Optimal. Leaf size=137 \[ -\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{2 b (a+b x)^2}+\frac {B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac {B d^2 n \log (c+d x)}{2 b (b c-a d)^2}+\frac {B d n}{2 b (a+b x) (b c-a d)}-\frac {B n}{4 b (a+b x)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 149, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6742, 2492, 44} \[ -\frac {A}{2 b (a+b x)^2}+\frac {B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac {B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac {B d n}{2 b (a+b x) (b c-a d)}-\frac {B n}{4 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2492
Rule 6742
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx &=\int \left (\frac {A}{(a+b x)^3}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}\right ) \, dx\\ &=-\frac {A}{2 b (a+b x)^2}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx\\ &=-\frac {A}{2 b (a+b x)^2}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b}\\ &=-\frac {A}{2 b (a+b x)^2}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac {(B (b c-a d) 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}{2 b}\\ &=-\frac {A}{2 b (a+b x)^2}-\frac {B n}{4 b (a+b x)^2}+\frac {B d n}{2 b (b c-a d) (a+b x)}+\frac {B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac {B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 121, normalized size = 0.88 \[ -\frac {\frac {2 A}{(a+b x)^2}+B n \left (-\frac {2 d^2 \log (a+b x)}{(b c-a d)^2}+\frac {2 d^2 \log (c+d x)}{(b c-a d)^2}+\frac {\frac {2 d (a+b x)}{a d-b c}+1}{(a+b x)^2}\right )+\frac {2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 296, normalized size = 2.16 \[ -\frac {2 \, A b^{2} c^{2} - 4 \, A a b c d + 2 \, A a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} n x + {\left (B b^{2} c^{2} - 4 \, B a b c d + 3 \, B a^{2} d^{2}\right )} n - 2 \, {\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x - {\left (B b^{2} c^{2} - 2 \, B a b c d\right )} n\right )} \log \left (b x + a\right ) + 2 \, {\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x - {\left (B b^{2} c^{2} - 2 \, B a b c d\right )} n\right )} \log \left (d x + c\right ) + 2 \, {\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} \log \relax (e)}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 239, normalized size = 1.74 \[ \frac {B d^{2} n \log \left (b x + a\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac {B d^{2} n \log \left (d x + c\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac {B n \log \left (b x + a\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac {B n \log \left (d x + c\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac {2 \, B b d n x - B b c n + 3 \, B a d n - 2 \, A b c - 2 \, B b c + 2 \, A a d + 2 \, B a d}{4 \, {\left (b^{4} c x^{2} - a b^{3} d x^{2} + 2 \, a b^{3} c x - 2 \, a^{2} b^{2} d x + a^{2} b^{2} c - a^{3} b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.49, size = 1379, normalized size = 10.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 230, normalized size = 1.68 \[ \frac {{\left (\frac {2 \, d^{2} e n \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac {2 \, d^{2} e n \log \left (d x + c\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac {2 \, b d e n x - b c e n + 3 \, a d e n}{a^{2} b^{2} c - a^{3} b d + {\left (b^{4} c - a b^{3} d\right )} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x}\right )} B}{4 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} - \frac {A}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 192, normalized size = 1.40 \[ -\frac {\frac {2\,A\,a\,d-2\,A\,b\,c+3\,B\,a\,d\,n-B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b\,d\,n\,x}{a\,d-b\,c}}{2\,a^2\,b+4\,a\,b^2\,x+2\,b^3\,x^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{2\,b\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}-\frac {B\,d^2\,n\,\mathrm {atanh}\left (\frac {2\,b^3\,c^2-2\,a^2\,b\,d^2}{2\,b\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{b\,{\left (a\,d-b\,c\right )}^2} \]
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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