Optimal. Leaf size=177 \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{3 b g^4 (a+b x)^3}-\frac{2 B d^2}{3 b g^4 (a+b x) (b c-a d)^2}-\frac{2 B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac{2 B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac{B d}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac{2 B}{9 b g^4 (a+b x)^3} \]
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Rubi [A] time = 0.114985, antiderivative size = 177, 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} \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{3 b g^4 (a+b x)^3}-\frac{2 B d^2}{3 b g^4 (a+b x) (b c-a d)^2}-\frac{2 B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac{2 B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac{B d}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac{2 B}{9 b g^4 (a+b x)^3} \]
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)^2}{(c+d x)^2}\right )}{(a g+b g x)^4} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac{B \int \frac{2 (b c-a d)}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac{(2 B (b c-a d)) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac{(2 B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac{2 B}{9 b g^4 (a+b x)^3}+\frac{B d}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 \log (c+d x)}{3 b (b c-a d)^3 g^4}\\ \end{align*}
Mathematica [A] time = 0.134791, size = 140, normalized size = 0.79 \[ -\frac{3 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+\frac{B \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}}{9 b g^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 579, normalized size = 3.3 \begin{align*} -{\frac{{d}^{3}Ab}{{g}^{4} \left ( ad-bc \right ) ^{3}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-2}}+{\frac{{d}^{3}A}{{g}^{4} \left ( ad-bc \right ) ^{3}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+{\frac{{d}^{3}A{b}^{2}}{3\,{g}^{4} \left ( ad-bc \right ) ^{3}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}}-{\frac{11\,{d}^{3}B}{9\,{g}^{4}b \left ( dx+c \right ) ^{3}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}}+{\frac{{d}^{3}{b}^{2}B}{3\,{g}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}}-{\frac{2\,{d}^{3}Bb}{3\,{g}^{4} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) \left ( dx+c \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}}-{\frac{5\,{d}^{3}B}{3\,{g}^{4} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}}+{\frac{{d}^{3}B}{{g}^{4} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}}+{\frac{{d}^{3}Bb}{{g}^{4} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) \left ( dx+c \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23764, size = 648, normalized size = 3.66 \begin{align*} -\frac{1}{9} \, B{\left (\frac{6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x +{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac{3 \, \log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac{6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac{6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac{A}{3 \,{\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.07211, size = 869, normalized size = 4.91 \begin{align*} -\frac{{\left (3 \, A + 2 \, B\right )} b^{3} c^{3} - 9 \,{\left (A + B\right )} a b^{2} c^{2} d + 9 \,{\left (A + 2 \, B\right )} a^{2} b c d^{2} -{\left (3 \, A + 11 \, B\right )} a^{3} d^{3} + 6 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 3 \,{\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{9 \,{\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x +{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.49235, size = 677, normalized size = 3.82 \begin{align*} - \frac{B \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{3 a^{3} b g^{4} + 9 a^{2} b^{2} g^{4} x + 9 a b^{3} g^{4} x^{2} + 3 b^{4} g^{4} x^{3}} - \frac{2 B d^{3} \log{\left (x + \frac{- \frac{2 B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac{8 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac{12 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac{8 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + 2 B a d^{4} - \frac{2 B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + 2 B b c d^{3}}{4 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac{2 B d^{3} \log{\left (x + \frac{\frac{2 B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac{8 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac{12 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac{8 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + 2 B a d^{4} + \frac{2 B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + 2 B b c d^{3}}{4 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} - \frac{3 A a^{2} d^{2} - 6 A a b c d + 3 A b^{2} c^{2} + 11 B a^{2} d^{2} - 7 B a b c d + 2 B b^{2} c^{2} + 6 B b^{2} d^{2} x^{2} + x \left (15 B a b d^{2} - 3 B b^{2} c d\right )}{9 a^{5} b d^{2} g^{4} - 18 a^{4} b^{2} c d g^{4} + 9 a^{3} b^{3} c^{2} g^{4} + x^{3} \left (9 a^{2} b^{4} d^{2} g^{4} - 18 a b^{5} c d g^{4} + 9 b^{6} c^{2} g^{4}\right ) + x^{2} \left (27 a^{3} b^{3} d^{2} g^{4} - 54 a^{2} b^{4} c d g^{4} + 27 a b^{5} c^{2} g^{4}\right ) + x \left (27 a^{4} b^{2} d^{2} g^{4} - 54 a^{3} b^{3} c d g^{4} + 27 a^{2} b^{4} c^{2} g^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33325, size = 639, normalized size = 3.61 \begin{align*} -\frac{2 \, B d^{3} \log \left (b x + a\right )}{3 \,{\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} + \frac{2 \, B d^{3} \log \left (d x + c\right )}{3 \,{\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} - \frac{B \log \left (\frac{b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \,{\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac{6 \, B b^{2} d^{2} x^{2} - 3 \, B b^{2} c d x + 15 \, B a b d^{2} x + 3 \, A b^{2} c^{2} + 5 \, B b^{2} c^{2} - 6 \, A a b c d - 13 \, B a b c d + 3 \, A a^{2} d^{2} + 14 \, B a^{2} d^{2}}{9 \,{\left (b^{6} c^{2} g^{4} x^{3} - 2 \, a b^{5} c d g^{4} x^{3} + a^{2} b^{4} d^{2} g^{4} x^{3} + 3 \, a b^{5} c^{2} g^{4} x^{2} - 6 \, a^{2} b^{4} c d g^{4} x^{2} + 3 \, a^{3} b^{3} d^{2} g^{4} x^{2} + 3 \, a^{2} b^{4} c^{2} g^{4} x - 6 \, a^{3} b^{3} c d g^{4} x + 3 \, a^{4} b^{2} d^{2} g^{4} x + a^{3} b^{3} c^{2} g^{4} - 2 \, a^{4} b^{2} c d g^{4} + a^{5} b d^{2} g^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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