Optimal. Leaf size=147 \[ -\frac {2 B^2 i^2 (c+d x)^3}{27 (b c-a d) g^4 (a+b x)^3}-\frac {2 B i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d) g^4 (a+b x)^3} \]
[Out]
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Rubi [A]
time = 0.12, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2562, 2342,
2341} \begin {gather*} -\frac {i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {2 B i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)}-\frac {2 B^2 i^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2342
Rule 2562
Rubi steps
\begin {align*} \int \frac {(71 c+71 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx &=\int \left (\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^4}+\frac {10082 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^3}+\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac {\left (5041 d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^2 g^4}+\frac {(10082 d (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b^2 g^4}+\frac {\left (5041 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {(10082 B d (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B (b c-a d)^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2 (b c-a d)\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac {\left (10082 B d (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac {\left (10082 B (b c-a d)^3\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3 b^2 g^4}-\frac {\left (10082 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}+\frac {\left (10082 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}-\frac {(10082 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 b^2 g^4}+\frac {(10082 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^4}+\frac {\left (10082 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{3 b^2 g^4}\\ &=-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) g^4}-\frac {\left (10082 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{3 b^3 (b c-a d) g^4}-\frac {\left (5041 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (5041 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B^2 (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}\\ &=-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}-\frac {\left (5041 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (5041 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}+\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) e g^4}-\frac {\left (10082 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^3 (b c-a d) e g^4}\\ &=-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^2 (b c-a d)\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^3 g^4}-\frac {\left (5041 B^2 d (b c-a d)^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^3 g^4}+\frac {\left (5041 B^2 d (b c-a d)^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}{b^3 g^4}+\frac {\left (10082 B^2 (b c-a d)^3\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^3 g^4}+\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) e g^4}-\frac {\left (10082 B^2 d^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b^3 (b c-a d) e g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (10082 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (10082 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}+\frac {\left (10082 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d) g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 1.23, size = 1355, normalized size = 9.22 \begin {gather*} -\frac {i^2 \left (18 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+54 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-54 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+54 B d^2 (a+b x)^2 \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \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)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\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 )+27 B d (a+b x) \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (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 d^2 (a+b x)^2 \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)}{-b c+a d}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\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 )+B \left (12 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-18 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+36 d^2 (b c-a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+36 B d^2 (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 (a+b x) \left ((b c-a d)^2+2 d (-b c+a d) (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 \left (2 (b c-a d)^3-3 d (b c-a d)^2 (a+b x)+6 d^2 (b c-a d) (a+b x)^2+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 (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)}{-b c+a d}\right )\right )+18 B d^3 (a+b x)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\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 )}{54 b^3 (b c-a d) g^4 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs.
\(2(141)=282\).
time = 0.70, size = 367, normalized size = 2.50 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 5514 vs.
\(2 (134) = 268\).
time = 0.83, size = 5514, normalized size = 37.51 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 407 vs.
\(2 (134) = 268\).
time = 0.37, size = 407, normalized size = 2.77 \begin {gather*} \frac {{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{3} d^{3} + 3 \, {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c d^{2} - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} b^{3} c d^{2} x^{2} + 3 \, B^{2} b^{3} c^{2} d x + B^{2} b^{3} c^{3}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} + 3 \, {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{2} d - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (3 \, A B + B^{2}\right )} b^{3} d^{3} x^{3} + 3 \, {\left (3 \, A B + B^{2}\right )} b^{3} c d^{2} x^{2} + 3 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{2} d x + {\left (3 \, A B + B^{2}\right )} b^{3} c^{3}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{27 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1182 vs.
\(2 (131) = 262\).
time = 35.43, size = 1182, normalized size = 8.04 \begin {gather*} - \frac {2 B d^{3} i^{2} \cdot \left (3 A + B\right ) \log {\left (x + \frac {6 A B a d^{4} i^{2} + 6 A B b c d^{3} i^{2} + 2 B^{2} a d^{4} i^{2} + 2 B^{2} b c d^{3} i^{2} - \frac {2 B a^{2} d^{5} i^{2} \cdot \left (3 A + B\right )}{a d - b c} + \frac {4 B a b c d^{4} i^{2} \cdot \left (3 A + B\right )}{a d - b c} - \frac {2 B b^{2} c^{2} d^{3} i^{2} \cdot \left (3 A + B\right )}{a d - b c}}{12 A B b d^{4} i^{2} + 4 B^{2} b d^{4} i^{2}} \right )}}{9 b^{3} g^{4} \left (a d - b c\right )} + \frac {2 B d^{3} i^{2} \cdot \left (3 A + B\right ) \log {\left (x + \frac {6 A B a d^{4} i^{2} + 6 A B b c d^{3} i^{2} + 2 B^{2} a d^{4} i^{2} + 2 B^{2} b c d^{3} i^{2} + \frac {2 B a^{2} d^{5} i^{2} \cdot \left (3 A + B\right )}{a d - b c} - \frac {4 B a b c d^{4} i^{2} \cdot \left (3 A + B\right )}{a d - b c} + \frac {2 B b^{2} c^{2} d^{3} i^{2} \cdot \left (3 A + B\right )}{a d - b c}}{12 A B b d^{4} i^{2} + 4 B^{2} b d^{4} i^{2}} \right )}}{9 b^{3} g^{4} \left (a d - b c\right )} + \frac {\left (B^{2} c^{3} i^{2} + 3 B^{2} c^{2} d i^{2} x + 3 B^{2} c d^{2} i^{2} x^{2} + B^{2} d^{3} i^{2} x^{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{3 a^{4} d g^{4} - 3 a^{3} b c g^{4} + 9 a^{3} b d g^{4} x - 9 a^{2} b^{2} c g^{4} x + 9 a^{2} b^{2} d g^{4} x^{2} - 9 a b^{3} c g^{4} x^{2} + 3 a b^{3} d g^{4} x^{3} - 3 b^{4} c g^{4} x^{3}} + \frac {- 9 A^{2} a^{2} d^{2} i^{2} - 9 A^{2} a b c d i^{2} - 9 A^{2} b^{2} c^{2} i^{2} - 6 A B a^{2} d^{2} i^{2} - 6 A B a b c d i^{2} - 6 A B b^{2} c^{2} i^{2} - 2 B^{2} a^{2} d^{2} i^{2} - 2 B^{2} a b c d i^{2} - 2 B^{2} b^{2} c^{2} i^{2} + x^{2} \left (- 27 A^{2} b^{2} d^{2} i^{2} - 18 A B b^{2} d^{2} i^{2} - 6 B^{2} b^{2} d^{2} i^{2}\right ) + x \left (- 27 A^{2} a b d^{2} i^{2} - 27 A^{2} b^{2} c d i^{2} - 18 A B a b d^{2} i^{2} - 18 A B b^{2} c d i^{2} - 6 B^{2} a b d^{2} i^{2} - 6 B^{2} b^{2} c d i^{2}\right )}{27 a^{3} b^{3} g^{4} + 81 a^{2} b^{4} g^{4} x + 81 a b^{5} g^{4} x^{2} + 27 b^{6} g^{4} x^{3}} + \frac {\left (- 6 A B a^{2} d^{2} i^{2} - 6 A B a b c d i^{2} - 18 A B a b d^{2} i^{2} x - 6 A B b^{2} c^{2} i^{2} - 18 A B b^{2} c d i^{2} x - 18 A B b^{2} d^{2} i^{2} x^{2} - 2 B^{2} a^{2} d^{2} i^{2} - 2 B^{2} a b c d i^{2} - 6 B^{2} a b d^{2} i^{2} x - 2 B^{2} b^{2} c^{2} i^{2} - 6 B^{2} b^{2} c d i^{2} x - 6 B^{2} b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{9 a^{3} b^{3} g^{4} + 27 a^{2} b^{4} g^{4} x + 27 a b^{5} g^{4} x^{2} + 9 b^{6} g^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.90, size = 180, normalized size = 1.22 \begin {gather*} \frac {{\left (9 \, B^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 18 \, A B e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 6 \, B^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 9 \, A^{2} e^{4} + 6 \, A B e^{4} + 2 \, B^{2} e^{4}\right )} {\left (d x + c\right )}^{3} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{27 \, {\left (b x e + a e\right )}^{3} g^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.41, size = 1153, normalized size = 7.84 \begin {gather*} -\frac {x^2\,\left (9\,A^2\,b^2\,d^2\,i^2+6\,A\,B\,b^2\,d^2\,i^2+2\,B^2\,b^2\,d^2\,i^2\right )+x\,\left (9\,c\,A^2\,b^2\,d\,i^2+9\,a\,A^2\,b\,d^2\,i^2+6\,c\,A\,B\,b^2\,d\,i^2+6\,a\,A\,B\,b\,d^2\,i^2+2\,c\,B^2\,b^2\,d\,i^2+2\,a\,B^2\,b\,d^2\,i^2\right )+3\,A^2\,a^2\,d^2\,i^2+3\,A^2\,b^2\,c^2\,i^2+\frac {2\,B^2\,a^2\,d^2\,i^2}{3}+\frac {2\,B^2\,b^2\,c^2\,i^2}{3}+2\,A\,B\,a^2\,d^2\,i^2+2\,A\,B\,b^2\,c^2\,i^2+3\,A^2\,a\,b\,c\,d\,i^2+\frac {2\,B^2\,a\,b\,c\,d\,i^2}{3}+2\,A\,B\,a\,b\,c\,d\,i^2}{9\,a^3\,b^3\,g^4+27\,a^2\,b^4\,g^4\,x+27\,a\,b^5\,g^4\,x^2+9\,b^6\,g^4\,x^3}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {x\,\left (b\,\left (\frac {B^2\,c\,d\,i^2}{3\,b^3\,g^4}+\frac {B^2\,a\,d^2\,i^2}{3\,b^4\,g^4}\right )+\frac {2\,B^2\,c\,d\,i^2}{3\,b^2\,g^4}+\frac {2\,B^2\,a\,d^2\,i^2}{3\,b^3\,g^4}\right )+a\,\left (\frac {B^2\,c\,d\,i^2}{3\,b^3\,g^4}+\frac {B^2\,a\,d^2\,i^2}{3\,b^4\,g^4}\right )+\frac {B^2\,c^2\,i^2}{3\,b^2\,g^4}+\frac {B^2\,d^2\,i^2\,x^2}{b^2\,g^4}}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B^2\,d^3\,i^2}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (b\,\left (\frac {B\,i^2\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^4\,g^4}+\frac {2\,A\,B\,a\,d\,i^2}{3\,b^4\,g^4}\right )+\frac {2\,B\,i^2\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^3\,g^4}+\frac {2\,B^2\,d^3\,i^2\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}+\frac {4\,A\,B\,a\,d\,i^2}{3\,b^3\,g^4}\right )+x^2\,\left (\frac {2\,A\,B\,d\,i^2}{b^2\,g^4}-\frac {2\,B^2\,d^3\,i^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )+a\,\left (\frac {B\,i^2\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^4\,g^4}+\frac {2\,A\,B\,a\,d\,i^2}{3\,b^4\,g^4}\right )+\frac {2\,B\,i^2\,\left (-B\,a^2\,d^2+B\,a\,b\,c\,d+A\,b^2\,c^2\right )}{3\,b^4\,d\,g^4}+\frac {2\,B^2\,d^3\,i^2\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,i^2\,\mathrm {atan}\left (\frac {\left (\frac {9\,c\,b^4\,g^4+9\,a\,d\,b^3\,g^4}{9\,b^3\,g^4}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (3\,A+B\right )\,4{}\mathrm {i}}{9\,b^3\,g^4\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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