Optimal. Leaf size=427 \[ -\frac {b^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 (a+b x)^3 (b c-a d)^3}-\frac {2 b^2 B n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{9 (a+b x)^3 (b c-a d)^3}-\frac {d^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^3}-\frac {2 B d^2 n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^3}+\frac {b d (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x)^2 (b c-a d)^3}+\frac {b B d n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x)^2 (b c-a d)^3}-\frac {2 b^2 B^2 n^2 (c+d x)^3}{27 (a+b x)^3 (b c-a d)^3}-\frac {2 B^2 d^2 n^2 (c+d x)}{(a+b x) (b c-a d)^3}+\frac {b B^2 d n^2 (c+d x)^2}{2 (a+b x)^2 (b c-a d)^3} \]
[Out]
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Rubi [C] time = 1.21, antiderivative size = 730, normalized size of antiderivative = 1.71, number of steps used = 26, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 2492, 44, 2514, 2490, 32, 2488, 2411, 2343, 2333, 2315} \[ -\frac {2 B^2 d^3 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n^2 \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{3 b (b c-a d)^3}-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B d^2 n}{3 b (a+b x) (b c-a d)^2}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {A B d n}{3 b (a+b x)^2 (b c-a d)}-\frac {2 A B n}{9 b (a+b x)^3}+\frac {2 B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (a+b x) (b c-a d)^3}-\frac {11 B^2 d^2 n^2}{9 b (a+b x) (b c-a d)^2}-\frac {5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac {5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^2 (b c-a d)}-\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {5 B^2 d n^2}{18 b (a+b x)^2 (b c-a d)}-\frac {2 B^2 n^2}{27 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 32
Rule 44
Rule 2315
Rule 2333
Rule 2343
Rule 2411
Rule 2488
Rule 2490
Rule 2492
Rule 2514
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx &=\int \left (\frac {A^2}{(a+b x)^4}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}\right ) \, dx\\ &=-\frac {A^2}{3 b (a+b x)^3}+(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx+B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {(2 A B (b c-a d) n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b}+\frac {\left (2 B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {(2 A B (b c-a d) n) \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}{3 b}+\frac {\left (2 B^2 (b c-a d) n\right ) \int \left (\frac {b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^4}-\frac {b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B n}{9 b (a+b x)^3}+\frac {A B d n}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {1}{3} \left (2 B^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx-\frac {\left (2 B^2 d^3 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{3 (b c-a d)^3}+\frac {\left (2 B^2 d^4 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{3 b (b c-a d)^3}+\frac {\left (2 B^2 d^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2}-\frac {\left (2 B^2 d n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{3 (b c-a d)}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B n}{9 b (a+b x)^3}+\frac {A B d n}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {2 B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^2} \, dx}{3 (b c-a d)^2}-\frac {\left (2 B^2 d^3 n^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac {\left (2 B^2 d^3 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B n}{9 b (a+b x)^3}+\frac {A B d n}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {2 B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {2 B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {\left (B^2 d n^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}+\frac {\left (2 B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{3 b (b c-a d)^2}-\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{3 b^2 (b c-a d)^2}+\frac {\left (2 B^2 (b c-a d) n^2\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}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B n}{9 b (a+b x)^3}-\frac {2 B^2 n^2}{27 b (a+b x)^3}+\frac {A B d n}{3 b (b c-a d) (a+b x)^2}+\frac {5 B^2 d n^2}{18 b (b c-a d) (a+b x)^2}-\frac {2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {11 B^2 d^2 n^2}{9 b (b c-a d)^2 (a+b x)}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}-\frac {5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac {5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {2 B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {\left (2 B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{3 b (b c-a d)^2}+\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{3 b^2 (b c-a d)^2}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B n}{9 b (a+b x)^3}-\frac {2 B^2 n^2}{27 b (a+b x)^3}+\frac {A B d n}{3 b (b c-a d) (a+b x)^2}+\frac {5 B^2 d n^2}{18 b (b c-a d) (a+b x)^2}-\frac {2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {11 B^2 d^2 n^2}{9 b (b c-a d)^2 (a+b x)}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}-\frac {5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac {5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {2 B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {\left (2 B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{3 b (b c-a d)^2}+\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{3 b^2 (b c-a d)^2}\\ &=-\frac {A^2}{3 b (a+b x)^3}-\frac {2 A B n}{9 b (a+b x)^3}-\frac {2 B^2 n^2}{27 b (a+b x)^3}+\frac {A B d n}{3 b (b c-a d) (a+b x)^2}+\frac {5 B^2 d n^2}{18 b (b c-a d) (a+b x)^2}-\frac {2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {11 B^2 d^2 n^2}{9 b (b c-a d)^2 (a+b x)}-\frac {2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}-\frac {5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac {2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac {5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {2 B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^2 d^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b (b c-a d)^3}-\frac {2 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{3 b (b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 432, normalized size = 1.01 \[ \frac {-(b c-a d) \left (6 B \left (B n \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+6 A (b c-a d)^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A B n \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+B^2 n^2 \left (85 a^2 d^2+a b d (147 d x-23 c)+b^2 \left (4 c^2-15 c d x+66 d^2 x^2\right )\right )+18 A^2 (b c-a d)^2+18 B^2 (b c-a d)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B d^3 n (a+b x)^3 \log (c+d x) \left (6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A+11 B n\right )-6 B d^3 n (a+b x)^3 \log (a+b x) \left (6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A+6 B n \log (c+d x)+11 B n\right )+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(c+d x)+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(a+b x)}{54 b (a+b x)^3 (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 1635, normalized size = 3.83 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b x + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.70, size = 25057, normalized size = 58.68 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.43, size = 1500, normalized size = 3.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.84, size = 911, normalized size = 2.13 \[ \frac {\frac {18\,A^2\,a^2\,d^2-36\,A^2\,a\,b\,c\,d+18\,A^2\,b^2\,c^2+66\,A\,B\,a^2\,d^2\,n-42\,A\,B\,a\,b\,c\,d\,n+12\,A\,B\,b^2\,c^2\,n+85\,B^2\,a^2\,d^2\,n^2-23\,B^2\,a\,b\,c\,d\,n^2+4\,B^2\,b^2\,c^2\,n^2}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (-5\,c\,B^2\,b^2\,d\,n^2+49\,a\,B^2\,b\,d^2\,n^2-6\,A\,c\,B\,b^2\,d\,n+30\,A\,a\,B\,b\,d^2\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x^2\,\left (11\,d\,B^2\,b^2\,n^2+6\,A\,d\,B\,b^2\,n\right )}{a\,d-b\,c}}{x^3\,\left (9\,b^5\,c-9\,a\,b^4\,d\right )+x\,\left (27\,a^2\,b^3\,c-27\,a^3\,b^2\,d\right )-x^2\,\left (27\,a^2\,b^3\,d-27\,a\,b^4\,c\right )+9\,a^3\,b^2\,c-9\,a^4\,b\,d}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {B^2}{3\,b\,\left (a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3\right )}-\frac {B^2\,d^3}{3\,b\,\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 )-\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {2\,A\,B}{3\,\left (a^3\,b+3\,a^2\,b^2\,x+3\,a\,b^3\,x^2+b^4\,x^3\right )}+\frac {2\,B^2\,d^3\,\left (a\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{2\,d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{d}\right )+x\,\left (b\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{2\,d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{d}\right )+\frac {2\,a\,b^2\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{d^2}\right )+\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{d^3}+\frac {3\,b^3\,n\,x^2\,\left (a\,d-b\,c\right )}{d}\right )}{9\,b\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (a^3\,b+3\,a^2\,b^2\,x+3\,a\,b^3\,x^2+b^4\,x^3\right )}\right )-\frac {B\,d^3\,n\,\mathrm {atan}\left (\frac {B\,d^3\,n\,\left (6\,A+11\,B\,n\right )\,\left (\frac {a^3\,b\,d^3-a^2\,b^2\,c\,d^2-a\,b^3\,c^2\,d+b^4\,c^3}{a^2\,b\,d^2-2\,a\,b^2\,c\,d+b^3\,c^2}+2\,b\,d\,x\right )\,\left (a^2\,b\,d^2-2\,a\,b^2\,c\,d+b^3\,c^2\right )\,1{}\mathrm {i}}{b\,\left (11\,B^2\,d^3\,n^2+6\,A\,B\,d^3\,n\right )\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (6\,A+11\,B\,n\right )\,2{}\mathrm {i}}{9\,b\,{\left (a\,d-b\,c\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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