Optimal. Leaf size=908 \[ \frac {B^2 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^6}{36 b^4 d^3}+\frac {B g^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^6}{60 b^4 d^3}+\frac {11 B^2 g^2 i^3 \log (c+d x) (b c-a d)^6}{180 b^4 d^3}+\frac {B^2 g^2 i^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {7 B^2 g^2 i^3 x (b c-a d)^5}{180 b^3 d^2}+\frac {B g^2 i^3 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^5}{60 b^4 d^2}-\frac {7 B^2 g^2 i^3 (c+d x)^2 (b c-a d)^4}{360 b^2 d^3}-\frac {B g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4}{60 b^4 d}-\frac {B g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4}{10 b^2 d^3}-\frac {B^2 g^2 i^3 (c+d x)^3 (b c-a d)^3}{60 b d^3}+\frac {g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^3}{60 b^4}-\frac {B g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3}{30 b^4}+\frac {B g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3}{45 b d^3}+\frac {B^2 g^2 i^3 (c+d x)^4 (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2}{20 b^3}+\frac {7 B g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)}{10 b^2}-\frac {b B g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{15 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.02, antiderivative size = 825, normalized size of antiderivative = 0.91, number of steps used = 86, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2528, 2525, 12, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ \frac {B^2 g^2 i^3 \log ^2(a+b x) (b c-a d)^6}{60 b^4 d^3}-\frac {B^2 g^2 i^3 \log (a+b x) (b c-a d)^6}{45 b^4 d^3}-\frac {B g^2 i^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^6}{30 b^4 d^3}+\frac {B^2 g^2 i^3 \log (c+d x) (b c-a d)^6}{30 b^4 d^3}-\frac {B^2 g^2 i^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {B^2 g^2 i^3 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {B^2 g^2 i^3 x (b c-a d)^5}{45 b^3 d^2}-\frac {A B g^2 i^3 x (b c-a d)^5}{30 b^3 d^2}-\frac {B^2 g^2 i^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right ) (b c-a d)^5}{30 b^4 d^2}-\frac {7 B^2 g^2 i^3 (c+d x)^2 (b c-a d)^4}{360 b^2 d^3}-\frac {B g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4}{60 b^2 d^3}-\frac {B^2 g^2 i^3 (c+d x)^3 (b c-a d)^3}{60 b d^3}-\frac {B g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3}{90 b d^3}+\frac {B^2 g^2 i^3 (c+d x)^4 (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2}{4 d^3}+\frac {7 B g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2}{60 d^3}-\frac {2 b g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)}{5 d^3}-\frac {b B g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{15 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 d^3} \]
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 (75 c+75 d x)^3 (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 (75 c+75 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {2 b (b c-a d) g^2 (75 c+75 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{75 d^2}+\frac {b^2 g^2 (75 c+75 d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5625 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (75 c+75 d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{5625 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int (75 c+75 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{75 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (75 c+75 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{d^2}\\ &=\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}-\frac {\left (b^2 B g^2\right ) \int \frac {177978515625 (b c-a d) (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{1265625 d^3}+\frac {\left (4 b B (b c-a d) g^2\right ) \int \frac {2373046875 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{28125 d^3}-\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {31640625 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{150 d^3}\\ &=\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}-\frac {\left (140625 b^2 B (b c-a d) g^2\right ) \int \frac {(c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^3}+\frac {\left (337500 b B (b c-a d)^2 g^2\right ) \int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^3}-\frac {\left (421875 B (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{2 d^3}\\ &=\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}-\frac {\left (140625 b^2 B (b c-a d) g^2\right ) \int \left (\frac {d (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^5}+\frac {(b c-a d)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^3}+\frac {\left (337500 b B (b c-a d)^2 g^2\right ) \int \left (\frac {d (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac {(b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^3}-\frac {\left (421875 B (b c-a d)^3 g^2\right ) \int \left (\frac {d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {(b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{2 d^3}\\ &=\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}-\frac {\left (140625 b B (b c-a d) g^2\right ) \int (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}-\frac {\left (140625 B (b c-a d)^2 g^2\right ) \int (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}+\frac {\left (337500 B (b c-a d)^2 g^2\right ) \int (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}-\frac {\left (140625 B (b c-a d)^3 g^2\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^2}-\frac {\left (421875 B (b c-a d)^3 g^2\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{2 b d^2}+\frac {\left (337500 B (b c-a d)^3 g^2\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^2}-\frac {\left (140625 B (b c-a d)^4 g^2\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^2}-\frac {\left (421875 B (b c-a d)^4 g^2\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{2 b^2 d^2}+\frac {\left (337500 B (b c-a d)^4 g^2\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^2}-\frac {\left (140625 B (b c-a d)^5 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 d^2}-\frac {\left (421875 B (b c-a d)^5 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{2 b^3 d^2}+\frac {\left (337500 B (b c-a d)^5 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 d^2}-\frac {\left (140625 B (b c-a d)^6 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 d^3}-\frac {\left (421875 B (b c-a d)^6 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 b^3 d^3}+\frac {\left (337500 B (b c-a d)^6 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 d^3}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {\left (28125 b B^2 (b c-a d) g^2\right ) \int \frac {(b c-a d) (c+d x)^4}{a+b x} \, dx}{d^3}+\frac {\left (140625 B^2 (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{4 d^3}-\frac {\left (84375 B^2 (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{d^3}+\frac {\left (46875 B^2 (b c-a d)^3 g^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{b d^3}+\frac {\left (140625 B^2 (b c-a d)^3 g^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{2 b d^3}-\frac {\left (112500 B^2 (b c-a d)^3 g^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{b d^3}+\frac {\left (140625 B^2 (b c-a d)^4 g^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{2 b^2 d^3}+\frac {\left (421875 B^2 (b c-a d)^4 g^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{4 b^2 d^3}-\frac {\left (168750 B^2 (b c-a d)^4 g^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^3}-\frac {\left (140625 B^2 (b c-a d)^5 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 d^2}-\frac {\left (421875 B^2 (b c-a d)^5 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{2 b^3 d^2}+\frac {\left (337500 B^2 (b c-a d)^5 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 d^2}+\frac {\left (140625 B^2 (b c-a d)^6 g^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}{b^4 d^3}+\frac {\left (421875 B^2 (b c-a d)^6 g^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 b^4 d^3}-\frac {\left (337500 B^2 (b c-a d)^6 g^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}{b^4 d^3}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {28125 B^2 (b c-a d)^5 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {\left (28125 b B^2 (b c-a d)^2 g^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{d^3}+\frac {\left (140625 B^2 (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{4 d^3}-\frac {\left (84375 B^2 (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^3}+\frac {\left (46875 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{b d^3}+\frac {\left (140625 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{2 b d^3}-\frac {\left (112500 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{b d^3}+\frac {\left (140625 B^2 (b c-a d)^5 g^2\right ) \int \frac {c+d x}{a+b x} \, dx}{2 b^2 d^3}+\frac {\left (421875 B^2 (b c-a d)^5 g^2\right ) \int \frac {c+d x}{a+b x} \, dx}{4 b^2 d^3}-\frac {\left (168750 B^2 (b c-a d)^5 g^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^3}+\frac {\left (140625 B^2 (b c-a d)^6 g^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^2}+\frac {\left (421875 B^2 (b c-a d)^6 g^2\right ) \int \frac {1}{c+d x} \, dx}{2 b^4 d^2}-\frac {\left (337500 B^2 (b c-a d)^6 g^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^2}+\frac {\left (140625 B^2 (b c-a d)^6 g^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}{b^4 d^3 e}+\frac {\left (421875 B^2 (b c-a d)^6 g^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 b^4 d^3 e}-\frac {\left (337500 B^2 (b c-a d)^6 g^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}{b^4 d^3 e}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {28125 B^2 (b c-a d)^5 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log (c+d x)}{2 b^4 d^3}+\frac {\left (28125 b B^2 (b c-a d)^2 g^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{d^3}+\frac {\left (140625 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{4 d^3}-\frac {\left (84375 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{d^3}+\frac {\left (46875 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{b d^3}+\frac {\left (140625 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{2 b d^3}-\frac {\left (112500 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{b d^3}+\frac {\left (140625 B^2 (b c-a d)^5 g^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{2 b^2 d^3}+\frac {\left (421875 B^2 (b c-a d)^5 g^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{4 b^2 d^3}-\frac {\left (168750 B^2 (b c-a d)^5 g^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^3}+\frac {\left (140625 B^2 (b c-a d)^6 g^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}{b^4 d^3 e}+\frac {\left (421875 B^2 (b c-a d)^6 g^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 b^4 d^3 e}-\frac {\left (337500 B^2 (b c-a d)^6 g^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}{b^4 d^3 e}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {9375 B^2 (b c-a d)^5 g^2 x}{b^3 d^2}-\frac {65625 B^2 (b c-a d)^4 g^2 (c+d x)^2}{8 b^2 d^3}-\frac {28125 B^2 (b c-a d)^3 g^2 (c+d x)^3}{4 b d^3}+\frac {28125 B^2 (b c-a d)^2 g^2 (c+d x)^4}{4 d^3}-\frac {9375 B^2 (b c-a d)^6 g^2 \log (a+b x)}{b^4 d^3}-\frac {28125 B^2 (b c-a d)^5 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log (c+d x)}{2 b^4 d^3}+\frac {\left (140625 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^3}+\frac {\left (421875 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 b^3 d^3}-\frac {\left (337500 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^3}-\frac {\left (140625 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d^2}-\frac {\left (421875 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b^4 d^2}+\frac {\left (337500 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d^2}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {9375 B^2 (b c-a d)^5 g^2 x}{b^3 d^2}-\frac {65625 B^2 (b c-a d)^4 g^2 (c+d x)^2}{8 b^2 d^3}-\frac {28125 B^2 (b c-a d)^3 g^2 (c+d x)^3}{4 b d^3}+\frac {28125 B^2 (b c-a d)^2 g^2 (c+d x)^4}{4 d^3}-\frac {9375 B^2 (b c-a d)^6 g^2 \log (a+b x)}{b^4 d^3}-\frac {28125 B^2 (b c-a d)^5 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log (c+d x)}{2 b^4 d^3}-\frac {28125 B^2 (b c-a d)^6 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d^3}+\frac {\left (140625 B^2 (b c-a d)^6 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^3}+\frac {\left (421875 B^2 (b c-a d)^6 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 d^3}-\frac {\left (337500 B^2 (b c-a d)^6 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^3}+\frac {\left (140625 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^3}+\frac {\left (421875 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 b^3 d^3}-\frac {\left (337500 B^2 (b c-a d)^6 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^3}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {9375 B^2 (b c-a d)^5 g^2 x}{b^3 d^2}-\frac {65625 B^2 (b c-a d)^4 g^2 (c+d x)^2}{8 b^2 d^3}-\frac {28125 B^2 (b c-a d)^3 g^2 (c+d x)^3}{4 b d^3}+\frac {28125 B^2 (b c-a d)^2 g^2 (c+d x)^4}{4 d^3}-\frac {9375 B^2 (b c-a d)^6 g^2 \log (a+b x)}{b^4 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log ^2(a+b x)}{4 b^4 d^3}-\frac {28125 B^2 (b c-a d)^5 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log (c+d x)}{2 b^4 d^3}-\frac {28125 B^2 (b c-a d)^6 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d^3}+\frac {\left (140625 B^2 (b c-a d)^6 g^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 )}{b^4 d^3}+\frac {\left (421875 B^2 (b c-a d)^6 g^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^4 d^3}-\frac {\left (337500 B^2 (b c-a d)^6 g^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 )}{b^4 d^3}\\ &=-\frac {28125 A B (b c-a d)^5 g^2 x}{2 b^3 d^2}-\frac {9375 B^2 (b c-a d)^5 g^2 x}{b^3 d^2}-\frac {65625 B^2 (b c-a d)^4 g^2 (c+d x)^2}{8 b^2 d^3}-\frac {28125 B^2 (b c-a d)^3 g^2 (c+d x)^3}{4 b d^3}+\frac {28125 B^2 (b c-a d)^2 g^2 (c+d x)^4}{4 d^3}-\frac {9375 B^2 (b c-a d)^6 g^2 \log (a+b x)}{b^4 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log ^2(a+b x)}{4 b^4 d^3}-\frac {28125 B^2 (b c-a d)^5 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^2}-\frac {28125 B (b c-a d)^4 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^3}-\frac {9375 B (b c-a d)^3 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b d^3}+\frac {196875 B (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {28125 b B (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {28125 B (b c-a d)^6 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 d^3}+\frac {421875 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 d^3}-\frac {168750 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {140625 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {28125 B^2 (b c-a d)^6 g^2 \log (c+d x)}{2 b^4 d^3}-\frac {28125 B^2 (b c-a d)^6 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d^3}-\frac {28125 B^2 (b c-a d)^6 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b^4 d^3}\\ \end {align*}
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Mathematica [A] time = 1.38, size = 1555, normalized size = 1.71 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{2} d^{3} g^{2} i^{3} x^{5} + A^{2} a^{2} c^{3} g^{2} i^{3} + {\left (3 \, A^{2} b^{2} c d^{2} + 2 \, A^{2} a b d^{3}\right )} g^{2} i^{3} x^{4} + {\left (3 \, A^{2} b^{2} c^{2} d + 6 \, A^{2} a b c d^{2} + A^{2} a^{2} d^{3}\right )} g^{2} i^{3} x^{3} + {\left (A^{2} b^{2} c^{3} + 6 \, A^{2} a b c^{2} d + 3 \, A^{2} a^{2} c d^{2}\right )} g^{2} i^{3} x^{2} + {\left (2 \, A^{2} a b c^{3} + 3 \, A^{2} a^{2} c^{2} d\right )} g^{2} i^{3} x + {\left (B^{2} b^{2} d^{3} g^{2} i^{3} x^{5} + B^{2} a^{2} c^{3} g^{2} i^{3} + {\left (3 \, B^{2} b^{2} c d^{2} + 2 \, B^{2} a b d^{3}\right )} g^{2} i^{3} x^{4} + {\left (3 \, B^{2} b^{2} c^{2} d + 6 \, B^{2} a b c d^{2} + B^{2} a^{2} d^{3}\right )} g^{2} i^{3} x^{3} + {\left (B^{2} b^{2} c^{3} + 6 \, B^{2} a b c^{2} d + 3 \, B^{2} a^{2} c d^{2}\right )} g^{2} i^{3} x^{2} + {\left (2 \, B^{2} a b c^{3} + 3 \, B^{2} a^{2} c^{2} d\right )} g^{2} i^{3} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B b^{2} d^{3} g^{2} i^{3} x^{5} + A B a^{2} c^{3} g^{2} i^{3} + {\left (3 \, A B b^{2} c d^{2} + 2 \, A B a b d^{3}\right )} g^{2} i^{3} x^{4} + {\left (3 \, A B b^{2} c^{2} d + 6 \, A B a b c d^{2} + A B a^{2} d^{3}\right )} g^{2} i^{3} x^{3} + {\left (A B b^{2} c^{3} + 6 \, A B a b c^{2} d + 3 \, A B a^{2} c d^{2}\right )} g^{2} i^{3} x^{2} + {\left (2 \, A B a b c^{3} + 3 \, A B a^{2} c^{2} d\right )} g^{2} i^{3} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right ), x\right ) \]
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 = 2.81, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.86, size = 5196, normalized size = 5.72 \[ \text {result too large to display} \]
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 \[ \int {\left (a\,g+b\,g\,x\right )}^2\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \]
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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