Optimal. Leaf size=87 \[ \frac{(a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(f+g x) (b f-a g)}+\frac{B (b c-a d) \log \left (\frac{f+g x}{c+d x}\right )}{(b f-a g) (d f-c g)} \]
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Rubi [A] time = 0.105829, antiderivative size = 113, normalized size of antiderivative = 1.3, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2525, 12, 72} \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{g (f+g x)}+\frac{B (b c-a d) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac{b B \log (a+b x)}{g (b f-a g)}-\frac{B d \log (c+d x)}{g (d f-c g)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 72
Rubi steps
\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(f+g x)^2} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{g (f+g x)}+\frac{B \int \frac{b c-a d}{(a+b x) (c+d x) (f+g x)} \, dx}{g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{g (f+g x)}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x) (c+d x) (f+g x)} \, dx}{g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{g (f+g x)}+\frac{(B (b c-a d)) \int \left (\frac{b^2}{(b c-a d) (b f-a g) (a+b x)}+\frac{d^2}{(b c-a d) (-d f+c g) (c+d x)}+\frac{g^2}{(b f-a g) (d f-c g) (f+g x)}\right ) \, dx}{g}\\ &=\frac{b B \log (a+b x)}{g (b f-a g)}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{g (f+g x)}-\frac{B d \log (c+d x)}{g (d f-c g)}+\frac{B (b c-a d) \log (f+g x)}{(b f-a g) (d f-c g)}\\ \end{align*}
Mathematica [A] time = 0.151138, size = 105, normalized size = 1.21 \[ \frac{\frac{B (b \log (a+b x) (d f-c g)+\log (c+d x) (a d g-b d f)+g (b c-a d) \log (f+g x))}{(b f-a g) (d f-c g)}-\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{f+g x}}{g} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.149, size = 926, normalized size = 10.6 \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 = 1.12098, size = 186, normalized size = 2.14 \begin{align*} B{\left (\frac{b \log \left (b x + a\right )}{b f g - a g^{2}} - \frac{d \log \left (d x + c\right )}{d f g - c g^{2}} + \frac{{\left (b c - a d\right )} \log \left (g x + f\right )}{b d f^{2} + a c g^{2} -{\left (b c + a d\right )} f g} - \frac{\log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{g^{2} x + f g}\right )} - \frac{A}{g^{2} x + f g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 21.9531, size = 559, normalized size = 6.43 \begin{align*} -\frac{A b d f^{2} + A a c g^{2} -{\left (A b c + A a d\right )} f g -{\left (B b d f^{2} - B b c f g +{\left (B b d f g - B b c g^{2}\right )} x\right )} \log \left (b x + a\right ) +{\left (B b d f^{2} - B a d f g +{\left (B b d f g - B a d g^{2}\right )} x\right )} \log \left (d x + c\right ) -{\left ({\left (B b c - B a d\right )} g^{2} x +{\left (B b c - B a d\right )} f g\right )} \log \left (g x + f\right ) +{\left (B b d f^{2} + B a c g^{2} -{\left (B b c + B a d\right )} f g\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{b d f^{3} g + a c f g^{3} -{\left (b c + a d\right )} f^{2} g^{2} +{\left (b d f^{2} g^{2} + a c g^{4} -{\left (b c + a d\right )} f g^{3}\right )} x} \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 = 1.31311, size = 378, normalized size = 4.34 \begin{align*} \frac{{\left (B b c - B a d\right )} \log \left (g x + f\right )}{b d f^{2} - b c f g - a d f g + a c g^{2}} - \frac{B \log \left (\frac{b x + a}{d x + c}\right )}{g^{2} x + f g} - \frac{{\left (B b c - B a d\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{2 \,{\left (b d f^{2} - b c f g - a d f g + a c g^{2}\right )}} - \frac{A + B}{g^{2} x + f g} + \frac{{\left (2 \, B b^{2} c d f - 2 \, B a b d^{2} f - B b^{2} c^{2} g + B a^{2} d^{2} g\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | -b c + a d \right |}}{2 \, b d x + b c + a d +{\left | -b c + a d \right |}} \right |}\right )}{2 \,{\left (b d f^{2} g - b c f g^{2} - a d f g^{2} + a c g^{3}\right )}{\left | -b c + a d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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