a d) \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^2}+\frac {19044 B^2 d (b c-a d) \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}-\frac {\left (9522 B^2 c d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 g^2}-\frac {\left (9522 a B^2 d^2\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^3 g^2}+\frac {\left (9522 B^2 d (b c-a d)\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^3 g^2}+\frac {\left (9522 B^2 d (b c-a d)\right ) \text {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^2}\\ &=-\frac {9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac {4761 a B^2 d^2 \log ^2(a+b x)}{b^3 g^2}-\frac {9522 A B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}+\frac {4761 B^2 d (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \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^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {9522 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {9522 a 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^2}-\frac {9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {4761 d^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac {4761 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac {9522 B^2 c d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac {9522 B^2 d (b c-a d) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {9522 B c d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {9522 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {4761 B^2 c d \log ^2(c+d x)}{b^2 g^2}+\frac {4761 B^2 d (b c-a d) \log ^2(c+d x)}{b^3 g^2}+\frac {9522 a B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 A B d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {9522 a B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 A B d (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}-\frac {9522 B^2 d (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {9522 B^2 c d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {9522 B^2 d (b c-a d) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {19044 B^2 d (b c-a d) \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^2}+\frac {19044 B^2 d (b c-a d) \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2652\) vs. \(2(442)=884\).
time = 4.10, size = 2652, normalized size = 6.00 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (b g x +a g \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \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 \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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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