Optimal. Leaf size=78 \[ \frac{g (a+b x)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{2 b}-\frac{B g (b c-a d)^2 \log (c+d x)}{b d^2}+\frac{B g x (b c-a d)}{d} \]
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Rubi [A] time = 0.0523818, antiderivative size = 78, 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, 43} \[ \frac{g (a+b x)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{2 b}-\frac{B g (b c-a d)^2 \log (c+d x)}{b d^2}+\frac{B g x (b c-a d)}{d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (a g+b g x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx &=\frac{g (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}-\frac{B \int \frac{2 (b c-a d) g^2 (-a-b x)}{c+d x} \, dx}{2 b g}\\ &=\frac{g (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}-\frac{(B (b c-a d) g) \int \frac{-a-b x}{c+d x} \, dx}{b}\\ &=\frac{g (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}-\frac{(B (b c-a d) g) \int \left (-\frac{b}{d}+\frac{b c-a d}{d (c+d x)}\right ) \, dx}{b}\\ &=\frac{B (b c-a d) g x}{d}-\frac{B (b c-a d)^2 g \log (c+d x)}{b d^2}+\frac{g (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.037048, size = 72, normalized size = 0.92 \[ \frac{g \left ((a+b x)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-\frac{2 B (a d-b c) ((a d-b c) \log (c+d x)+b d x)}{d^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.242, size = 340, normalized size = 4.4 \begin{align*}{\frac{bA{x}^{2}g}{2}}+Axag+{\frac{A{a}^{2}g}{2\,b}}+{\frac{bB{x}^{2}g}{2}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+B\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) xag+{\frac{gB{a}^{2}}{2\,b}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+{\frac{gB\ln \left ( \left ( bx+a \right ) ^{-1} \right ){a}^{2}}{b}}-2\,{\frac{gB\ln \left ( \left ( bx+a \right ) ^{-1} \right ) ac}{d}}+{\frac{bgB\ln \left ( \left ( bx+a \right ) ^{-1} \right ){c}^{2}}{{d}^{2}}}-Bxag-{\frac{gB{a}^{2}}{b}}+{\frac{bgBcx}{d}}+{\frac{gBac}{d}}-{\frac{gB{a}^{2}}{b}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) }+2\,{\frac{gBac}{d}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) }-{\frac{bgB{c}^{2}}{{d}^{2}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.24947, size = 338, normalized size = 4.33 \begin{align*} \frac{1}{2} \, A b g x^{2} +{\left (x \log \left (\frac{d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac{2 \, a \log \left (b x + a\right )}{b} + \frac{2 \, c \log \left (d x + c\right )}{d}\right )} B a g + \frac{1}{2} \,{\left (x^{2} \log \left (\frac{d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac{2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac{2 \, c^{2} \log \left (d x + c\right )}{d^{2}} + \frac{2 \,{\left (b c - a d\right )} x}{b d}\right )} B b g + A a g x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10473, size = 329, normalized size = 4.22 \begin{align*} \frac{A b^{2} d^{2} g x^{2} - 2 \, B a^{2} d^{2} g \log \left (b x + a\right ) + 2 \,{\left (B b^{2} c d +{\left (A - B\right )} a b d^{2}\right )} g x - 2 \,{\left (B b^{2} c^{2} - 2 \, B a b c d\right )} g \log \left (d x + c\right ) +{\left (B b^{2} d^{2} g x^{2} + 2 \, B a b d^{2} g x\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{2 \, b d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.54484, size = 253, normalized size = 3.24 \begin{align*} \frac{A b g x^{2}}{2} - \frac{B a^{2} g \log{\left (x + \frac{\frac{B a^{3} d^{2} g}{b} + 2 B a^{2} c d g - B a b c^{2} g}{B a^{2} d^{2} g + 2 B a b c d g - B b^{2} c^{2} g} \right )}}{b} + \frac{B c g \left (2 a d - b c\right ) \log{\left (x + \frac{3 B a^{2} c d g - B a b c^{2} g - B a c g \left (2 a d - b c\right ) + \frac{B b c^{2} g \left (2 a d - b c\right )}{d}}{B a^{2} d^{2} g + 2 B a b c d g - B b^{2} c^{2} g} \right )}}{d^{2}} + \left (B a g x + \frac{B b g x^{2}}{2}\right ) \log{\left (\frac{e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )} + \frac{x \left (A a d g - B a d g + B b c g\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.74365, size = 173, normalized size = 2.22 \begin{align*} -\frac{B a^{2} g \log \left (b x + a\right )}{b} + \frac{1}{2} \,{\left (A b g + B b g\right )} x^{2} + \frac{1}{2} \,{\left (B b g x^{2} + 2 \, B a g x\right )} \log \left (\frac{d^{2} x^{2} + 2 \, c d x + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac{{\left (B b c g + A a d g\right )} x}{d} - \frac{{\left (B b c^{2} g - 2 \, B a c d g\right )} \log \left (-d x - c\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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