Optimal. Leaf size=1208 \[ \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^4}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {f+g x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n^2 \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-3 a d g) n^2 (c+d x)}{4 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^3 g^3 n^2 (c+d x)}{6 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 (b f-a g)^2 (d f-c g)^4 (f+g x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.55, antiderivative size = 1968, normalized size of antiderivative = 1.63, number of steps used = 41, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, 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 12
Rule 72
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x) (f+g x)^4} \, dx}{2 g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {(B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x) (f+g x)^4} \, dx}{2 g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (b f-a g)^4 (a+b x)}-\frac {d^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (-d f+c g)^4 (c+d x)}+\frac {g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {\left (b^5 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{2 g (b f-a g)^4}-\frac {\left (B d^5 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{2 g (d f-c g)^4}+\frac {(B (b c-a d) g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{2 g (b f-a g)^4}+\frac {\left (B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{2 g (d f-c g)^4}+\frac {\left (B^2 (b c-a d) n^2\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) n^2\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 ) n^2\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 ) n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (f+g x)}{a+b x} \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^4 B^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{2 g (b f-a g)^4}+\frac {\left (B^2 d^4 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{2 g (d f-c g)^4}+\frac {\left (B^2 (b c-a d)^2 n^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) n^2\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 ) n^2\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 ) n^2\right ) \int \left (\frac {b \log (f+g x)}{a+b x}-\frac {d \log (f+g x)}{c+d x}\right ) \, dx}{2 (b f-a g)^4 (d f-c g)^4}\\ &=-\frac {B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (b^5 B^2 n^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 n^2\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 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 g (d f-c g)^4}-\frac {\left (B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 g (d f-c g)^4}+\frac {\left (B^2 (b c-a d)^2 n^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) n^2\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 ) n^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}{2 (b f-a g)^3 (d f-c g)^3}+\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 ) n^2\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 ) n^2\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 n^2}{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) n^2}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) n^2 \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) n^2 \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 ) n^2 \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) n^2 \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) n^2 \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 ) n^2 \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}+\frac {B^2 d^4 n^2 \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 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}+\frac {b^4 B^2 n^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 n^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 ) n^2 \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 ) n^2 \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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n^2 \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 n^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 n^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 n^2\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 n^2\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 ) n^2\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 ) n^2\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 n^2}{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) n^2}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) n^2 \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) n^2 \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 ) n^2 \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {b^4 B^2 n^2 \log ^2(a+b x)}{4 g (b f-a g)^4}-\frac {B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) n^2 \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) n^2 \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 ) n^2 \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}+\frac {B^2 d^4 n^2 \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 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B^2 d^4 n^2 \log ^2(c+d x)}{4 g (d f-c g)^4}+\frac {b^4 B^2 n^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 n^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 ) n^2 \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 ) n^2 \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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n^2 \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 n^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 n^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 )}{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 ) n^2\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 ) n^2\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 n^2}{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) n^2}{12 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^3 B^2 (b c-a d) n^2 \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) n^2 \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 ) n^2 \log (a+b x)}{2 (b f-a g)^4 (d f-c g)^3}-\frac {b^4 B^2 n^2 \log ^2(a+b x)}{4 g (b f-a g)^4}-\frac {B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 d^3 (b c-a d) n^2 \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) n^2 \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 ) n^2 \log (c+d x)}{2 (b f-a g)^3 (d f-c g)^4}+\frac {B^2 d^4 n^2 \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 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 g (d f-c g)^4}-\frac {B^2 d^4 n^2 \log ^2(c+d x)}{4 g (d f-c g)^4}+\frac {b^4 B^2 n^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 n^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 ) n^2 \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 ) n^2 \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 ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n^2 \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 n^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 n^2 \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 ) n^2 \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 ) n^2 \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{2 (b f-a g)^4 (d f-c g)^4}\\ \end {align*}
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Mathematica [A] time = 7.32, size = 1476, normalized size = 1.22 \[ \frac {B (b c-a d) n \left (\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^4}{(b c-a d) (b f-a g)^4}-\frac {B n \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 {Li}_2\left (-\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 (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b f-a g) (d f-c g) (f+g x)^3}-\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\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 ) n \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) n \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 n \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 n \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 {Li}_2\left (\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 ) n \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 {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )-\text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )\right )}{(b f-a g)^4 (d f-c g)^4}\right )}{2 g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2}}{g^{5} x^{5} + 5 \, f g^{4} x^{4} + 10 \, f^{2} g^{3} x^{3} + 10 \, f^{3} g^{2} x^{2} + 5 \, f^{4} g x + f^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (g x +f \right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^5} \,d x \]
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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