Optimal. Leaf size=380 \[ \frac {B d^3 n (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 B d n (c+d x)}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 B n (c+d x)^2}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2}-\frac {d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^3 i^2}-\frac {3 b B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^3 i^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2561, 45, 2372,
12, 14, 2338} \begin {gather*} -\frac {b^3 (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^2 (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 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^2 (a+b x) (b c-a d)^4}-\frac {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^2 (c+d x) (b c-a d)^4}+\frac {3 b 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^2 (b c-a d)^4}-\frac {b^3 B n (c+d x)^2}{4 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 B d n (c+d x)}{g^3 i^2 (a+b x) (b c-a d)^4}+\frac {B d^3 n (a+b x)}{g^3 i^2 (c+d x) (b c-a d)^4}-\frac {3 b B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i^2 (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
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 )}{(149 c+149 d x)^2 (a g+b g x)^3} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^2 g^3 (a+b x)^3}-\frac {2 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 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 )}{22201 (b c-a d)^3 g^3 (c+d x)^2}-\frac {3 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (3 b^2 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}{22201 (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} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (2 b^2 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}{22201 (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)^2} \, dx}{22201 (b c-a d)^3 g^3}+\frac {b^2 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{22201 (b c-a d)^2 g^3}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 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 )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b 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}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b 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}{22201 (b c-a d)^4 g^3}-\frac {(2 b B d n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{22201 (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)^2} \, dx}{22201 (b c-a d)^3 g^3}+\frac {(b B n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{44402 (b c-a d)^2 g^3}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 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 )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b 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}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b 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}{22201 (b c-a d)^4 g^3}-\frac {(2 b B d n) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{22201 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{22201 (b c-a d)^2 g^3}+\frac {(b B n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{44402 (b c-a d) g^3}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 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 )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b^2 B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b^2 B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {(2 b B d n) \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}{22201 (b c-a d)^2 g^3}-\frac {\left (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}{22201 (b c-a d)^2 g^3}+\frac {(b B n) \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}{44402 (b c-a d) g^3}\\ &=-\frac {b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac {5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 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 )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b^2 B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}\\ &=-\frac {b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac {5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(a+b x)}{44402 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 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 )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b 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 )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b 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 )}{22201 (b c-a d)^4 g^3}\\ &=-\frac {b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac {5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(a+b x)}{44402 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 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 )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.49, size = 478, normalized size = 1.26 \begin {gather*} \frac {-\frac {b B (b c-a d)^2 n}{(a+b x)^2}+\frac {8 b^2 B c d n}{a+b x}-\frac {8 a b B d^2 n}{a+b x}+\frac {2 b B d (b c-a d) n}{a+b x}-\frac {4 b B c d^2 n}{c+d x}+\frac {4 a B d^3 n}{c+d x}+6 b B d^2 n \log (a+b x)-\frac {2 b (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 {8 b 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 {4 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}+12 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 b B d^2 n \log (c+d x)-12 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-6 b 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 )+6 b 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)^4 g^3 i^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )^{2}}\, dx\]
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. 1623 vs.
\(2 (354) = 708\).
time = 0.42, size = 1623, normalized size = 4.27 \begin {gather*} -\frac {1}{2} \, B {\left (\frac {6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} g^{3} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} g^{3} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} g^{3} x + {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3}\right )} g^{3}} + \frac {6 \, b d^{2} \log \left (b x + a\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}} - \frac {6 \, b d^{2} \log \left (d x + c\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {{\left (b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )^{2} + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )^{2} - 3 \, {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x - 2 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{4 \, {\left (a^{2} b^{4} c^{5} g^{3} - 4 \, a^{3} b^{3} c^{4} d g^{3} + 6 \, a^{4} b^{2} c^{3} d^{2} g^{3} - 4 \, a^{5} b c^{2} d^{3} g^{3} + a^{6} c d^{4} g^{3} + {\left (b^{6} c^{4} d g^{3} - 4 \, a b^{5} c^{3} d^{2} g^{3} + 6 \, a^{2} b^{4} c^{2} d^{3} g^{3} - 4 \, a^{3} b^{3} c d^{4} g^{3} + a^{4} b^{2} d^{5} g^{3}\right )} x^{3} + {\left (b^{6} c^{5} g^{3} - 2 \, a b^{5} c^{4} d g^{3} - 2 \, a^{2} b^{4} c^{3} d^{2} g^{3} + 8 \, a^{3} b^{3} c^{2} d^{3} g^{3} - 7 \, a^{4} b^{2} c d^{4} g^{3} + 2 \, a^{5} b d^{5} g^{3}\right )} x^{2} + {\left (2 \, a b^{5} c^{5} g^{3} - 7 \, a^{2} b^{4} c^{4} d g^{3} + 8 \, a^{3} b^{3} c^{3} d^{2} g^{3} - 2 \, a^{4} b^{2} c^{2} d^{3} g^{3} - 2 \, a^{5} b c d^{4} g^{3} + a^{6} d^{5} g^{3}\right )} x\right )}} - \frac {1}{2} \, A {\left (\frac {6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} g^{3} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} g^{3} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} g^{3} x + {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3}\right )} g^{3}} + \frac {6 \, b d^{2} \log \left (b x + a\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}} - \frac {6 \, b d^{2} \log \left (d x + c\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}}\right )} \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. 776 vs.
\(2 (354) = 708\).
time = 0.41, size = 776, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (A + B\right )} b^{3} c^{3} - 12 \, {\left (A + B\right )} a b^{2} c^{2} d + 6 \, {\left (A + B\right )} a^{2} b c d^{2} + 4 \, {\left (A + B\right )} a^{3} d^{3} - 6 \, {\left (2 \, {\left (A + B\right )} b^{3} c d^{2} - 2 \, {\left (A + B\right )} a b^{2} d^{3} + {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} - 6 \, {\left (B b^{3} d^{3} n x^{3} + B a^{2} b c d^{2} n + {\left (B b^{3} c d^{2} + 2 \, B a b^{2} d^{3}\right )} n x^{2} + {\left (2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{3} c^{3} - 12 \, B a b^{2} c^{2} d + 15 \, B a^{2} b c d^{2} - 4 \, B a^{3} d^{3}\right )} n - 3 \, {\left (2 \, {\left (A + B\right )} b^{3} c^{2} d + 4 \, {\left (A + B\right )} a b^{2} c d^{2} - 6 \, {\left (A + B\right )} a^{2} b d^{3} + {\left (3 \, B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} - B a^{2} b d^{3}\right )} n\right )} x - 2 \, {\left (6 \, {\left (A + B\right )} a^{2} b c d^{2} + 3 \, {\left (B b^{3} d^{3} n + 2 \, {\left (A + B\right )} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (3 \, B b^{3} c d^{2} n + 2 \, {\left (A + B\right )} b^{3} c d^{2} + 4 \, {\left (A + B\right )} a b^{2} d^{3}\right )} x^{2} - {\left (B b^{3} c^{3} - 6 \, B a b^{2} c^{2} d + 2 \, B a^{3} d^{3}\right )} n + 3 \, {\left (4 \, {\left (A + B\right )} a b^{2} c d^{2} + 2 \, {\left (A + B\right )} a^{2} b d^{3} + {\left (B b^{3} c^{2} d + 4 \, B a b^{2} c d^{2} - 2 \, B a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} g^{3} x^{3} + {\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} g^{3} x^{2} + {\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} g^{3} x + {\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4}\right )} g^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A]
time = 124.71, size = 222, normalized size = 0.58 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, {\left (B b n - \frac {2 \, {\left (b x + a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {B b n - \frac {4 \, {\left (b x + a\right )} B d n}{d x + c} + 2 \, A b + 2 \, B b - \frac {4 \, {\left (b x + a\right )} A d}{d x + c} - \frac {4 \, {\left (b x + a\right )} B d}{d x + c}}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.38, size = 1016, normalized size = 2.67 \begin {gather*} \frac {3\,B\,b\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^3\,i^2\,n\,{\left (a\,d-b\,c\right )}^4}-\frac {\frac {4\,A\,a^2\,d^2-2\,A\,b^2\,c^2-4\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+10\,A\,a\,b\,c\,d+11\,B\,a\,b\,c\,d\,n}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x^2\,\left (2\,A\,b^2\,d^2+B\,b^2\,d^2\,n\right )}{a\,d-b\,c}+\frac {3\,x\,\left (6\,A\,a\,b\,d^2+2\,A\,b^2\,c\,d+B\,a\,b\,d^2\,n+3\,B\,b^2\,c\,d\,n\right )}{2\,\left (a\,d-b\,c\right )}}{x\,\left (2\,a^4\,d^3\,g^3\,i^2-6\,a^2\,b^2\,c^2\,d\,g^3\,i^2+4\,a\,b^3\,c^3\,g^3\,i^2\right )+x^2\,\left (4\,a^3\,b\,d^3\,g^3\,i^2-6\,a^2\,b^2\,c\,d^2\,g^3\,i^2+2\,b^4\,c^3\,g^3\,i^2\right )+x^3\,\left (2\,a^2\,b^2\,d^3\,g^3\,i^2-4\,a\,b^3\,c\,d^2\,g^3\,i^2+2\,b^4\,c^2\,d\,g^3\,i^2\right )+2\,a^2\,b^2\,c^3\,g^3\,i^2+2\,a^4\,c\,d^2\,g^3\,i^2-4\,a^3\,b\,c^2\,d\,g^3\,i^2}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {\frac {B\,\left (2\,a\,d+b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B\,b\,d\,x}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (d\,a^2\,g^3\,i^2+2\,b\,c\,a\,g^3\,i^2\right )+x^2\,\left (c\,b^2\,g^3\,i^2+2\,a\,d\,b\,g^3\,i^2\right )+a^2\,c\,g^3\,i^2+b^2\,d\,g^3\,i^2\,x^3}+\frac {3\,B\,b\,d^2\,\left (b\,g^3\,i^2\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,c\,g^3\,i^2\,n\,\left (a\,d-b\,c\right )}{d}+\frac {g^3\,i^2\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )}{g^3\,i^2\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (d\,a^2\,g^3\,i^2+2\,b\,c\,a\,g^3\,i^2\right )+x^2\,\left (c\,b^2\,g^3\,i^2+2\,a\,d\,b\,g^3\,i^2\right )+a^2\,c\,g^3\,i^2+b^2\,d\,g^3\,i^2\,x^3\right )}\right )-\frac {b\,d^2\,\mathrm {atan}\left (\frac {b\,d^2\,\left (2\,A+B\,n\right )\,\left (\frac {a^4\,d^4\,g^3\,i^2-2\,a^3\,b\,c\,d^3\,g^3\,i^2+2\,a\,b^3\,c^3\,d\,g^3\,i^2-b^4\,c^4\,g^3\,i^2}{a^3\,d^3\,g^3\,i^2-3\,a^2\,b\,c\,d^2\,g^3\,i^2+3\,a\,b^2\,c^2\,d\,g^3\,i^2-b^3\,c^3\,g^3\,i^2}+2\,b\,d\,x\right )\,\left (a^3\,d^3\,g^3\,i^2-3\,a^2\,b\,c\,d^2\,g^3\,i^2+3\,a\,b^2\,c^2\,d\,g^3\,i^2-b^3\,c^3\,g^3\,i^2\right )\,3{}\mathrm {i}}{g^3\,i^2\,\left (6\,A\,b\,d^2+3\,B\,b\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,A+B\,n\right )\,3{}\mathrm {i}}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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