Optimal. Leaf size=144 \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{2 b g^3 (a+b x)^2}-\frac{B d^2 \log (a+b x)}{2 b g^3 (b c-a d)^2}+\frac{B d^2 \log (c+d x)}{2 b g^3 (b c-a d)^2}-\frac{B d}{2 b g^3 (a+b x) (b c-a d)}+\frac{B}{4 b g^3 (a+b x)^2} \]
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Rubi [A] time = 0.104102, 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} \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{2 b g^3 (a+b x)^2}-\frac{B d^2 \log (a+b x)}{2 b g^3 (b c-a d)^2}+\frac{B d^2 \log (c+d x)}{2 b g^3 (b c-a d)^2}-\frac{B d}{2 b g^3 (a+b x) (b c-a d)}+\frac{B}{4 b g^3 (a+b x)^2} \]
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 (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b g^3 (a+b x)^2}+\frac{B \int \frac{-b c+a d}{g^2 (a+b x)^3 (c+d x)} \, dx}{2 b g}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b g^3 (a+b x)^2}-\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b g^3}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b g^3 (a+b x)^2}-\frac{(B (b c-a d)) \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 g^3}\\ &=\frac{B}{4 b g^3 (a+b x)^2}-\frac{B d}{2 b (b c-a d) g^3 (a+b x)}-\frac{B d^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}+\frac{B d^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b g^3 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0983657, size = 128, normalized size = 0.89 \[ -\frac{(b c-a d) \left (-2 a A d+2 B (b c-a d) \log \left (\frac{e (c+d x)}{a+b x}\right )+3 a B d+2 A b c-b B c+2 b B d x\right )-2 B d^2 (a+b x)^2 \log (c+d x)+2 B d^2 (a+b x)^2 \log (a+b x)}{4 b g^3 (a+b x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 753, 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 = 1.24118, size = 344, normalized size = 2.39 \begin{align*} -\frac{1}{4} \, B{\left (\frac{2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x +{\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac{2 \, \log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac{2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac{2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac{A}{2 \,{\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02353, size = 455, normalized size = 3.16 \begin{align*} -\frac{{\left (2 \, A - B\right )} b^{2} c^{2} - 4 \,{\left (A - B\right )} a b c d +{\left (2 \, A - 3 \, 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 a b d^{2} x - B b^{2} c^{2} + 2 \, B a b c d\right )} \log \left (\frac{d e x + c e}{b x + a}\right )}{4 \,{\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x +{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.23947, size = 422, normalized size = 2.93 \begin{align*} - \frac{B \log{\left (\frac{e \left (c + d x\right )}{a + b x} \right )}}{2 a^{2} b g^{3} + 4 a b^{2} g^{3} x + 2 b^{3} g^{3} x^{2}} + \frac{B d^{2} \log{\left (x + \frac{- \frac{B a^{3} d^{5}}{\left (a d - b c\right )^{2}} + \frac{3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} + \frac{B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} - \frac{B d^{2} \log{\left (x + \frac{\frac{B a^{3} d^{5}}{\left (a d - b c\right )^{2}} - \frac{3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} - \frac{B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} + \frac{- 2 A a d + 2 A b c + 3 B a d - B b c + 2 B b d x}{4 a^{3} b d g^{3} - 4 a^{2} b^{2} c g^{3} + x^{2} \left (4 a b^{3} d g^{3} - 4 b^{4} c g^{3}\right ) + x \left (8 a^{2} b^{2} d g^{3} - 8 a b^{3} c g^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3898, size = 327, normalized size = 2.27 \begin{align*} -\frac{B d^{2} \log \left (b x + a\right )}{2 \,{\left (b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}\right )}} + \frac{B d^{2} \log \left (d x + c\right )}{2 \,{\left (b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}\right )}} - \frac{B \log \left (\frac{d x + c}{b x + a}\right )}{2 \,{\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac{2 \, B b d x + 2 \, A b c + B b c - 2 \, A a d + B a d}{4 \,{\left (b^{4} c g^{3} x^{2} - a b^{3} d g^{3} x^{2} + 2 \, a b^{3} c g^{3} x - 2 \, a^{2} b^{2} d g^{3} x + a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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