Optimal. Leaf size=166 \[ -\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{3 b (a+b x)^3}-\frac{B d^2 n}{3 b (a+b x) (b c-a d)^2}-\frac{B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac{B d n}{6 b (a+b x)^2 (b c-a d)}-\frac{B n}{9 b (a+b x)^3} \]
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Rubi [A] time = 0.169881, antiderivative size = 178, normalized size of antiderivative = 1.07, 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} \[ -\frac{A}{3 b (a+b x)^3}-\frac{B d^2 n}{3 b (a+b x) (b c-a d)^2}-\frac{B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{B d n}{6 b (a+b x)^2 (b c-a d)}-\frac{B n}{9 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2492
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx &=\int \left (\frac{A}{(a+b x)^4}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}\right ) \, dx\\ &=-\frac{A}{3 b (a+b x)^3}+B \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx\\ &=-\frac{A}{3 b (a+b x)^3}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b}\\ &=-\frac{A}{3 b (a+b x)^3}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{(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{A}{3 b (a+b x)^3}-\frac{B n}{9 b (a+b x)^3}+\frac{B d n}{6 b (b c-a d) (a+b x)^2}-\frac{B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.400264, size = 143, normalized size = 0.86 \[ -\frac{\frac{6 A}{(a+b x)^3}+B n \left (\frac{\frac{6 d^2 (a+b x)^2}{(b c-a d)^2}+\frac{3 d (a+b x)}{a d-b c}+2}{(a+b x)^3}+\frac{6 d^3 \log (a+b x)}{(b c-a d)^3}-\frac{6 d^3 \log (c+d x)}{(b c-a d)^3}\right )+\frac{6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}}{18 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.475, size = 1976, normalized size = 11.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.25239, size = 540, normalized size = 3.25 \begin{align*} -\frac{{\left (\frac{6 \, d^{3} e n \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{6 \, d^{3} e n \log \left (d x + c\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac{6 \, b^{2} d^{2} e n x^{2} + 2 \, b^{2} c^{2} e n - 7 \, a b c d e n + 11 \, a^{2} d^{2} e n - 3 \,{\left (b^{2} c d e n - 5 \, a b d^{2} e n\right )} x}{a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} + 3 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x}\right )} B}{18 \, e} - \frac{B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac{A}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.13249, size = 1123, normalized size = 6.77 \begin{align*} -\frac{6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} + 6 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \,{\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} n x +{\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3}\right )} n + 6 \,{\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x +{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (b x + a\right ) - 6 \,{\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x +{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (d x + c\right ) + 6 \,{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \left (e\right )}{18 \,{\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} +{\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 )} 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 )} 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 )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29256, size = 605, normalized size = 3.64 \begin{align*} -\frac{B d^{3} n \log \left (b x + a\right )}{3 \,{\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 )}} + \frac{B d^{3} n \log \left (d x + c\right )}{3 \,{\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 )}} - \frac{B n \log \left (b x + a\right )}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac{B n \log \left (d x + c\right )}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac{6 \, B b^{2} d^{2} n x^{2} - 3 \, B b^{2} c d n x + 15 \, B a b d^{2} n x + 2 \, B b^{2} c^{2} n - 7 \, B a b c d n + 11 \, B a^{2} d^{2} n + 6 \, A b^{2} c^{2} + 6 \, B b^{2} c^{2} - 12 \, A a b c d - 12 \, B a b c d + 6 \, A a^{2} d^{2} + 6 \, B a^{2} d^{2}}{18 \,{\left (b^{6} c^{2} x^{3} - 2 \, a b^{5} c d x^{3} + a^{2} b^{4} d^{2} x^{3} + 3 \, a b^{5} c^{2} x^{2} - 6 \, a^{2} b^{4} c d x^{2} + 3 \, a^{3} b^{3} d^{2} x^{2} + 3 \, a^{2} b^{4} c^{2} x - 6 \, a^{3} b^{3} c d x + 3 \, a^{4} b^{2} d^{2} x + a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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