Optimal. Leaf size=475 \[ -\frac {b^2 i (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac {b^2 B i n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^3}-\frac {d^2 i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^3}-\frac {B d^2 i n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b d i (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^3}+\frac {4 b B d i n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B^2 i n^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^3}-\frac {B^2 d^2 i n^2 (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac {4 b B^2 d i n^2 (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^3} \]
[Out]
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Rubi [C] time = 2.87, antiderivative size = 892, normalized size of antiderivative = 1.88, number of steps used = 70, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 i n^2 \log ^2(a+b x) d^4}{12 b^2 (b c-a d)^3 g^5}+\frac {B^2 i n^2 \log ^2(c+d x) d^4}{12 b^2 (b c-a d)^3 g^5}-\frac {13 B^2 i n^2 \log (a+b x) d^4}{72 b^2 (b c-a d)^3 g^5}-\frac {B i n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^4}{6 b^2 (b c-a d)^3 g^5}+\frac {13 B^2 i n^2 \log (c+d x) d^4}{72 b^2 (b c-a d)^3 g^5}-\frac {B^2 i n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{6 b^2 (b c-a d)^3 g^5}+\frac {B i n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac {B^2 i n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac {B^2 i n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac {B^2 i n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac {B i n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^3}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {13 B^2 i n^2 d^3}{72 b^2 (b c-a d)^2 g^5 (a+b x)}+\frac {B i n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^2}{12 b^2 (b c-a d) g^5 (a+b x)^2}+\frac {B^2 i n^2 d^2}{144 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {i \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d}{3 b^2 g^5 (a+b x)^3}-\frac {B i n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d}{18 b^2 g^5 (a+b x)^3}+\frac {5 B^2 i n^2 d}{216 b^2 g^5 (a+b x)^3}-\frac {(b c-a d) i \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {B (b c-a d) i n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {B^2 (b c-a d) i n^2}{32 b^2 g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(167 c+167 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx &=\int \left (\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g^5 (a+b x)^5}+\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g^5 (a+b x)^4}\right ) \, dx\\ &=\frac {(167 d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4} \, dx}{b g^5}+\frac {(167 (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5} \, dx}{b g^5}\\ &=-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {(334 B d 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)^4 (c+d x)} \, dx}{3 b^2 g^5}+\frac {(167 B (b c-a d) 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)^5 (c+d x)} \, dx}{2 b^2 g^5}\\ &=-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {(334 B d (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)^4 (c+d x)} \, dx}{3 b^2 g^5}+\frac {\left (167 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b^2 g^5}\\ &=-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {(334 B d (b c-a d) n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^2 g^5}+\frac {\left (167 B (b c-a d)^2 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (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)^5 (c+d x)}\right ) \, dx}{2 b^2 g^5}\\ &=-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}-\frac {(167 B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{2 b g^5}+\frac {(334 B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3 b g^5}+\frac {\left (167 B d^4 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 b (b c-a d)^3 g^5}-\frac {\left (334 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3 b (b c-a d)^3 g^5}-\frac {\left (167 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 b^2 (b c-a d)^3 g^5}+\frac {\left (334 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}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{2 b (b c-a d)^2 g^5}+\frac {\left (334 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3 b (b c-a d)^2 g^5}+\frac {\left (167 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{2 b (b c-a d) g^5}-\frac {\left (334 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3 b (b c-a d) g^5}+\frac {(167 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)^5} \, dx}{2 b g^5}\\ &=-\frac {167 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {167 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18 b^2 g^5 (a+b x)^3}+\frac {167 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {167 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {167 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {167 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)}{6 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6 b^2 g^5}+\frac {\left (334 B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^2 g^5}-\frac {\left (167 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 (a+b x)}{a+b x} \, dx}{2 b^2 (b c-a d)^3 g^5}+\frac {\left (167 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 b^2 (b c-a d)^3 g^5}+\frac {\left (334 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 (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (334 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}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b^2 (b c-a d)^2 g^5}+\frac {\left (334 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^2 (b c-a d)^2 g^5}+\frac {\left (167 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b^2 (b c-a d) g^5}-\frac {\left (167 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^2 (b c-a d) g^5}+\frac {\left (167 B^2 (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b^2 g^5}\\ &=-\frac {167 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {167 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18 b^2 g^5 (a+b x)^3}+\frac {167 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {167 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {167 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {167 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)}{6 b^2 (b c-a d)^3 g^5}+\frac {\left (167 B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{4 b^2 g^5}-\frac {\left (167 B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^2 g^5}-\frac {\left (167 B^2 d^4 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 b^2 (b c-a d)^3 g^5}+\frac {\left (167 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 b^2 (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^4 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}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (334 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}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2 b^2 (b c-a d) g^5}+\frac {\left (334 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^2 (b c-a d) g^5}-\frac {\left (167 B^2 d (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6 b^2 g^5}+\frac {\left (334 B^2 d (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^2 g^5}+\frac {\left (167 B^2 (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{8 b^2 g^5}\\ &=-\frac {167 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {167 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18 b^2 g^5 (a+b x)^3}+\frac {167 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {167 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {167 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {167 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)}{6 b^2 (b c-a d)^3 g^5}+\frac {\left (167 B^2 d^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{4 b^2 g^5}-\frac {\left (167 B^2 d^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b^2 g^5}-\frac {\left (167 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 b (b c-a d)^3 g^5}+\frac {\left (167 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 b (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b (b c-a d)^3 g^5}-\frac {\left (334 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b (b c-a d)^3 g^5}+\frac {\left (167 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b^2 (b c-a d)^3 g^5}-\frac {\left (334 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^2 (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{2 b^2 (b c-a d) g^5}+\frac {\left (334 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b^2 (b c-a d) g^5}-\frac {\left (167 B^2 d (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{6 b^2 g^5}+\frac {\left (334 B^2 d (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b^2 g^5}+\frac {\left (167 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b^2 g^5}\\ &=-\frac {167 B^2 (b c-a d) n^2}{32 b^2 g^5 (a+b x)^4}+\frac {835 B^2 d n^2}{216 b^2 g^5 (a+b x)^3}+\frac {167 B^2 d^2 n^2}{144 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {2171 B^2 d^3 n^2}{72 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {2171 B^2 d^4 n^2 \log (a+b x)}{72 b^2 (b c-a d)^3 g^5}-\frac {167 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {167 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18 b^2 g^5 (a+b x)^3}+\frac {167 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {167 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {167 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {2171 B^2 d^4 n^2 \log (c+d x)}{72 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{6 b^2 (b c-a d)^3 g^5}+\frac {167 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)}{6 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b^2 (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^2 (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 b (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b (b c-a d)^3 g^5}-\frac {\left (167 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 b^2 (b c-a d)^3 g^5}+\frac {\left (334 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}{3 b^2 (b c-a d)^3 g^5}\\ &=-\frac {167 B^2 (b c-a d) n^2}{32 b^2 g^5 (a+b x)^4}+\frac {835 B^2 d n^2}{216 b^2 g^5 (a+b x)^3}+\frac {167 B^2 d^2 n^2}{144 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {2171 B^2 d^3 n^2}{72 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {2171 B^2 d^4 n^2 \log (a+b x)}{72 b^2 (b c-a d)^3 g^5}+\frac {167 B^2 d^4 n^2 \log ^2(a+b x)}{12 b^2 (b c-a d)^3 g^5}-\frac {167 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {167 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18 b^2 g^5 (a+b x)^3}+\frac {167 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {167 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {167 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {2171 B^2 d^4 n^2 \log (c+d x)}{72 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{6 b^2 (b c-a d)^3 g^5}+\frac {167 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)}{6 b^2 (b c-a d)^3 g^5}+\frac {167 B^2 d^4 n^2 \log ^2(c+d x)}{12 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {\left (167 B^2 d^4 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 b^2 (b c-a d)^3 g^5}-\frac {\left (167 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 b^2 (b c-a d)^3 g^5}+\frac {\left (334 B^2 d^4 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 )}{3 b^2 (b c-a d)^3 g^5}+\frac {\left (334 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 )}{3 b^2 (b c-a d)^3 g^5}\\ &=-\frac {167 B^2 (b c-a d) n^2}{32 b^2 g^5 (a+b x)^4}+\frac {835 B^2 d n^2}{216 b^2 g^5 (a+b x)^3}+\frac {167 B^2 d^2 n^2}{144 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {2171 B^2 d^3 n^2}{72 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {2171 B^2 d^4 n^2 \log (a+b x)}{72 b^2 (b c-a d)^3 g^5}+\frac {167 B^2 d^4 n^2 \log ^2(a+b x)}{12 b^2 (b c-a d)^3 g^5}-\frac {167 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac {167 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18 b^2 g^5 (a+b x)^3}+\frac {167 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {167 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {167 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac {167 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2 g^5 (a+b x)^3}+\frac {2171 B^2 d^4 n^2 \log (c+d x)}{72 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{6 b^2 (b c-a d)^3 g^5}+\frac {167 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)}{6 b^2 (b c-a d)^3 g^5}+\frac {167 B^2 d^4 n^2 \log ^2(c+d x)}{12 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{6 b^2 (b c-a d)^3 g^5}-\frac {167 B^2 d^4 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{6 b^2 (b c-a d)^3 g^5}\\ \end {align*}
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Mathematica [C] time = 1.36, size = 1392, normalized size = 2.93 \[ -\frac {i \left (216 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^4-288 d (a d-b c)^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+16 B d n (a+b x) \left (12 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3-18 d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2+36 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+36 B d^2 n (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d n (a+b x) \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B n \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 n (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+18 B d^3 n (a+b x)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+3 B n \left (36 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4+72 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2+144 d^3 (a d-b c) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+48 d (a d-b c)^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+144 d^4 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-144 B d^3 n (a+b x)^3 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+36 B d^2 n (a+b x)^2 \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-8 B d n (a+b x) \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )+3 B n \left (3 (b c-a d)^4+6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a d-b c) (a+b x)^3+4 d (a d-b c)^3 (a+b x)-12 d^4 (a+b x)^4 \log (a+b x)+12 d^4 (a+b x)^4 \log (c+d x)\right )+72 B d^4 n (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-72 B d^4 n (a+b x)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{864 b^2 (b c-a d)^3 g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.14, size = 1868, normalized size = 3.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 38.00, size = 841, normalized size = 1.77 \[ -\frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b^{2} i n^{2} - \frac {8 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{d x + c} + \frac {6 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (9 \, B^{2} b^{2} i n^{2} - \frac {32 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{d x + c} + \frac {36 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (d x + c\right )}^{2}} + 36 \, A B b^{2} i n + 36 \, B^{2} b^{2} i n - \frac {96 \, {\left (b x + a\right )} A B b d i n}{d x + c} - \frac {96 \, {\left (b x + a\right )} B^{2} b d i n}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} A B d^{2} i n}{{\left (d x + c\right )}^{2}} + \frac {72 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {27 \, B^{2} b^{2} i n^{2} - \frac {128 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{d x + c} + \frac {216 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (d x + c\right )}^{2}} + 108 \, A B b^{2} i n + 108 \, B^{2} b^{2} i n - \frac {384 \, {\left (b x + a\right )} A B b d i n}{d x + c} - \frac {384 \, {\left (b x + a\right )} B^{2} b d i n}{d x + c} + \frac {432 \, {\left (b x + a\right )}^{2} A B d^{2} i n}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n}{{\left (d x + c\right )}^{2}} + 216 \, A^{2} b^{2} i + 432 \, A B b^{2} i + 216 \, B^{2} b^{2} i - \frac {576 \, {\left (b x + a\right )} A^{2} b d i}{d x + c} - \frac {1152 \, {\left (b x + a\right )} A B b d i}{d x + c} - \frac {576 \, {\left (b x + a\right )} B^{2} b d i}{d x + c} + \frac {432 \, {\left (b x + a\right )}^{2} A^{2} d^{2} i}{{\left (d x + c\right )}^{2}} + \frac {864 \, {\left (b x + a\right )}^{2} A B d^{2} i}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 4.87, size = 4838, normalized size = 10.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.74, size = 1794, normalized size = 3.78 \[ \text {result too large to display} \]
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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