Optimal. Leaf size=175 \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{3 b g^4 (a+b x)^3}-\frac{B d^2}{3 b g^4 (a+b x) (b c-a d)^2}-\frac{B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac{B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac{B d}{6 b g^4 (a+b x)^2 (b c-a d)}-\frac{B}{9 b g^4 (a+b x)^3} \]
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Rubi [A] time = 0.129579, antiderivative size = 175, 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} \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{3 b g^4 (a+b x)^3}-\frac{B d^2}{3 b g^4 (a+b x) (b c-a d)^2}-\frac{B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac{B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac{B d}{6 b g^4 (a+b x)^2 (b c-a d)}-\frac{B}{9 b g^4 (a+b x)^3} \]
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)}{c+d x}\right )}{(a g+b g x)^4} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b g^4 (a+b x)^3}+\frac{B \int \frac{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)}{c+d x}\right )}{3 b g^4 (a+b x)^3}+\frac{(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)}{c+d x}\right )}{3 b g^4 (a+b x)^3}+\frac{(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{B}{9 b g^4 (a+b x)^3}+\frac{B d}{6 b (b c-a d) g^4 (a+b x)^2}-\frac{B d^2}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{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)}{c+d x}\right )}{3 b g^4 (a+b x)^3}+\frac{B d^3 \log (c+d x)}{3 b (b c-a d)^3 g^4}\\ \end{align*}
Mathematica [A] time = 0.162892, size = 141, normalized size = 0.81 \[ -\frac{\frac{B \left ((b c-a d) \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )-6 d^3 (a+b x)^3 \log (c+d x)+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}+6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{18 b g^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 1191, normalized size = 6.8 \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 [B] time = 1.22489, size = 578, normalized size = 3.3 \begin{align*} -\frac{1}{18} \, 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{6 \, \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\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 = 1.05122, size = 826, normalized size = 4.72 \begin{align*} -\frac{2 \,{\left (3 \, A + B\right )} b^{3} c^{3} - 9 \,{\left (2 \, A + B\right )} a b^{2} c^{2} d + 18 \,{\left (A + B\right )} a^{2} b c d^{2} -{\left (6 \, 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 + 6 \,{\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 e x + a e}{d x + c}\right )}{18 \,{\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 = 6.86449, size = 656, normalized size = 3.75 \begin{align*} - \frac{B \log{\left (\frac{e \left (a + b x\right )}{c + d x} \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{B d^{3} \log{\left (x + \frac{- \frac{B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac{4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac{6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac{4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} - \frac{B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac{B d^{3} \log{\left (x + \frac{\frac{B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac{4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac{6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac{4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} + \frac{B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} - \frac{6 A a^{2} d^{2} - 12 A a b c d + 6 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 )}{18 a^{5} b d^{2} g^{4} - 36 a^{4} b^{2} c d g^{4} + 18 a^{3} b^{3} c^{2} g^{4} + x^{3} \left (18 a^{2} b^{4} d^{2} g^{4} - 36 a b^{5} c d g^{4} + 18 b^{6} c^{2} g^{4}\right ) + x^{2} \left (54 a^{3} b^{3} d^{2} g^{4} - 108 a^{2} b^{4} c d g^{4} + 54 a b^{5} c^{2} g^{4}\right ) + x \left (54 a^{4} b^{2} d^{2} g^{4} - 108 a^{3} b^{3} c d g^{4} + 54 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 = 1.41307, size = 609, normalized size = 3.48 \begin{align*} -\frac{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{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 x + a}{d x + c}\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 + 6 \, A b^{2} c^{2} + 8 \, B b^{2} c^{2} - 12 \, A a b c d - 19 \, B a b c d + 6 \, A a^{2} d^{2} + 17 \, B a^{2} d^{2}}{18 \,{\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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