Optimal. Leaf size=483 \[ -\frac {B d^4 n (a+b x)^2}{4 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {4 b B d^3 n (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 B d n (c+d x)}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B n (c+d x)^2}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {d^4 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^3 i^3}-\frac {3 b^2 B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^3 i^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 483, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2561, 45, 2372,
2338} \begin {gather*} -\frac {b^4 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac {6 b^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}+\frac {d^4 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac {4 b d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac {b^4 B n (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 B d n (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac {3 b^2 B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}-\frac {B d^4 n (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5}+\frac {4 b B d^3 n (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2338
Rule 2372
Rule 2561
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(157 c+157 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (a+b x)^3}-\frac {3 b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)^2}+\frac {6 b^3 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (c+d x)^3}-\frac {3 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)^2}-\frac {6 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (6 b^3 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^3 d\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3869893 (b c-a d)^4 g^3}-\frac {\left (3 b d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac {b^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3869893 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3869893 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2 n\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}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac {\left (b^2 B n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^3 g^3}+\frac {\left (b^2 B n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^2 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\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}{3869893 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2 n\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3869893 (b c-a d)^3 g^3}+\frac {\left (b^2 B n\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}{7739786 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7739786 (b c-a d)^2 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.81, size = 561, normalized size = 1.16 \begin {gather*} -\frac {\frac {b^2 B (b c-a d)^2 n}{(a+b x)^2}-\frac {12 b^3 B c d n}{a+b x}+\frac {12 a b^2 B d^2 n}{a+b x}-\frac {2 b^2 B d (b c-a d) n}{a+b x}+\frac {B d^2 (b c-a d)^2 n}{(c+d x)^2}+\frac {12 b^2 B c d^2 n}{c+d x}-\frac {12 a b B d^3 n}{c+d x}+\frac {2 b B d^2 (b c-a d) n}{c+d x}+\frac {2 b^2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}-\frac {12 b^2 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}-\frac {2 d^2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2}-\frac {12 b d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}-24 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+12 b^2 B d^2 n \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 )-12 b^2 B d^2 n \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 )}{4 (b c-a d)^5 g^3 i^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2300 vs. \(2 (450) = 900\).
time = 0.55, size = 2300, normalized size = 4.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1265 vs. \(2 (450) = 900\).
time = 0.41, size = 1265, normalized size = 2.62 \begin {gather*} \frac {2 \, {\left (-i \, A - i \, B\right )} b^{4} c^{4} + 16 \, {\left (i \, A + i \, B\right )} a b^{3} c^{3} d + 16 \, {\left (-i \, A - i \, B\right )} a^{3} b c d^{3} + 2 \, {\left (i \, A + i \, B\right )} a^{4} d^{4} + 24 \, {\left ({\left (i \, A + i \, B\right )} b^{4} c d^{3} + {\left (-i \, A - i \, B\right )} a b^{3} d^{4}\right )} x^{3} + 12 \, {\left (3 \, {\left (i \, A + i \, B\right )} b^{4} c^{2} d^{2} + 3 \, {\left (-i \, A - i \, B\right )} a^{2} b^{2} d^{4} + {\left (i \, B b^{4} c^{2} d^{2} - 2 i \, B a b^{3} c d^{3} + i \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + 12 \, {\left (i \, B b^{4} d^{4} n x^{4} + i \, B a^{2} b^{2} c^{2} d^{2} n + 2 \, {\left (i \, B b^{4} c d^{3} + i \, B a b^{3} d^{4}\right )} n x^{3} + {\left (i \, B b^{4} c^{2} d^{2} + 4 i \, B a b^{3} c d^{3} + i \, B a^{2} b^{2} d^{4}\right )} n x^{2} + 2 \, {\left (i \, B a b^{3} c^{2} d^{2} + i \, B a^{2} b^{2} c d^{3}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (i \, B b^{4} c^{4} - 16 i \, B a b^{3} c^{3} d + 30 i \, B a^{2} b^{2} c^{2} d^{2} - 16 i \, B a^{3} b c d^{3} + i \, B a^{4} d^{4}\right )} n + 4 \, {\left (2 \, {\left (i \, A + i \, B\right )} b^{4} c^{3} d + 12 \, {\left (i \, A + i \, B\right )} a b^{3} c^{2} d^{2} + 12 \, {\left (-i \, A - i \, B\right )} a^{2} b^{2} c d^{3} + 2 \, {\left (-i \, A - i \, B\right )} a^{3} b d^{4} + 3 \, {\left (i \, B b^{4} c^{3} d - i \, B a b^{3} c^{2} d^{2} - i \, B a^{2} b^{2} c d^{3} + i \, B a^{3} b d^{4}\right )} n\right )} x + 2 \, {\left (12 \, {\left (i \, A + i \, B\right )} b^{4} d^{4} x^{4} + 12 \, {\left (i \, A + i \, B\right )} a^{2} b^{2} c^{2} d^{2} + 12 \, {\left (2 \, {\left (i \, A + i \, B\right )} b^{4} c d^{3} + 2 \, {\left (i \, A + i \, B\right )} a b^{3} d^{4} + {\left (i \, B b^{4} c d^{3} - i \, B a b^{3} d^{4}\right )} n\right )} x^{3} + 6 \, {\left (2 \, {\left (i \, A + i \, B\right )} b^{4} c^{2} d^{2} + 8 \, {\left (i \, A + i \, B\right )} a b^{3} c d^{3} + 2 \, {\left (i \, A + i \, B\right )} a^{2} b^{2} d^{4} + 3 \, {\left (i \, B b^{4} c^{2} d^{2} - i \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + {\left (-i \, B b^{4} c^{4} + 8 i \, B a b^{3} c^{3} d - 8 i \, B a^{3} b c d^{3} + i \, B a^{4} d^{4}\right )} n + 4 \, {\left (6 \, {\left (i \, A + i \, B\right )} a b^{3} c^{2} d^{2} + 6 \, {\left (i \, A + i \, B\right )} a^{2} b^{2} c d^{3} + {\left (i \, B b^{4} c^{3} d + 6 i \, B a b^{3} c^{2} d^{2} - 6 i \, B a^{2} b^{2} c d^{3} - i \, B a^{3} b d^{4}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} g^{3} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} g^{3} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} g^{3} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} g^{3} x + {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5}\right )} g^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 7.93, size = 1341, normalized size = 2.78 \begin {gather*} \frac {\frac {2\,x\,\left (2\,A\,a^2\,b\,d^3+2\,A\,b^3\,c^2\,d+14\,A\,a\,b^2\,c\,d^2-3\,B\,a^2\,b\,d^3\,n+3\,B\,b^3\,c^2\,d\,n\right )}{a\,d-b\,c}-\frac {2\,A\,a^3\,d^3+2\,A\,b^3\,c^3-B\,a^3\,d^3\,n+B\,b^3\,c^3\,n-14\,A\,a\,b^2\,c^2\,d-14\,A\,a^2\,b\,c\,d^2-15\,B\,a\,b^2\,c^2\,d\,n+15\,B\,a^2\,b\,c\,d^2\,n}{2\,\left (a\,d-b\,c\right )}+\frac {6\,x^2\,\left (3\,A\,a\,b^2\,d^3+3\,A\,b^3\,c\,d^2-B\,a\,b^2\,d^3\,n+B\,b^3\,c\,d^2\,n\right )}{a\,d-b\,c}+\frac {12\,A\,b^3\,d^3\,x^3}{a\,d-b\,c}}{x^4\,\left (2\,a^3\,b^2\,d^5\,g^3\,i^3-6\,a^2\,b^3\,c\,d^4\,g^3\,i^3+6\,a\,b^4\,c^2\,d^3\,g^3\,i^3-2\,b^5\,c^3\,d^2\,g^3\,i^3\right )-x\,\left (-4\,a^5\,c\,d^4\,g^3\,i^3+8\,a^4\,b\,c^2\,d^3\,g^3\,i^3-8\,a^2\,b^3\,c^4\,d\,g^3\,i^3+4\,a\,b^4\,c^5\,g^3\,i^3\right )+x^3\,\left (4\,a^4\,b\,d^5\,g^3\,i^3-8\,a^3\,b^2\,c\,d^4\,g^3\,i^3+8\,a\,b^4\,c^3\,d^2\,g^3\,i^3-4\,b^5\,c^4\,d\,g^3\,i^3\right )+x^2\,\left (2\,a^5\,d^5\,g^3\,i^3+2\,a^4\,b\,c\,d^4\,g^3\,i^3-16\,a^3\,b^2\,c^2\,d^3\,g^3\,i^3+16\,a^2\,b^3\,c^3\,d^2\,g^3\,i^3-2\,a\,b^4\,c^4\,d\,g^3\,i^3-2\,b^5\,c^5\,g^3\,i^3\right )-2\,a^2\,b^3\,c^5\,g^3\,i^3+2\,a^5\,c^2\,d^3\,g^3\,i^3+6\,a^3\,b^2\,c^4\,d\,g^3\,i^3-6\,a^4\,b\,c^3\,d^2\,g^3\,i^3}+\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (x\,\left (\frac {3\,B\,b\,d\,{\left (a\,d+b\,c\right )}^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}-\frac {B\,b\,d}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {6\,B\,a\,b^2\,c\,d^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )-\frac {B\,\left (a\,d+b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {6\,B\,b^3\,d^3\,x^3}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {9\,B\,b^2\,d^2\,x^2\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {3\,B\,a\,b\,c\,d\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )}{x\,\left (2\,d\,a^2\,c\,g^3\,i^3+2\,b\,a\,c^2\,g^3\,i^3\right )+x^3\,\left (2\,c\,b^2\,d\,g^3\,i^3+2\,a\,b\,d^2\,g^3\,i^3\right )+x^2\,\left (a^2\,d^2\,g^3\,i^3+4\,a\,b\,c\,d\,g^3\,i^3+b^2\,c^2\,g^3\,i^3\right )+a^2\,c^2\,g^3\,i^3+b^2\,d^2\,g^3\,i^3\,x^4}-\frac {3\,B\,b^2\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g^3\,i^3\,n\,{\left (a\,d-b\,c\right )}^5}+\frac {A\,b^2\,d^2\,\mathrm {atan}\left (\frac {\left (a^5\,d^5\,g^3\,i^3-3\,a^4\,b\,c\,d^4\,g^3\,i^3+2\,a^3\,b^2\,c^2\,d^3\,g^3\,i^3+2\,a^2\,b^3\,c^3\,d^2\,g^3\,i^3-3\,a\,b^4\,c^4\,d\,g^3\,i^3+b^5\,c^5\,g^3\,i^3\right )\,1{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}+\frac {b\,d\,x\,\left (a^4\,d^4\,g^3\,i^3-4\,a^3\,b\,c\,d^3\,g^3\,i^3+6\,a^2\,b^2\,c^2\,d^2\,g^3\,i^3-4\,a\,b^3\,c^3\,d\,g^3\,i^3+b^4\,c^4\,g^3\,i^3\right )\,2{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}\right )\,12{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________