Optimal. Leaf size=874 \[ \frac{B^2 g^3 \log \left (\frac{a+b x}{c+d x}\right ) (b c-a d)^4}{6 b^4 d^4}+\frac{B^2 g^3 \log (c+d x) (b c-a d)^4}{6 b^4 d^4}+\frac{B^2 g^3 x (b c-a d)^3}{6 b^3 d^3}+\frac{B^2 g^2 (4 b d f-3 b c g-a d g) \log \left (\frac{a+b x}{c+d x}\right ) (b c-a d)^3}{4 b^4 d^4}+\frac{B^2 g^2 (4 b d f-3 b c g-a d g) \log (c+d x) (b c-a d)^3}{4 b^4 d^4}+\frac{B^2 g^3 (c+d x)^2 (b c-a d)^2}{12 b^2 d^4}+\frac{B^2 g^2 (4 b d f-3 b c g-a d g) x (b c-a d)^2}{4 b^3 d^3}+\frac{B^2 g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x) (b c-a d)^2}{2 b^4 d^4}-\frac{B g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{6 b d^4}-\frac{B g^2 (4 b d f-3 b c g-a d g) (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{4 b^2 d^4}-\frac{B g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{2 b^4 d^3}-\frac{B (2 b d f-b c g-a d g) \left (-\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2+2 a d^2 f g b-a^2 d^2 g^2\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{2 b^4 d^4}-\frac{B^2 (2 b d f-b c g-a d g) \left (-\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2+2 a d^2 f g b-a^2 d^2 g^2\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{2 b^4 d^4}-\frac{(b f-a g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g} \]
[Out]
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Rubi [A] time = 1.74393, antiderivative size = 994, normalized size of antiderivative = 1.14, number of steps used = 33, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {2525, 12, 2528, 2486, 31, 72, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B^2 \log ^2(a+b x) (b f-a g)^4}{4 b^4 g}-\frac{B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b f-a g)^4}{2 b^4 g}-\frac{B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) (b f-a g)^4}{2 b^4 g}-\frac{B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) (b f-a g)^4}{2 b^4 g}+\frac{B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B^2 (d f-c g)^4 \log ^2(c+d x)}{4 d^4 g}-\frac{B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{4 b^3 d^3}-\frac{A B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) x}{2 b^3 d^3}-\frac{a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 b^4 d}+\frac{a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{4 b^4 d^2}-\frac{B^2 (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4 d^3}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^2}+\frac{B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 b d^4}-\frac{B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{4 b^2 d^4}+\frac{B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x)}{2 b^4 d^4}-\frac{B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}-\frac{B^2 (d f-c g)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 d^4 g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 72
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int (f+g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}-\frac{B \int \frac{(b c-a d) (f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\right )\right )}{b^2 d^2}+\frac{g^4 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b d}+\frac{(b f-a g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right )}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{2 g}\\ &=\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \, dx}{2 b d}-\frac{\left (B (b f-a g)^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 b^3 g}+\frac{\left (B (d f-c g)^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{2 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)}{c+d x}\right )\right ) \, dx}{2 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)}{c+d x}\right )\right ) \, dx}{2 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}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac{\left (B^2 (b c-a d) g^3\right ) \int \frac{(b c-a d) x^3}{(a+b x) (c+d x)} \, dx}{6 b d}+\frac{\left (B^2 (b f-a g)^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 (a+b x)}{e (a+b x)} \, dx}{2 b^4 g}-\frac{\left (B^2 (d f-c g)^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 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{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{4 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)}{c+d x}\right ) \, dx}{2 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}{2 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac{\left (B^2 (b c-a d)^2 g^3\right ) \int \frac{x^3}{(a+b x) (c+d x)} \, dx}{6 b d}+\frac{\left (B^2 (b f-a g)^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 (a+b x)}{a+b x} \, dx}{2 b^4 e g}-\frac{\left (B^2 (d f-c g)^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 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}{4 b^2 d^2}+\frac{\left (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}{2 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}{2 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{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)}{2 b^4 d^4}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac{\left (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}{6 b d}+\frac{\left (B^2 (b f-a g)^4\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 b^4 e g}-\frac{\left (B^2 (d f-c g)^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 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}{4 b^2 d^2}\\ &=-\frac{B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac{a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac{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)}{2 b^4 d^4}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac{\left (B^2 (b f-a g)^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 b^3 g}-\frac{\left (B^2 d (b f-a g)^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b^4 g}-\frac{\left (b B^2 (d f-c g)^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 d^4 g}+\frac{\left (B^2 (d f-c g)^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 d^3 g}\\ &=-\frac{B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac{a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac{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)}{2 b^4 d^4}-\frac{B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}-\frac{B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 g}+\frac{\left (B^2 (b f-a g)^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 g}+\frac{\left (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}{2 b^3 g}+\frac{\left (B^2 (d f-c g)^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 d^4 g}+\frac{\left (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}{2 d^3 g}\\ &=-\frac{B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac{a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 b^4 d^2}+\frac{B^2 (b f-a g)^4 \log ^2(a+b x)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac{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)}{2 b^4 d^4}-\frac{B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac{B^2 (d f-c g)^4 \log ^2(c+d x)}{4 d^4 g}-\frac{B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 g}+\frac{\left (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 )}{2 b^4 g}+\frac{\left (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 )}{2 d^4 g}\\ &=-\frac{B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac{a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 b^4 d^2}+\frac{B^2 (b f-a g)^4 \log ^2(a+b x)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac{B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac{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)}{2 b^4 d^4}-\frac{B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac{B^2 (d f-c g)^4 \log ^2(c+d x)}{4 d^4 g}-\frac{B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 g}-\frac{B^2 (b f-a g)^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b^4 g}-\frac{B^2 (d f-c g)^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 d^4 g}\\ \end{align*}
Mathematica [A] time = 0.972346, size = 733, normalized size = 0.84 \[ \frac{(f+g x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2-\frac{B \left (3 b^4 B (d f-c g)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-3 B d^4 (b f-a g)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+6 A b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+6 B d g^2 (a+b x) (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )-6 B g^2 (b c-a d)^2 \log (c+d x) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+B g^4 (b c-a d) \left (2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (2 a d+2 b c-b d x)-2 b^3 c^3 \log (c+d x)\right )-3 B g^3 (b c-a d) (a d g+b c g-4 b d f) \left (b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )-a^2 d^2 \log (a+b x)\right )+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 b^3 d^3 g^4 x^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-6 b^4 (d f-c g)^4 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+6 d^4 (b f-a g)^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )\right )}{3 b^4 d^4}}{4 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.677, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{3} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.80513, size = 2889, normalized size = 3.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} 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 e x + a e}{d x + c}\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 e x + a e}{d x + c}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x + f\right )}^{3}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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