Optimal. Leaf size=208 \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3}{2 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}+\frac{B d}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac{B}{8 b g^5 (a+b x)^4} \]
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Rubi [A] time = 0.144397, antiderivative size = 208, 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} \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3}{2 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}+\frac{B d}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac{B}{8 b g^5 (a+b x)^4} \]
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)^5} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{2 (b c-a d)}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{4 b g^5 (a+b x)^4}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{4 b g^5 (a+b x)^4}+\frac{(B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}\\ &=-\frac{B}{8 b g^5 (a+b x)^4}+\frac{B d}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \log (a+b x)}{2 b (b c-a d)^4 g^5}-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{4 b g^5 (a+b x)^4}-\frac{B d^4 \log (c+d x)}{2 b (b c-a d)^4 g^5}\\ \end{align*}
Mathematica [A] time = 0.2014, size = 162, normalized size = 0.78 \[ -\frac{6 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+\frac{B \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )}{(b c-a d)^4}}{24 b g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.207, size = 833, normalized size = 4. \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 = 1.3979, size = 944, normalized size = 4.54 \begin{align*} \frac{1}{24} \, B{\left (\frac{12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x +{\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} - \frac{6 \, \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^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac{12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac{12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac{A}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.08479, size = 1328, normalized size = 6.38 \begin{align*} -\frac{3 \,{\left (2 \, A + B\right )} b^{4} c^{4} - 8 \,{\left (3 \, A + 2 \, B\right )} a b^{3} c^{3} d + 36 \,{\left (A + B\right )} a^{2} b^{2} c^{2} d^{2} - 24 \,{\left (A + 2 \, B\right )} a^{3} b c d^{3} +{\left (6 \, A + 25 \, B\right )} a^{4} d^{4} - 12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 6 \,{\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\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 )}{24 \,{\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x +{\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.26937, size = 947, normalized size = 4.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.44465, size = 566, normalized size = 2.72 \begin{align*} -\frac{B d^{4} \log \left (-\frac{b c g}{b g x + a g} + \frac{a d g}{b g x + a g} - d\right )}{2 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} + \frac{B d^{3}}{2 \,{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )}{\left (b g x + a g\right )} b g} - \frac{B d^{2}}{4 \,{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )}{\left (b g x + a g\right )}^{2} b g^{2}} - \frac{B \log \left (\frac{b^{2}}{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{4 \,{\left (b g x + a g\right )}^{4} b g} + \frac{B d}{6 \,{\left (b g x + a g\right )}^{3}{\left (b c - a d\right )} b g^{2}} - \frac{2 \, A b^{3} g^{3} + 3 \, B b^{3} g^{3}}{8 \,{\left (b g x + a g\right )}^{4} b^{4} g^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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