Optimal. Leaf size=766 \[ -\frac {B g^3 i^2 n (b c-a d)^6 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{180 b^3 d^4}-\frac {B g^3 i^2 n (a+b x) (b c-a d)^5 \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{180 b^3 d^3}+\frac {B g^3 i^2 n (a+b x)^2 (b c-a d)^4 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{180 b^3 d^2}-\frac {B g^3 i^2 n (a+b x)^3 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{90 b^3 d}+\frac {g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{60 b^3}-\frac {B g^3 i^2 n (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^3}+\frac {g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{15 b^2}-\frac {B g^3 i^2 n (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^2}+\frac {g^3 i^2 (a+b x)^4 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b}-\frac {B^2 g^3 i^2 n^2 (b c-a d)^6 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4}-\frac {B^2 g^3 i^2 n^2 (b c-a d)^6 \log (c+d x)}{20 b^3 d^4}+\frac {B^2 g^3 i^2 n^2 (a+b x)^4 (b c-a d)^2}{60 b^3}+\frac {3 B^2 g^3 i^2 n^2 x (b c-a d)^5}{20 b^2 d^3}-\frac {3 B^2 g^3 i^2 n^2 (c+d x)^2 (b c-a d)^4}{40 b d^4}+\frac {B^2 g^3 i^2 n^2 (c+d x)^3 (b c-a d)^3}{60 d^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.35, antiderivative size = 848, normalized size of antiderivative = 1.11, number of steps used = 83, number of rules used = 13, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B^2 g^3 i^2 n^2 \log ^2(c+d x) (b c-a d)^6}{60 b^3 d^4}+\frac {B^2 g^3 i^2 n^2 \log (c+d x) (b c-a d)^6}{90 b^3 d^4}-\frac {B^2 g^3 i^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^6}{30 b^3 d^4}+\frac {B g^3 i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^6}{30 b^3 d^4}-\frac {B^2 g^3 i^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^6}{30 b^3 d^4}+\frac {B^2 g^3 i^2 n^2 x (b c-a d)^5}{45 b^2 d^3}-\frac {A B g^3 i^2 n x (b c-a d)^5}{30 b^2 d^3}-\frac {B^2 g^3 i^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) (b c-a d)^5}{30 b^3 d^3}-\frac {7 B^2 g^3 i^2 n^2 (a+b x)^2 (b c-a d)^4}{360 b^3 d^2}+\frac {B g^3 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{60 b^3 d^2}+\frac {B^2 g^3 i^2 n^2 (a+b x)^3 (b c-a d)^3}{60 b^3 d}-\frac {B g^3 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{90 b^3 d}+\frac {B^2 g^3 i^2 n^2 (a+b x)^4 (b c-a d)^2}{60 b^3}+\frac {g^3 i^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{4 b^3}-\frac {7 B g^3 i^2 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{60 b^3}+\frac {2 d g^3 i^2 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{5 b^3}-\frac {B d g^3 i^2 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{15 b^3}+\frac {d^2 g^3 i^2 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 43
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int (168 c+168 d x)^2 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac {28224 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}+\frac {56448 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {28224 d^2 (a g+b g x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}\right ) \, dx\\ &=\frac {\left (28224 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2}+\frac {\left (28224 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g^2}+\frac {(56448 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g}\\ &=\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac {\left (9408 B d^2 n\right ) \int \frac {(b c-a d) g^6 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3 g^3}-\frac {(112896 B d (b c-a d) n) \int \frac {(b c-a d) g^5 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{5 b^3 g^2}-\frac {\left (14112 B (b c-a d)^2 n\right ) \int \frac {(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3 g}\\ &=\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac {\left (9408 B d^2 (b c-a d) g^3 n\right ) \int \frac {(a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3}-\frac {\left (112896 B d (b c-a d)^2 g^3 n\right ) \int \frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{5 b^3}-\frac {\left (14112 B (b c-a d)^3 g^3 n\right ) \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3}\\ &=\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac {\left (9408 B d^2 (b c-a d) g^3 n\right ) \int \left (\frac {b (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^5}-\frac {b (b c-a d)^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^5 (c+d x)}\right ) \, dx}{b^3}-\frac {\left (112896 B d (b c-a d)^2 g^3 n\right ) \int \left (-\frac {b (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac {\left (14112 B (b c-a d)^3 g^3 n\right ) \int \left (\frac {b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^3}\\ &=\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac {\left (9408 B d (b c-a d) g^3 n\right ) \int (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2}+\frac {\left (9408 B (b c-a d)^2 g^3 n\right ) \int (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2}-\frac {\left (112896 B (b c-a d)^2 g^3 n\right ) \int (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2}-\frac {\left (9408 B (b c-a d)^3 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}-\frac {\left (14112 B (b c-a d)^3 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}+\frac {\left (112896 B (b c-a d)^3 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d}+\frac {\left (9408 B (b c-a d)^4 g^3 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d^2}+\frac {\left (14112 B (b c-a d)^4 g^3 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d^2}-\frac {\left (112896 B (b c-a d)^4 g^3 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d^2}-\frac {\left (9408 B (b c-a d)^5 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d^3}-\frac {\left (14112 B (b c-a d)^5 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d^3}+\frac {\left (112896 B (b c-a d)^5 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d^3}+\frac {\left (9408 B (b c-a d)^6 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 d^3}+\frac {\left (14112 B (b c-a d)^6 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 d^3}-\frac {\left (112896 B (b c-a d)^6 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac {2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac {1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac {16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac {9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}-\frac {\left (9408 B^2 (b c-a d)^5 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d^3}-\frac {\left (14112 B^2 (b c-a d)^5 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d^3}+\frac {\left (112896 B^2 (b c-a d)^5 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^2 d^3}+\frac {\left (9408 B^2 d (b c-a d) g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)^4}{c+d x} \, dx}{5 b^3}-\frac {\left (2352 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)^3}{c+d x} \, dx}{b^3}+\frac {\left (28224 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac {\left (3136 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{b^3 d}+\frac {\left (4704 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{b^3 d}-\frac {\left (37632 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{5 b^3 d}-\frac {\left (4704 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{b^3 d^2}-\frac {\left (7056 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{b^3 d^2}+\frac {\left (56448 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{5 b^3 d^2}-\frac {\left (9408 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}-\frac {\left (14112 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}+\frac {\left (112896 B^2 (b c-a d)^6 g^3 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}{5 b^3 d^4}\\ &=-\frac {4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}-\frac {4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac {2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac {1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac {16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac {9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac {\left (9408 B^2 d (b c-a d)^2 g^3 n^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{5 b^3}-\frac {\left (2352 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{b^3}+\frac {\left (28224 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac {\left (3136 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{b^3 d}+\frac {\left (4704 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{b^3 d}-\frac {\left (37632 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{5 b^3 d}-\frac {\left (4704 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{b^3 d^2}-\frac {\left (7056 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{b^3 d^2}+\frac {\left (56448 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{5 b^3 d^2}-\frac {\left (9408 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}-\frac {\left (14112 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}+\frac {\left (112896 B^2 (b c-a d)^6 g^3 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}{5 b^3 d^4}+\frac {\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^3}+\frac {\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^3}-\frac {\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}-\frac {4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac {2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac {1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac {16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac {9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {4704 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}+\frac {4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac {\left (9408 B^2 d (b c-a d)^2 g^3 n^2\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac {\left (2352 B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{b^3}+\frac {\left (28224 B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{5 b^3}+\frac {\left (3136 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{b^3 d}+\frac {\left (4704 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{b^3 d}-\frac {\left (37632 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{5 b^3 d}-\frac {\left (4704 B^2 (b c-a d)^5 g^3 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b^3 d^2}-\frac {\left (7056 B^2 (b c-a d)^5 g^3 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b^3 d^2}+\frac {\left (56448 B^2 (b c-a d)^5 g^3 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{5 b^3 d^2}-\frac {\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 d^4}-\frac {\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac {\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 b^2 d^4}+\frac {\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 d^3}+\frac {\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 d^3}-\frac {\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac {3136 B^2 (b c-a d)^5 g^3 n^2 x}{5 b^2 d^3}-\frac {2744 B^2 (b c-a d)^4 g^3 n^2 (a+b x)^2}{5 b^3 d^2}+\frac {2352 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^3}{5 b^3 d}+\frac {2352 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^4}{5 b^3}-\frac {4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac {2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac {1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac {16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac {9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {1568 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}-\frac {4704 B^2 (b c-a d)^6 g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 d^4}+\frac {4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac {\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 d^4}+\frac {\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 d^4}-\frac {\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 d^4}+\frac {\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 d^3}+\frac {\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 d^3}-\frac {\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac {3136 B^2 (b c-a d)^5 g^3 n^2 x}{5 b^2 d^3}-\frac {2744 B^2 (b c-a d)^4 g^3 n^2 (a+b x)^2}{5 b^3 d^2}+\frac {2352 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^3}{5 b^3 d}+\frac {2352 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^4}{5 b^3}-\frac {4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac {2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac {1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac {16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac {9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {1568 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}-\frac {4704 B^2 (b c-a d)^6 g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 d^4}+\frac {4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac {2352 B^2 (b c-a d)^6 g^3 n^2 \log ^2(c+d x)}{5 b^3 d^4}+\frac {\left (9408 B^2 (b c-a d)^6 g^3 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 )}{b^3 d^4}+\frac {\left (14112 B^2 (b c-a d)^6 g^3 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 )}{b^3 d^4}-\frac {\left (112896 B^2 (b c-a d)^6 g^3 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 )}{5 b^3 d^4}\\ &=-\frac {4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac {3136 B^2 (b c-a d)^5 g^3 n^2 x}{5 b^2 d^3}-\frac {2744 B^2 (b c-a d)^4 g^3 n^2 (a+b x)^2}{5 b^3 d^2}+\frac {2352 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^3}{5 b^3 d}+\frac {2352 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^4}{5 b^3}-\frac {4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac {2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac {1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac {16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac {9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac {7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac {4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac {1568 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}-\frac {4704 B^2 (b c-a d)^6 g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 d^4}+\frac {4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac {2352 B^2 (b c-a d)^6 g^3 n^2 \log ^2(c+d x)}{5 b^3 d^4}-\frac {4704 B^2 (b c-a d)^6 g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^3 d^4}\\ \end {align*}
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Mathematica [B] time = 1.54, size = 1634, normalized size = 2.13 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{3} d^{2} g^{3} i^{2} x^{5} + A^{2} a^{3} c^{2} g^{3} i^{2} + {\left (2 \, A^{2} b^{3} c d + 3 \, A^{2} a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} + {\left (A^{2} b^{3} c^{2} + 6 \, A^{2} a b^{2} c d + 3 \, A^{2} a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} + {\left (3 \, A^{2} a b^{2} c^{2} + 6 \, A^{2} a^{2} b c d + A^{2} a^{3} d^{2}\right )} g^{3} i^{2} x^{2} + {\left (3 \, A^{2} a^{2} b c^{2} + 2 \, A^{2} a^{3} c d\right )} g^{3} i^{2} x + {\left (B^{2} b^{3} d^{2} g^{3} i^{2} x^{5} + B^{2} a^{3} c^{2} g^{3} i^{2} + {\left (2 \, B^{2} b^{3} c d + 3 \, B^{2} a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} + {\left (B^{2} b^{3} c^{2} + 6 \, B^{2} a b^{2} c d + 3 \, B^{2} a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} + {\left (3 \, B^{2} a b^{2} c^{2} + 6 \, B^{2} a^{2} b c d + B^{2} a^{3} d^{2}\right )} g^{3} i^{2} x^{2} + {\left (3 \, B^{2} a^{2} b c^{2} + 2 \, B^{2} a^{3} c d\right )} g^{3} i^{2} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{3} d^{2} g^{3} i^{2} x^{5} + A B a^{3} c^{2} g^{3} i^{2} + {\left (2 \, A B b^{3} c d + 3 \, A B a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} + {\left (A B b^{3} c^{2} + 6 \, A B a b^{2} c d + 3 \, A B a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} + {\left (3 \, A B a b^{2} c^{2} + 6 \, A B a^{2} b c d + A B a^{3} d^{2}\right )} g^{3} i^{2} x^{2} + {\left (3 \, A B a^{2} b c^{2} + 2 \, A B a^{3} c d\right )} g^{3} i^{2} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]
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 = 0.46, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 8.10, size = 5952, normalized size = 7.77 \[ \text {result too large to display} \]
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 \[ \int {\left (a\,g+b\,g\,x\right )}^3\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
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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