Optimal. Leaf size=1159 \[ -\frac {B^2 (b c-a d)^2 g^3 (c+d x)^2}{12 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}-\frac {B^2 (b c-a d)^3 g^3 (c+d x)}{6 (b f-a g)^3 (d f-c g)^4 (f+g x)}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-3 a d g) (c+d x)}{4 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^4 g^3 \log \left (\frac {a+b x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) \log \left (\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B (b c-a d) g^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}+\frac {b^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}+\frac {B^2 (b c-a d)^4 g^3 \log \left (\frac {f+g x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) \log \left (\frac {f+g x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.67, antiderivative size = 1159, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2554, 2398,
2404, 2338, 2356, 46, 2351, 31, 2354, 2438} \begin {gather*} \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^4}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) \log \left (\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^4 g^3 \log \left (\frac {a+b x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}+\frac {B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}-\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) \log \left (\frac {f+g x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^4 g^3 \log \left (\frac {f+g x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \text {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-3 a d g) (c+d x)}{4 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^3 g^3 (c+d x)}{6 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^2 g^3 (c+d x)^2}{12 (b f-a g)^2 (d f-c g)^4 (f+g x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 46
Rule 2338
Rule 2351
Rule 2354
Rule 2356
Rule 2398
Rule 2404
Rule 2438
Rule 2554
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^5} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}+\frac {B \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x) (f+g x)^4} \, dx}{2 g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}+\frac {(B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x) (f+g x)^4} \, dx}{2 g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}+\frac {(B (b c-a d)) \int \left (\frac {b^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (b f-a g)^4 (a+b x)}-\frac {d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (-d f+c g)^4 (c+d x)}+\frac {g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g) (d f-c g) (f+g x)^4}-\frac {g^2 (-2 b d f+b c g+a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g)^2 (d f-c g)^2 (f+g x)^3}+\frac {g^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)^2}+\frac {g^2 (2 b d f-b c g-a d g) \left (2 b^2 d^2 f^2-2 b^2 c d f g-2 a b d^2 f g+b^2 c^2 g^2+a^2 d^2 g^2\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g)^4 (d f-c g)^4 (f+g x)}\right ) \, dx}{2 g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}+\frac {\left (b^5 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 g (b f-a g)^4}-\frac {\left (B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{2 g (d f-c g)^4}+\frac {(B (b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx}{2 (b f-a g) (d f-c g)}+\frac {(B (b c-a d) g (2 b d f-b c g-a d g)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx}{2 (b f-a g)^2 (d f-c g)^2}+\frac {\left (B (b c-a d) g \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^2} \, dx}{2 (b f-a g)^3 (d f-c g)^3}-\frac {\left (B (b c-a d) g (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^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}{2 g (b f-a g)^4}+\frac {\left (B^2 d^4\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}{2 g (d f-c g)^4}+\frac {\left (B^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x) (c+d x) (f+g x)^3} \, dx}{6 (b f-a g) (d f-c g)}+\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g)\right ) \int \frac {b c-a d}{(a+b x) (c+d x) (f+g x)^2} \, dx}{4 (b f-a g)^2 (d f-c g)^2}+\frac {\left (B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )\right ) \int \frac {b c-a d}{(a+b x) (c+d x) (f+g x)} \, dx}{2 (b f-a g)^3 (d f-c g)^3}+\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\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 (f+g x)}{e (a+b x)} \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^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}{2 e g (b f-a g)^4}+\frac {\left (B^2 d^4\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}{2 e g (d f-c g)^4}+\frac {\left (B^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x) (c+d x) (f+g x)^3} \, dx}{6 (b f-a g) (d f-c g)}+\frac {\left (B^2 (b c-a d)^2 (2 b d f-b c g-a d g)\right ) \int \frac {1}{(a+b x) (c+d x) (f+g x)^2} \, dx}{4 (b f-a g)^2 (d f-c g)^2}+\frac {\left (B^2 (b c-a d)^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )\right ) \int \frac {1}{(a+b x) (c+d x) (f+g x)} \, dx}{2 (b f-a g)^3 (d f-c g)^3}+\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\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 (f+g x)}{a+b x} \, dx}{2 e (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^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}{2 e g (b f-a g)^4}+\frac {\left (B^2 d^4\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}{2 e g (d f-c g)^4}+\frac {\left (B^2 (b c-a d)^2\right ) \int \left (\frac {b^4}{(b c-a d) (b f-a g)^3 (a+b x)}+\frac {d^4}{(b c-a d) (-d f+c g)^3 (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^3}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)^2}+\frac {g^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}\right ) \, dx}{6 (b f-a g) (d f-c g)}+\frac {\left (B^2 (b c-a d)^2 (2 b d f-b c g-a d g)\right ) \int \left (\frac {b^3}{(b c-a d) (b f-a g)^2 (a+b x)}-\frac {d^3}{(b c-a d) (-d f+c g)^2 (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^2}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}\right ) \, dx}{4 (b f-a g)^2 (d f-c g)^2}+\frac {\left (B^2 (b c-a d)^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )\right ) \int \left (\frac {b^2}{(b c-a d) (b f-a g) (a+b x)}+\frac {d^2}{(b c-a d) (-d f+c g) (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)}\right ) \, dx}{2 (b f-a g)^3 (d f-c g)^3}+\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \int \left (\frac {b e \log (f+g x)}{a+b x}-\frac {d e \log (f+g x)}{c+d x}\right ) \, dx}{2 e (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B^2 (b c-a d)^2 g}{12 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {5 B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) \log (a+b x)}{6 (b f-a g)^4 (d f-c g)}+\frac {b^2 B^2 (b c-a d) (2 b d f-b c g-a d g) \log (a+b x)}{4 (b f-a g)^4 (d f-c g)^2}+\frac {b B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) \log (c+d x)}{6 (b f-a g) (d f-c g)^4}-\frac {B^2 d^2 (b c-a d) (2 b d f-b c g-a d g) \log (c+d x)}{4 (b f-a g)^2 (d f-c g)^4}-\frac {B^2 d (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)^2 \log (f+g x)}{4 (b f-a g)^4 (d f-c g)^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^5 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 g (b f-a g)^4}+\frac {\left (b^4 B^2 d\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 g (b f-a g)^4}+\frac {\left (b B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 g (d f-c g)^4}-\frac {\left (B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 g (d f-c g)^4}+\frac {\left (b B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \int \frac {\log (f+g x)}{a+b x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (B^2 d (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \int \frac {\log (f+g x)}{c+d x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B^2 (b c-a d)^2 g}{12 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {5 B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) \log (a+b x)}{6 (b f-a g)^4 (d f-c g)}+\frac {b^2 B^2 (b c-a d) (2 b d f-b c g-a d g) \log (a+b x)}{4 (b f-a g)^4 (d f-c g)^2}+\frac {b B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) \log (c+d x)}{6 (b f-a g) (d f-c g)^4}-\frac {B^2 d^2 (b c-a d) (2 b d f-b c g-a d g) \log (c+d x)}{4 (b f-a g)^2 (d f-c g)^4}-\frac {B^2 d (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}+\frac {b^4 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 g (b f-a g)^4}+\frac {B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)^2 \log (f+g x)}{4 (b f-a g)^4 (d f-c g)^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 g (b f-a g)^4}-\frac {\left (b^5 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 g (b f-a g)^4}-\frac {\left (B^2 d^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 g (d f-c g)^4}-\frac {\left (B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 g (d f-c g)^4}-\frac {\left (B^2 (b c-a d) g (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}+\frac {\left (B^2 (b c-a d) g (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B^2 (b c-a d)^2 g}{12 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {5 B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) \log (a+b x)}{6 (b f-a g)^4 (d f-c g)}+\frac {b^2 B^2 (b c-a d) (2 b d f-b c g-a d g) \log (a+b x)}{4 (b f-a g)^4 (d f-c g)^2}+\frac {b B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {b^4 B^2 \log ^2(a+b x)}{4 g (b f-a g)^4}-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) \log (c+d x)}{6 (b f-a g) (d f-c g)^4}-\frac {B^2 d^2 (b c-a d) (2 b d f-b c g-a d g) \log (c+d x)}{4 (b f-a g)^2 (d f-c g)^4}-\frac {B^2 d (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B^2 d^4 \log ^2(c+d x)}{4 g (d f-c g)^4}+\frac {b^4 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 g (b f-a g)^4}+\frac {B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)^2 \log (f+g x)}{4 (b f-a g)^4 (d f-c g)^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^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 )}{2 g (b f-a g)^4}-\frac {\left (B^2 d^4\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 g (d f-c g)^4}-\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B^2 (b c-a d)^2 g}{12 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {5 B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) \log (a+b x)}{6 (b f-a g)^4 (d f-c g)}+\frac {b^2 B^2 (b c-a d) (2 b d f-b c g-a d g) \log (a+b x)}{4 (b f-a g)^4 (d f-c g)^2}+\frac {b B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {b^4 B^2 \log ^2(a+b x)}{4 g (b f-a g)^4}-\frac {B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) \log (c+d x)}{6 (b f-a g) (d f-c g)^4}-\frac {B^2 d^2 (b c-a d) (2 b d f-b c g-a d g) \log (c+d x)}{4 (b f-a g)^2 (d f-c g)^4}-\frac {B^2 d (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B^2 d^4 \log ^2(c+d x)}{4 g (d f-c g)^4}+\frac {b^4 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 g (b f-a g)^4}+\frac {B^2 (b c-a d)^2 g (2 b d f-b c g-a d g)^2 \log (f+g x)}{4 (b f-a g)^4 (d f-c g)^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}+\frac {b^4 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 g (b f-a g)^4}+\frac {B^2 d^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 g (d f-c g)^4}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{2 (b f-a g)^4 (d f-c g)^4}\\ \end {align*}
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Mathematica [A]
time = 4.82, size = 1301, normalized size = 1.12 \begin {gather*} -\frac {3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B (f+g x) \left (2 (b c-a d) g (b f-a g)^3 (d f-c g)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-3 (b c-a d) g (b f-a g)^2 (d f-c g)^2 (-2 b d f+b c g+a d g) (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d) g (b f-a g) (d f-c g) \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 b^4 (d f-c g)^4 (f+g x)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 d^4 (b f-a g)^4 (f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+6 (b c-a d) g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) (f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)-6 B (b c-a d) g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) (f+g x)^3 (b (d f-c g) \log (a+b x)+(-b d f+a d g) \log (c+d x)+(b c-a d) g \log (f+g x))+3 B (b c-a d) g (2 b d f-b c g-a d g) (f+g x)^2 \left ((b c-a d) g (b f-a g) (d f-c g)-b^2 (d f-c g)^2 (f+g x) \log (a+b x)+d^2 (b f-a g)^2 (f+g x) \log (c+d x)+(b c-a d) g (-2 b d f+b c g+a d g) (f+g x) \log (f+g x)\right )+B (b c-a d) g (f+g x) \left ((b c-a d) g (b f-a g)^2 (d f-c g)^2+2 (b c-a d) g (b f-a g) (-d f+c g) (-2 b d f+b c g+a d g) (f+g x)-2 b^3 (d f-c g)^3 (f+g x)^2 \log (a+b x)+2 d^3 (b f-a g)^3 (f+g x)^2 \log (c+d x)-2 (b c-a d) g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) (f+g x)^2 \log (f+g x)\right )+3 b^4 B (d f-c g)^4 (f+g x)^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 )-3 B d^4 (b f-a g)^4 (f+g x)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-6 B (b c-a d) g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) (f+g x)^3 \left (\left (\log \left (\frac {g (a+b x)}{-b f+a g}\right )-\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)+\text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )-\text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g)^4 (d f-c g)^4}}{12 g (f+g x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.45, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (g x +f \right )^{5}}\, dx\]
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 (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (f+g\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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