Optimal. Leaf size=381 \[ \frac{4 B^2 (b c-a d) (-a d g-b c g+2 b d f) \text{PolyLog}\left (2,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac{b^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g (b f-a g)^2}+\frac{2 B g (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(f+g x) (b f-a g)^2 (d f-c g)}+\frac{2 B (b c-a d) (-a d g-b c g+2 b d f) \log \left (1-\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g (f+g x)^2}+\frac{4 B^2 g (b c-a d)^2 \log \left (\frac{f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 1.64416, antiderivative size = 899, normalized size of antiderivative = 2.36, number of steps used = 36, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 72} \[ -\frac{2 B^2 \log ^2(a+b x) b^2}{g (b f-a g)^2}+\frac{2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) b^2}{g (b f-a g)^2}+\frac{4 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) b^2}{g (b f-a g)^2}+\frac{4 B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) b^2}{g (b f-a g)^2}+\frac{4 B^2 (b c-a d) \log (a+b x) b}{(b f-a g)^2 (d f-c g)}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{2 B^2 d^2 \log ^2(c+d x)}{g (d f-c g)^2}-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{4 B^2 d (b c-a d) \log (c+d x)}{(b f-a g) (d f-c g)^2}-\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{2 B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d)^2 g \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 d^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g (d f-c g)^2}-\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )}{(b f-a g)^2 (d f-c g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 12
Rule 2528
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 72
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^3} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}+\frac{B \int \frac{2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x) (f+g x)^2} \, dx}{g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}+\frac{(2 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) (c+d x) (f+g x)^2} \, dx}{g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}+\frac{(2 B (b c-a d)) \int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (b f-a g)^2 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (-d f+c g)^2 (c+d x)}+\frac{g^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(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) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g)^2 (d f-c g)^2 (f+g x)}\right ) \, dx}{g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}+\frac{\left (2 b^3 B\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{g (b f-a g)^2}-\frac{\left (2 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{g (d f-c g)^2}+\frac{(2 B (b c-a d) g) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^2} \, dx}{(b f-a g) (d f-c g)}+\frac{(2 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)^2}{(c+d x)^2}\right )}{f+g x} \, dx}{(b f-a g)^2 (d f-c g)^2}\\ &=-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (2 b^2 B^2\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}{g (b f-a g)^2}+\frac{\left (2 B^2 d^2\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}{g (d f-c g)^2}+\frac{\left (2 B^2 (b c-a d)\right ) \int \frac{2 (b c-a d)}{(a+b x) (c+d x) (f+g x)} \, dx}{(b f-a g) (d f-c g)}-\frac{\left (2 B^2 (b c-a d) (2 b d f-b c g-a d g)\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 (f+g x)}{e (a+b x)^2} \, dx}{(b f-a g)^2 (d f-c g)^2}\\ &=-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (2 b^2 B^2\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}{e g (b f-a g)^2}+\frac{\left (2 B^2 d^2\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}{e g (d f-c g)^2}+\frac{\left (4 B^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x) (c+d x) (f+g x)} \, dx}{(b f-a g) (d f-c g)}-\frac{\left (2 B^2 (b c-a d) (2 b d f-b c g-a d g)\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 (f+g x)}{(a+b x)^2} \, dx}{e (b f-a g)^2 (d f-c g)^2}\\ &=-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (2 b^2 B^2\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}{e g (b f-a g)^2}+\frac{\left (2 B^2 d^2\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}{e g (d f-c g)^2}+\frac{\left (4 B^2 (b c-a d)^2\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}{(b f-a g) (d f-c g)}-\frac{\left (2 B^2 (b c-a d) (2 b d f-b c g-a d g)\right ) \int \left (\frac{2 b e \log (f+g x)}{a+b x}-\frac{2 d e \log (f+g x)}{c+d x}\right ) \, dx}{e (b f-a g)^2 (d f-c g)^2}\\ &=\frac{4 b B^2 (b c-a d) \log (a+b x)}{(b f-a g)^2 (d f-c g)}-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{4 B^2 d (b c-a d) \log (c+d x)}{(b f-a g) (d f-c g)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}+\frac{4 B^2 (b c-a d)^2 g \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (4 b^3 B^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{g (b f-a g)^2}+\frac{\left (4 b^2 B^2 d\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{g (b f-a g)^2}+\frac{\left (4 b B^2 d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{g (d f-c g)^2}-\frac{\left (4 B^2 d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{g (d f-c g)^2}-\frac{\left (4 b B^2 (b c-a d) (2 b d f-b c g-a d g)\right ) \int \frac{\log (f+g x)}{a+b x} \, dx}{(b f-a g)^2 (d f-c g)^2}+\frac{\left (4 B^2 d (b c-a d) (2 b d f-b c g-a d g)\right ) \int \frac{\log (f+g x)}{c+d x} \, dx}{(b f-a g)^2 (d f-c g)^2}\\ &=\frac{4 b B^2 (b c-a d) \log (a+b x)}{(b f-a g)^2 (d f-c g)}-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{4 B^2 d (b c-a d) \log (c+d x)}{(b f-a g) (d f-c g)^2}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}+\frac{4 b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g (b f-a g)^2}+\frac{4 B^2 (b c-a d)^2 g \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (4 b^2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{g (b f-a g)^2}-\frac{\left (4 b^3 B^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{g (b f-a g)^2}-\frac{\left (4 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{g (d f-c g)^2}-\frac{\left (4 B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{g (d f-c g)^2}+\frac{\left (4 B^2 (b c-a d) g (2 b d f-b c g-a d g)\right ) \int \frac{\log \left (\frac{g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (4 B^2 (b c-a d) g (2 b d f-b c g-a d g)\right ) \int \frac{\log \left (\frac{g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx}{(b f-a g)^2 (d f-c g)^2}\\ &=\frac{4 b B^2 (b c-a d) \log (a+b x)}{(b f-a g)^2 (d f-c g)}-\frac{2 b^2 B^2 \log ^2(a+b x)}{g (b f-a g)^2}-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{4 B^2 d (b c-a d) \log (c+d x)}{(b f-a g) (d f-c g)^2}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{2 B^2 d^2 \log ^2(c+d x)}{g (d f-c g)^2}+\frac{4 b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g (b f-a g)^2}+\frac{4 B^2 (b c-a d)^2 g \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (4 b^2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{g (b f-a g)^2}-\frac{\left (4 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{g (d f-c g)^2}+\frac{\left (4 B^2 (b c-a d) (2 b d f-b c g-a d g)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{(b f-a g)^2 (d f-c g)^2}-\frac{\left (4 B^2 (b c-a d) (2 b d f-b c g-a d g)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{(b f-a g)^2 (d f-c g)^2}\\ &=\frac{4 b B^2 (b c-a d) \log (a+b x)}{(b f-a g)^2 (d f-c g)}-\frac{2 b^2 B^2 \log ^2(a+b x)}{g (b f-a g)^2}-\frac{2 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}+\frac{2 b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g (f+g x)^2}-\frac{4 B^2 d (b c-a d) \log (c+d x)}{(b f-a g) (d f-c g)^2}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{2 B^2 d^2 \log ^2(c+d x)}{g (d f-c g)^2}+\frac{4 b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g (b f-a g)^2}+\frac{4 B^2 (b c-a d)^2 g \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{4 b^2 B^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{g (b f-a g)^2}+\frac{4 B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{g (d f-c g)^2}-\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \text{Li}_2\left (\frac{b (f+g x)}{b f-a g}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac{4 B^2 (b c-a d) (2 b d f-b c g-a d g) \text{Li}_2\left (\frac{d (f+g x)}{d f-c g}\right )}{(b f-a g)^2 (d f-c g)^2}\\ \end{align*}
Mathematica [A] time = 1.9621, size = 603, normalized size = 1.58 \[ -\frac{\frac{4 B (f+g x) \left (b^2 B (f+g x) (d f-c g)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-B d^2 (f+g x) (b f-a g)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B g (f+g x) (b c-a d) (a d g+b c g-2 b d f) \left (\text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )-\text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )+\log (f+g x) \left (\log \left (\frac{g (a+b x)}{a g-b f}\right )-\log \left (\frac{g (c+d x)}{c g-d f}\right )\right )\right )-b^2 (f+g x) \log (a+b x) (d f-c g)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+d^2 (f+g x) (b f-a g)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+g (b c-a d) (b f-a g) (d f-c g) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+g (f+g x) (b c-a d) \log (f+g x) (a d g+b c g-2 b d f) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-2 B g (f+g x) (b c-a d) (b \log (a+b x) (d f-c g)+\log (c+d x) (a d g-b d f)+g (b c-a d) \log (f+g x))\right )}{(b f-a g)^2 (d f-c g)^2}+\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g (f+g x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.223, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{3}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (\frac{2 \, b^{2} \log \left (b x + a\right )}{b^{2} f^{2} g - 2 \, a b f g^{2} + a^{2} g^{3}} - \frac{2 \, d^{2} \log \left (d x + c\right )}{d^{2} f^{2} g - 2 \, c d f g^{2} + c^{2} g^{3}} + \frac{2 \,{\left (2 \,{\left (b^{2} c d - a b d^{2}\right )} f -{\left (b^{2} c^{2} - a^{2} d^{2}\right )} g\right )} \log \left (g x + f\right )}{b^{2} d^{2} f^{4} + a^{2} c^{2} g^{4} - 2 \,{\left (b^{2} c d + a b d^{2}\right )} f^{3} g +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} - 2 \,{\left (a b c^{2} + a^{2} c d\right )} f g^{3}} - \frac{2 \,{\left (b c - a d\right )}}{b d f^{3} + a c f g^{2} -{\left (b c + a d\right )} f^{2} g +{\left (b d f^{2} g + a c g^{3} -{\left (b c + a d\right )} f g^{2}\right )} x} - \frac{\log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g}\right )} A B - B^{2}{\left (\frac{2 \, \log \left (d x + c\right )^{2}}{g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g} + \int -\frac{d g x \log \left (e\right )^{2} + c g \log \left (e\right )^{2} + 4 \,{\left (d g x + c g\right )} \log \left (b x + a\right )^{2} + 4 \,{\left (d g x \log \left (e\right ) + c g \log \left (e\right )\right )} \log \left (b x + a\right ) - 4 \,{\left ({\left (g \log \left (e\right ) - g\right )} d x + c g \log \left (e\right ) - d f + 2 \,{\left (d g x + c g\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{d g^{4} x^{4} + c f^{3} g +{\left (3 \, d f g^{3} + c g^{4}\right )} x^{3} + 3 \,{\left (d f^{2} g^{2} + c f g^{3}\right )} x^{2} +{\left (d f^{3} g + 3 \, c f^{2} g^{2}\right )} x}\,{d x}\right )} - \frac{A^{2}}{2 \,{\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \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 (\frac{B^{2} \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 \, A B \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 ) + A^{2}}{g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, 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 \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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