Optimal. Leaf size=144 \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{2 d i^3 (c+d x)^2}+\frac{b^2 B \log (a+b x)}{2 d i^3 (b c-a d)^2}-\frac{b^2 B \log (c+d x)}{2 d i^3 (b c-a d)^2}+\frac{b B}{2 d i^3 (c+d x) (b c-a d)}+\frac{B}{4 d i^3 (c+d x)^2} \]
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Rubi [A] time = 0.0993907, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{2 d i^3 (c+d x)^2}+\frac{b^2 B \log (a+b x)}{2 d i^3 (b c-a d)^2}-\frac{b^2 B \log (c+d x)}{2 d i^3 (b c-a d)^2}+\frac{b B}{2 d i^3 (c+d x) (b c-a d)}+\frac{B}{4 d i^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(50 c+50 d x)^3} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{250000 d (c+d x)^2}+\frac{B \int \frac{b c-a d}{2500 (a+b x) (c+d x)^3} \, dx}{100 d}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{250000 d (c+d x)^2}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{250000 d}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{250000 d (c+d x)^2}+\frac{(B (b c-a d)) \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}{250000 d}\\ &=\frac{B}{500000 d (c+d x)^2}+\frac{b B}{250000 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x)}{250000 d (b c-a d)^2}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{250000 d (c+d x)^2}-\frac{b^2 B \log (c+d x)}{250000 d (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.11675, size = 111, normalized size = 0.77 \[ \frac{\frac{B \left (2 b^2 (c+d x)^2 \log (a+b x)+(b c-a d) (-a d+3 b c+2 b d x)-2 b^2 (c+d x)^2 \log (c+d x)\right )}{(b c-a d)^2}-2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d i^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 746, normalized size = 5.2 \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 [A] time = 1.19764, size = 344, normalized size = 2.39 \begin{align*} \frac{1}{4} \, B{\left (\frac{2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x +{\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} - \frac{2 \, \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}} + \frac{2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac{2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac{A}{2 \,{\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.486076, size = 455, normalized size = 3.16 \begin{align*} -\frac{{\left (2 \, A - 3 \, B\right )} b^{2} c^{2} - 4 \,{\left (A - B\right )} a b c d +{\left (2 \, A - B\right )} a^{2} d^{2} - 2 \,{\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \,{\left (B b^{2} d^{2} x^{2} + 2 \, B b^{2} c d x + 2 \, B a b c d - B a^{2} d^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} i^{3} x +{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} i^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.99029, size = 422, normalized size = 2.93 \begin{align*} - \frac{B b^{2} \log{\left (x + \frac{- \frac{B a^{3} b^{2} d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 B a^{2} b^{3} c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 B a b^{4} c^{2} d}{\left (a d - b c\right )^{2}} + B a b^{2} d + \frac{B b^{5} c^{3}}{\left (a d - b c\right )^{2}} + B b^{3} c}{2 B b^{3} d} \right )}}{2 d i^{3} \left (a d - b c\right )^{2}} + \frac{B b^{2} \log{\left (x + \frac{\frac{B a^{3} b^{2} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 B a^{2} b^{3} c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 B a b^{4} c^{2} d}{\left (a d - b c\right )^{2}} + B a b^{2} d - \frac{B b^{5} c^{3}}{\left (a d - b c\right )^{2}} + B b^{3} c}{2 B b^{3} d} \right )}}{2 d i^{3} \left (a d - b c\right )^{2}} - \frac{B \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d i^{3} + 4 c d^{2} i^{3} x + 2 d^{3} i^{3} x^{2}} - \frac{2 A a d - 2 A b c - B a d + 3 B b c + 2 B b d x}{4 a c^{2} d^{2} i^{3} - 4 b c^{3} d i^{3} + x^{2} \left (4 a d^{4} i^{3} - 4 b c d^{3} i^{3}\right ) + x \left (8 a c d^{3} i^{3} - 8 b c^{2} d^{2} i^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31786, size = 282, normalized size = 1.96 \begin{align*} -\frac{B b^{2} \log \left (b x + a\right )}{2 \,{\left (b^{2} c^{2} d i - 2 \, a b c d^{2} i + a^{2} d^{3} i\right )}} + \frac{B b^{2} \log \left (d x + c\right )}{2 \,{\left (b^{2} c^{2} d i - 2 \, a b c d^{2} i + a^{2} d^{3} i\right )}} - \frac{B i \log \left (\frac{b x + a}{d x + c}\right )}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} + \frac{2 \, B b d i x - 2 \, A b c i + B b c i + 2 \, A a d i + B a d i}{4 \,{\left (b c d^{3} x^{2} - a d^{4} x^{2} + 2 \, b c^{2} d^{2} x - 2 \, a c d^{3} x + b c^{3} d - a c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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