Optimal. Leaf size=139 \[ -\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A}{2 b g^3 (a+b x)^2}-\frac{B d^2 \log (a+b x)}{b g^3 (b c-a d)^2}+\frac{B d^2 \log (c+d x)}{b g^3 (b c-a d)^2}-\frac{B d}{b g^3 (a+b x) (b c-a d)}+\frac{B}{2 b g^3 (a+b x)^2} \]
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Rubi [A] time = 0.100561, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A}{2 b g^3 (a+b x)^2}-\frac{B d^2 \log (a+b x)}{b g^3 (b c-a d)^2}+\frac{B d^2 \log (c+d x)}{b g^3 (b c-a d)^2}-\frac{B d}{b g^3 (a+b x) (b c-a d)}+\frac{B}{2 b g^3 (a+b x)^2} \]
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)^2}{(a+b x)^2}\right )}{(a g+b g x)^3} \, dx &=-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac{B \int \frac{2 (-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)^2}{(a+b x)^2}\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}{b g^3}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\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}{b g^3}\\ &=\frac{B}{2 b g^3 (a+b x)^2}-\frac{B d}{b (b c-a d) g^3 (a+b x)}-\frac{B d^2 \log (a+b x)}{b (b c-a d)^2 g^3}+\frac{B d^2 \log (c+d x)}{b (b c-a d)^2 g^3}-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{2 b g^3 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0962121, size = 128, normalized size = 0.92 \[ -\frac{(b c-a d) \left (-a A d+B (b c-a d) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+3 a B d+A b c-b B c+2 b B d x\right )-2 B d^2 (a+b x)^2 \log (c+d x)+2 B d^2 (a+b x)^2 \log (a+b x)}{2 b g^3 (a+b x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 300, normalized size = 2.2 \begin{align*} -{\frac{A}{2\,b \left ( bx+a \right ) ^{2}{g}^{3}}}-{\frac{B}{2\,b \left ( bx+a \right ) ^{2}{g}^{3}}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+{\frac{B{a}^{2}{d}^{2}}{2\,b{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}-{\frac{Badc}{{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{bB{c}^{2}}{2\,{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{B{d}^{2}a}{b{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-{\frac{Bdc}{{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+{\frac{B{d}^{3}a}{b{g}^{3} \left ( ad-bc \right ) ^{3}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) }-{\frac{B{d}^{2}c}{{g}^{3} \left ( ad-bc \right ) ^{3}}\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 = 1.21986, size = 413, normalized size = 2.97 \begin{align*} -\frac{1}{2} \, 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{\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 )}{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 = 1.00497, size = 495, normalized size = 3.56 \begin{align*} -\frac{{\left (A - B\right )} b^{2} c^{2} - 2 \,{\left (A - 2 \, B\right )} a b c d +{\left (A - 3 \, B\right )} a^{2} d^{2} + 2 \,{\left (B b^{2} c d - B a b d^{2}\right )} x -{\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^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{2 \,{\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 = 3.26856, size = 418, normalized size = 3.01 \begin{align*} - \frac{B \log{\left (\frac{e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \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 )}}{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 )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac{- A a d + A b c + 3 B a d - B b c + 2 B b d x}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 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 = 1.34159, size = 350, normalized size = 2.52 \begin{align*} -\frac{B d^{2} \log \left (b x + a\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} + \frac{B d^{2} \log \left (d x + c\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac{B \log \left (\frac{d^{2} x^{2} + 2 \, c d x + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\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 + A b c - A a d + 2 \, B a d}{2 \,{\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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