Optimal. Leaf size=429 \[ -\frac {8 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {2 b B^2 d (c+d x)^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {8 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}-\frac {4 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}+\frac {2 b B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {4 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2550, 2395,
2342, 2341} \begin {gather*} -\frac {b^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^3}-\frac {4 b^2 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac {d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^4 (a+b x) (b c-a d)^3}-\frac {4 B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}+\frac {b d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^4 (a+b x)^2 (b c-a d)^3}+\frac {2 b B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac {8 b^2 B^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}-\frac {8 B^2 d^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac {2 b B^2 d (c+d x)^2}{g^4 (a+b x)^2 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2342
Rule 2395
Rule 2550
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(2 B) \int \frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(4 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(4 B (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\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)^2}{(c+d x)^2}\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)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(4 B) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^4} \, dx}{3 g^4}-\frac {\left (4 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac {\left (4 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (4 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac {(4 B d) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^3} \, dx}{3 (b c-a d) g^4}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {\left (4 B^2\right ) \int \frac {2 (b c-a d)}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac {\left (4 B^2 d^3\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{3 b (b c-a d)^3 g^4}-\frac {\left (4 B^2 d^3\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (4 B^2 d^2\right ) \int \frac {2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)^2 g^4}-\frac {\left (2 B^2 d\right ) \int \frac {2 (b c-a d)}{(a+b x)^3 (c+d x)} \, dx}{3 b (b c-a d) g^4}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {\left (4 B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b g^4}+\frac {\left (8 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d) g^4}+\frac {\left (8 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac {\left (4 B^2 d^3\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{3 b (b c-a d)^3 e g^4}-\frac {\left (4 B^2 d^3\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{3 b (b c-a d)^3 e g^4}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {\left (4 B^2 d\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 g^4}+\frac {\left (8 B^2 d^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 (b c-a d) g^4}+\frac {\left (8 B^2 (b c-a d)\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 g^4}+\frac {\left (4 B^2 d^3\right ) \int \left (\frac {2 b e \log (a+b x)}{a+b x}-\frac {2 d e \log (a+b x)}{c+d x}\right ) \, dx}{3 b (b c-a d)^3 e g^4}-\frac {\left (4 B^2 d^3\right ) \int \left (\frac {2 b e \log (c+d x)}{a+b x}-\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b (b c-a d)^3 e g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (8 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (8 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 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 c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 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 (b c-a d)^3 g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 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 (b c-a d)^3 g^4}+\frac {\left (8 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 (b c-a d)^3 g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 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 = 0.42, size = 598, normalized size = 1.39 \begin {gather*} -\frac {9 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {2 B \left (6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-9 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+18 d^2 (b c-a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+18 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-18 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\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 )}{(b c-a d)^3}}{27 b 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. \(1018\) vs.
\(2(423)=846\).
time = 0.90, size = 1019, normalized size = 2.38 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. 1584 vs.
\(2 (429) = 858\).
time = 0.45, size = 1584, normalized size = 3.69 \begin {gather*} -\frac {2}{27} \, {\left (3 \, {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {4 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 108 \, a^{2} b c d^{2} - 85 \, a^{3} d^{3} + 66 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 18 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )^{2} - 18 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )^{2} - 3 \, {\left (5 \, b^{3} c^{2} d - 54 \, a b^{2} c d^{2} + 49 \, a^{2} b d^{3}\right )} x + 66 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (11 \, b^{3} d^{3} x^{3} + 33 \, a b^{2} d^{3} x^{2} + 33 \, a^{2} b d^{3} x + 11 \, a^{3} d^{3} - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{3} b^{4} c^{3} g^{4} - 3 \, a^{4} b^{3} c^{2} d g^{4} + 3 \, a^{5} b^{2} c d^{2} g^{4} - a^{6} b d^{3} g^{4} + {\left (b^{7} c^{3} g^{4} - 3 \, a b^{6} c^{2} d g^{4} + 3 \, a^{2} b^{5} c d^{2} g^{4} - a^{3} b^{4} d^{3} g^{4}\right )} x^{3} + 3 \, {\left (a b^{6} c^{3} g^{4} - 3 \, a^{2} b^{5} c^{2} d g^{4} + 3 \, a^{3} b^{4} c d^{2} g^{4} - a^{4} b^{3} d^{3} g^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} g^{4} - 3 \, a^{3} b^{4} c^{2} d g^{4} + 3 \, a^{4} b^{3} c d^{2} g^{4} - a^{5} b^{2} d^{3} g^{4}\right )} x}\right )} B^{2} - \frac {2}{9} \, A B {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {3 \, \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {B^{2} \log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac {A^{2}}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 715, normalized size = 1.67 \begin {gather*} -\frac {{\left (9 \, A^{2} + 12 \, A B + 8 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b^{2} c^{2} d + 27 \, {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a^{2} b c d^{2} - {\left (9 \, A^{2} + 66 \, A B + 170 \, B^{2}\right )} a^{3} d^{3} + 12 \, {\left ({\left (3 \, A B + 11 \, B^{2}\right )} b^{3} c d^{2} - {\left (3 \, A B + 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} - 6 \, {\left ({\left (3 \, A B + 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (A B + 3 \, B^{2}\right )} a b^{2} c d^{2} + {\left (15 \, A B + 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (3 \, A B + 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + {\left (3 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 9 \, {\left (A B + B^{2}\right )} a b^{2} c^{2} d + 9 \, {\left (A B + 2 \, B^{2}\right )} a^{2} b c d^{2} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (A B + 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} - 3 \, {\left (A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{27 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\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. 1561 vs.
\(2 (406) = 812\).
time = 16.28, size = 1561, normalized size = 3.64 \begin {gather*} - \frac {4 B d^{3} \cdot \left (3 A + 11 B\right ) \log {\left (x + \frac {12 A B a d^{4} + 12 A B b c d^{3} + 44 B^{2} a d^{4} + 44 B^{2} b c d^{3} - \frac {4 B a^{4} d^{7} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {16 B a^{3} b c d^{6} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {24 B a^{2} b^{2} c^{2} d^{5} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {16 B a b^{3} c^{3} d^{4} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 B b^{4} c^{4} d^{3} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}}}{24 A B b d^{4} + 88 B^{2} b d^{4}} \right )}}{9 b g^{4} \left (a d - b c\right )^{3}} + \frac {4 B d^{3} \cdot \left (3 A + 11 B\right ) \log {\left (x + \frac {12 A B a d^{4} + 12 A B b c d^{3} + 44 B^{2} a d^{4} + 44 B^{2} b c d^{3} + \frac {4 B a^{4} d^{7} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {16 B a^{3} b c d^{6} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {24 B a^{2} b^{2} c^{2} d^{5} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {16 B a b^{3} c^{3} d^{4} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 B b^{4} c^{4} d^{3} \cdot \left (3 A + 11 B\right )}{\left (a d - b c\right )^{3}}}{24 A B b d^{4} + 88 B^{2} b d^{4}} \right )}}{9 b g^{4} \left (a d - b c\right )^{3}} + \frac {\left (3 B^{2} a^{2} c d^{2} + 3 B^{2} a^{2} d^{3} x - 3 B^{2} a b c^{2} d + 3 B^{2} a b d^{3} x^{2} + B^{2} b^{2} c^{3} + B^{2} b^{2} d^{3} x^{3}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{3 a^{6} d^{3} g^{4} - 9 a^{5} b c d^{2} g^{4} + 9 a^{5} b d^{3} g^{4} x + 9 a^{4} b^{2} c^{2} d g^{4} - 27 a^{4} b^{2} c d^{2} g^{4} x + 9 a^{4} b^{2} d^{3} g^{4} x^{2} - 3 a^{3} b^{3} c^{3} g^{4} + 27 a^{3} b^{3} c^{2} d g^{4} x - 27 a^{3} b^{3} c d^{2} g^{4} x^{2} + 3 a^{3} b^{3} d^{3} g^{4} x^{3} - 9 a^{2} b^{4} c^{3} g^{4} x + 27 a^{2} b^{4} c^{2} d g^{4} x^{2} - 9 a^{2} b^{4} c d^{2} g^{4} x^{3} - 9 a b^{5} c^{3} g^{4} x^{2} + 9 a b^{5} c^{2} d g^{4} x^{3} - 3 b^{6} c^{3} g^{4} x^{3}} + \frac {\left (- 6 A B a^{2} d^{2} + 12 A B a b c d - 6 A B b^{2} c^{2} - 22 B^{2} a^{2} d^{2} + 14 B^{2} a b c d - 30 B^{2} a b d^{2} x - 4 B^{2} b^{2} c^{2} + 6 B^{2} b^{2} c d x - 12 B^{2} b^{2} d^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{9 a^{5} b d^{2} g^{4} - 18 a^{4} b^{2} c d g^{4} + 27 a^{4} b^{2} d^{2} g^{4} x + 9 a^{3} b^{3} c^{2} g^{4} - 54 a^{3} b^{3} c d g^{4} x + 27 a^{3} b^{3} d^{2} g^{4} x^{2} + 27 a^{2} b^{4} c^{2} g^{4} x - 54 a^{2} b^{4} c d g^{4} x^{2} + 9 a^{2} b^{4} d^{2} g^{4} x^{3} + 27 a b^{5} c^{2} g^{4} x^{2} - 18 a b^{5} c d g^{4} x^{3} + 9 b^{6} c^{2} g^{4} x^{3}} - \frac {9 A^{2} a^{2} d^{2} - 18 A^{2} a b c d + 9 A^{2} b^{2} c^{2} + 66 A B a^{2} d^{2} - 42 A B a b c d + 12 A B b^{2} c^{2} + 170 B^{2} a^{2} d^{2} - 46 B^{2} a b c d + 8 B^{2} b^{2} c^{2} + x^{2} \cdot \left (36 A B b^{2} d^{2} + 132 B^{2} b^{2} d^{2}\right ) + x \left (90 A B a b d^{2} - 18 A B b^{2} c d + 294 B^{2} a b d^{2} - 30 B^{2} b^{2} c d\right )}{27 a^{5} b d^{2} g^{4} - 54 a^{4} b^{2} c d g^{4} + 27 a^{3} b^{3} c^{2} g^{4} + x^{3} \cdot \left (27 a^{2} b^{4} d^{2} g^{4} - 54 a b^{5} c d g^{4} + 27 b^{6} c^{2} g^{4}\right ) + x^{2} \cdot \left (81 a^{3} b^{3} d^{2} g^{4} - 162 a^{2} b^{4} c d g^{4} + 81 a b^{5} c^{2} g^{4}\right ) + x \left (81 a^{4} b^{2} d^{2} g^{4} - 162 a^{3} b^{3} c d g^{4} + 81 a^{2} b^{4} c^{2} g^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.67, size = 1069, normalized size = 2.49 \begin {gather*} \frac {\frac {9\,A^2\,a^2\,d^2-18\,A^2\,a\,b\,c\,d+9\,A^2\,b^2\,c^2+66\,A\,B\,a^2\,d^2-42\,A\,B\,a\,b\,c\,d+12\,A\,B\,b^2\,c^2+170\,B^2\,a^2\,d^2-46\,B^2\,a\,b\,c\,d+8\,B^2\,b^2\,c^2}{3\,\left (a\,d-b\,c\right )}+\frac {2\,x\,\left (-5\,c\,B^2\,b^2\,d+49\,a\,B^2\,b\,d^2-3\,A\,c\,B\,b^2\,d+15\,A\,a\,B\,b\,d^2\right )}{a\,d-b\,c}+\frac {4\,d\,x^2\,\left (11\,d\,B^2\,b^2+3\,A\,d\,B\,b^2\right )}{a\,d-b\,c}}{x\,\left (27\,a^2\,b^3\,c\,g^4-27\,a^3\,b^2\,d\,g^4\right )-x^2\,\left (27\,a^2\,b^3\,d\,g^4-27\,a\,b^4\,c\,g^4\right )+x^3\,\left (9\,b^5\,c\,g^4-9\,a\,b^4\,d\,g^4\right )+9\,a^3\,b^2\,c\,g^4-9\,a^4\,b\,d\,g^4}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{3\,b^2\,g^4\,\left (3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2\right )}-\frac {B^2\,d^3}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {2\,A\,B}{3\,b^2\,d\,g^4}+\frac {2\,B^2\,d^3\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,b\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {2\,\left (3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{3\,b\,d^4}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {2\,B^2\,d^3\,x^2\,\left (\frac {2\,\left (b^2\,c-a\,b\,d\right )}{3\,d^2}-\frac {4\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {2\,B^2\,d^3\,x\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,b\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {2\,\left (3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{3\,d^3}+\frac {4\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\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\,\mathrm {atan}\left (\frac {B\,d^3\,\left (\frac {a^3\,b\,d^3\,g^4-a^2\,b^2\,c\,d^2\,g^4-a\,b^3\,c^2\,d\,g^4+b^4\,c^3\,g^4}{a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4}+2\,b\,d\,x\right )\,\left (3\,A+11\,B\right )\,\left (a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4\right )\,4{}\mathrm {i}}{b\,g^4\,{\left (a\,d-b\,c\right )}^3\,\left (44\,B^2\,d^3+12\,A\,B\,d^3\right )}\right )\,\left (3\,A+11\,B\right )\,8{}\mathrm {i}}{9\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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