Optimal. Leaf size=1159 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 3.4031, antiderivative size = 1881, normalized size of antiderivative = 1.62, number of steps used = 44, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 72} \[ \text{result too large to display} \]
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)}{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 7.31149, size = 1448, normalized size = 1.25 \[ \frac{B (b c-a d) \left (\frac{\log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) b^4}{(b c-a d) (b f-a g)^4}-\frac{B \left (\log ^2(a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a+b x)-2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )\right ) b^4}{2 (b c-a d) (b f-a g)^4}-\frac{g \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+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)^3 (d f-c g)^3 (f+g x)}-\frac{g (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 )}{2 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac{g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 (b f-a g) (d f-c g) (f+g x)^3}-\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (d f-c g)^4}+\frac{g (2 b d f-b c g-a d g) \left (2 d^2 f^2 b^2+c^2 g^2 b^2-2 c d f g b^2-2 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 (f+g x)}{(b f-a g)^4 (d f-c g)^4}+\frac{B (b c-a d) g \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right ) \left (\frac{b \log (a+b x)}{(b c-a d) (b f-a g)}-\frac{d \log (c+d x)}{(b c-a d) (d f-c g)}+\frac{g \log (f+g x)}{(b f-a g) (d f-c g)}\right )}{(b f-a g)^3 (d f-c g)^3}-\frac{B (b c-a d) g (2 b d f-b c g-a d g) \left (-\frac{\log (a+b x) b^2}{(b c-a d) (b f-a g)^2}+\frac{d^2 \log (c+d x)}{(b c-a d) (d f-c g)^2}-\frac{g (2 b d f-b c g-a d g) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{g}{(b f-a g) (d f-c g) (f+g x)}\right )}{2 (b f-a g)^2 (d f-c g)^2}-\frac{B (b c-a d) g \left (-\frac{2 \log (a+b x) b^3}{(b c-a d) (b f-a g)^3}+\frac{2 d^3 \log (c+d x)}{(b c-a d) (d f-c g)^3}-\frac{2 g \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right ) \log (f+g x)}{(b f-a g)^3 (d f-c g)^3}+\frac{2 g (2 b d f-b c g-a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac{g}{(b f-a g) (d f-c g) (f+g x)^2}\right )}{6 (b f-a g) (d f-c g)}+\frac{B d^4 \left (-\log ^2(c+d x)+2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )}{2 (b c-a d) (d f-c g)^4}-\frac{B g (2 b d f-b c g-a d g) \left (2 d^2 f^2 b^2+c^2 g^2 b^2-2 c d f g b^2-2 a d^2 f g b+a^2 d^2 g^2\right ) \left (\log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)-\log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)+\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 )\right )}{(b f-a g)^4 (d f-c g)^4}\right )}{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} \]
Antiderivative was successfully verified.
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Maple [F] time = 6.561, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{5}} \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 [F] time = 0., size = 0, normalized size = 0. \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 (\frac{B^{2} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac{b e x + a e}{d x + c}\right ) + A^{2}}{g^{5} x^{5} + 5 \, f g^{4} x^{4} + 10 \, f^{2} g^{3} x^{3} + 10 \, f^{3} g^{2} x^{2} + 5 \, f^{4} g x + f^{5}}, 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 )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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