Optimal. Leaf size=869 \[ \frac {2 B^2 g^3 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^4}{3 b^4 d^4}+\frac {2 B^2 g^3 \log (c+d x) (b c-a d)^4}{3 b^4 d^4}+\frac {2 B^2 g^3 x (b c-a d)^3}{3 b^3 d^3}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^3}{b^4 d^4}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) \log (c+d x) (b c-a d)^3}{b^4 d^4}+\frac {B^2 g^3 (c+d x)^2 (b c-a d)^2}{3 b^2 d^4}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) x (b c-a d)^2}{b^3 d^3}+\frac {2 B^2 g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x) (b c-a d)^2}{b^4 d^4}-\frac {B g^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) (b c-a d)}{3 b d^4}-\frac {B g^2 (4 b d f-3 b c g-a d g) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) (b c-a d)}{2 b^2 d^4}-\frac {B g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) (b c-a d)}{b^4 d^3}-\frac {B (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) (b c-a d)}{b^4 d^4}-\frac {2 B^2 (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{b^4 d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.81, antiderivative size = 973, normalized size of antiderivative = 1.12, number of steps used = 33, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2525, 12, 2528, 2486, 31, 72, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 \log ^2(a+b x) (b f-a g)^4}{b^4 g}-\frac {B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) (b f-a g)^4}{b^4 g}-\frac {2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) (b f-a g)^4}{b^4 g}-\frac {2 B^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) (b f-a g)^4}{b^4 g}+\frac {B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {B^2 (d f-c g)^4 \log ^2(c+d x)}{d^4 g}-\frac {2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac {A B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) x}{b^3 d^3}-\frac {2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac {a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}-\frac {B^2 (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}+\frac {2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac {B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac {2 B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x)}{b^4 d^4}-\frac {2 B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}-\frac {2 B^2 (d f-c g)^4 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^4 g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 72
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int (f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac {B \int \frac {2 (b c-a d) (f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx}{2 g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac {(B (b c-a d)) \int \frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac {(B (b c-a d)) \int \left (\frac {g^2 \left (a^2 d^2 g^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 ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^3 d^3}+\frac {g^3 (4 b d f-b c g-a d g) x \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 d^2}+\frac {g^4 x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b d}+\frac {(b f-a g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^3 (b c-a d) (a+b x)}+\frac {(d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac {\left (B (b c-a d) g^3\right ) \int x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b d}-\frac {\left (B (b f-a g)^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{b^3 g}+\frac {\left (B (d f-c g)^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{d^3 g}-\frac {\left (B (b c-a d) g^2 (4 b d f-b c g-a d g)\right ) \int x \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b^2 d^2}-\frac {\left (B (b c-a d) g \left (a^2 d^2 g^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 )\right ) \int \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b^3 d^3}\\ &=-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac {\left (B^2 (b c-a d) g^3\right ) \int \frac {2 (b c-a d) x^3}{(a+b x) (c+d x)} \, dx}{3 b d}+\frac {\left (B^2 (b f-a g)^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{b^4 g}-\frac {\left (B^2 (d f-c g)^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{d^4 g}+\frac {\left (B^2 (b c-a d) g^2 (4 b d f-b c g-a d g)\right ) \int \frac {2 (b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{2 b^2 d^2}-\frac {\left (B^2 (b c-a d) g \left (a^2 d^2 g^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 )\right ) \int \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \, dx}{b^3 d^3}\\ &=-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}-\frac {B^2 (b c-a d) g \left (a^2 d^2 g^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) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac {\left (2 B^2 (b c-a d)^2 g^3\right ) \int \frac {x^3}{(a+b x) (c+d x)} \, dx}{3 b d}+\frac {\left (B^2 (b f-a g)^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{b^4 e g}-\frac {\left (B^2 (d f-c g)^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{d^4 e g}+\frac {\left (B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g)\right ) \int \frac {x^2}{(a+b x) (c+d x)} \, dx}{b^2 d^2}+\frac {\left (2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^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 )\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^3}\\ &=-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}-\frac {B^2 (b c-a d) g \left (a^2 d^2 g^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) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^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 (c+d x)}{b^4 d^4}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac {\left (2 B^2 (b c-a d)^2 g^3\right ) \int \left (\frac {-b c-a d}{b^2 d^2}+\frac {x}{b d}-\frac {a^3}{b^2 (b c-a d) (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{3 b d}+\frac {\left (B^2 (b f-a g)^4\right ) \int \left (\frac {2 b e \log (a+b x)}{a+b x}-\frac {2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 e g}-\frac {\left (B^2 (d f-c g)^4\right ) \int \left (\frac {2 b e \log (c+d x)}{a+b x}-\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{d^4 e g}+\frac {\left (B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g)\right ) \int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{b^2 d^2}\\ &=-\frac {2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac {2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac {a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}-\frac {B^2 (b c-a d) g \left (a^2 d^2 g^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) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac {B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^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 (c+d x)}{b^4 d^4}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac {\left (2 B^2 (b f-a g)^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g}-\frac {\left (2 B^2 d (b f-a g)^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g}-\frac {\left (2 b B^2 (d f-c g)^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d^4 g}+\frac {\left (2 B^2 (d f-c g)^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{d^3 g}\\ &=-\frac {2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac {2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac {a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}-\frac {B^2 (b c-a d) g \left (a^2 d^2 g^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) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac {B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^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 (c+d x)}{b^4 d^4}-\frac {2 B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}-\frac {2 B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g}+\frac {\left (2 B^2 (b f-a g)^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g}+\frac {\left (2 B^2 (b f-a g)^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g}+\frac {\left (2 B^2 (d f-c g)^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{d^4 g}+\frac {\left (2 B^2 (d f-c g)^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{d^3 g}\\ &=-\frac {2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac {2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac {a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}+\frac {B^2 (b f-a g)^4 \log ^2(a+b x)}{b^4 g}-\frac {B^2 (b c-a d) g \left (a^2 d^2 g^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) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac {B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^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 (c+d x)}{b^4 d^4}-\frac {2 B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac {B^2 (d f-c g)^4 \log ^2(c+d x)}{d^4 g}-\frac {2 B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g}+\frac {\left (2 B^2 (b f-a g)^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g}+\frac {\left (2 B^2 (d f-c g)^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d^4 g}\\ &=-\frac {2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac {A B (b c-a d) g \left (a^2 d^2 g^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 ) x}{b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac {2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac {a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}+\frac {B^2 (b f-a g)^4 \log ^2(a+b x)}{b^4 g}-\frac {B^2 (b c-a d) g \left (a^2 d^2 g^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) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac {2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac {B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac {2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^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 (c+d x)}{b^4 d^4}-\frac {2 B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac {B^2 (d f-c g)^4 \log ^2(c+d x)}{d^4 g}-\frac {2 B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g}-\frac {2 B^2 (b f-a g)^4 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g}-\frac {2 B^2 (d f-c g)^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d^4 g}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 746, normalized size = 0.86 \[ \frac {(f+g x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2-\frac {2 B \left (2 B g^4 (b c-a d) \left (2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (2 a d+2 b c-b d x)-2 b^3 c^3 \log (c+d x)\right )+6 A b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+6 B d g^2 (a+b x) (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right ) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-12 B g^2 (b c-a d)^2 \log (c+d x) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )-6 B g^3 (b c-a d) (a d g+b c g-4 b d f) \left (b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )-a^2 d^2 \log (a+b x)\right )-6 b^4 (d f-c g)^4 \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+2 b^3 d^3 g^4 x^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+6 d^4 (b f-a g)^4 \log (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+6 b^4 B (d f-c g)^4 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-6 B d^4 (b f-a g)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )\right )}{3 b^4 d^4}}{4 g} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} g^{3} x^{3} + 3 \, A^{2} f g^{2} x^{2} + 3 \, A^{2} f^{2} g x + A^{2} f^{3} + {\left (B^{2} g^{3} x^{3} + 3 \, B^{2} f g^{2} x^{2} + 3 \, B^{2} f^{2} g x + B^{2} f^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, {\left (A B g^{3} x^{3} + 3 \, A B f g^{2} x^{2} + 3 \, A B f^{2} g x + A B f^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ), x\right ) \]
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 \[ \int {\left (g x + f\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.35, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right )^{2} e}{\left (d x +c \right )^{2}}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.96, size = 2351, normalized size = 2.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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