Optimal. Leaf size=206 \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{4 b g^5 (a+b x)^4}-\frac{B d^3}{4 b g^5 (a+b x) (b c-a d)^3}+\frac{B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}+\frac{B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}-\frac{B d}{12 b g^5 (a+b x)^3 (b c-a d)}+\frac{B}{16 b g^5 (a+b x)^4} \]
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Rubi [A] time = 0.14864, antiderivative size = 206, 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} \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{4 b g^5 (a+b x)^4}-\frac{B d^3}{4 b g^5 (a+b x) (b c-a d)^3}+\frac{B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}+\frac{B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}-\frac{B d}{12 b g^5 (a+b x)^3 (b c-a d)}+\frac{B}{16 b g^5 (a+b x)^4} \]
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 (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{-b c+a d}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\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}{4 b g^5}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\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}{4 b g^5}\\ &=\frac{B}{16 b g^5 (a+b x)^4}-\frac{B d}{12 b (b c-a d) g^5 (a+b x)^3}+\frac{B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}-\frac{B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}-\frac{B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}+\frac{B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.202056, size = 166, normalized size = 0.81 \[ \frac{\frac{B (a d-b c) \left (\frac{12 d^3 (b c-a d)}{a+b x}-\frac{6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac{4 d (b c-a d)^3}{(a+b x)^3}-\frac{3 (b c-a d)^4}{(a+b x)^4}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^5}-\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{(a+b x)^4}}{4 b g^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 1306, normalized size = 6.3 \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 = 1.1692, size = 873, normalized size = 4.24 \begin{align*} -\frac{1}{48} \, 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{12 \, \log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\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 = 1.10845, size = 1284, normalized size = 6.23 \begin{align*} -\frac{3 \,{\left (4 \, A - B\right )} b^{4} c^{4} - 16 \,{\left (3 \, A - B\right )} a b^{3} c^{3} d + 36 \,{\left (2 \, A - B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \,{\left (A - B\right )} a^{3} b c d^{3} +{\left (12 \, 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 - 12 \,{\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{d e x + c e}{b x + a}\right )}{48 \,{\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 = 7.93603, size = 944, normalized size = 4.58 \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 = 1.35681, size = 967, normalized size = 4.69 \begin{align*} -\frac{B d^{4} \log \left (b x + a\right )}{4 \,{\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^{4} \log \left (d x + c\right )}{4 \,{\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 \log \left (\frac{d x + c}{b x + a}\right )}{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 )}} - \frac{12 \, B b^{3} d^{3} x^{3} - 6 \, B b^{3} c d^{2} x^{2} + 42 \, B a b^{2} d^{3} x^{2} + 4 \, B b^{3} c^{2} d x - 20 \, B a b^{2} c d^{2} x + 52 \, B a^{2} b d^{3} x + 12 \, A b^{3} c^{3} + 9 \, B b^{3} c^{3} - 36 \, A a b^{2} c^{2} d - 23 \, B a b^{2} c^{2} d + 36 \, A a^{2} b c d^{2} + 13 \, B a^{2} b c d^{2} - 12 \, A a^{3} d^{3} + 13 \, B a^{3} d^{3}}{48 \,{\left (b^{8} c^{3} g^{5} x^{4} - 3 \, a b^{7} c^{2} d g^{5} x^{4} + 3 \, a^{2} b^{6} c d^{2} g^{5} x^{4} - a^{3} b^{5} d^{3} g^{5} x^{4} + 4 \, a b^{7} c^{3} g^{5} x^{3} - 12 \, a^{2} b^{6} c^{2} d g^{5} x^{3} + 12 \, a^{3} b^{5} c d^{2} g^{5} x^{3} - 4 \, a^{4} b^{4} d^{3} g^{5} x^{3} + 6 \, a^{2} b^{6} c^{3} g^{5} x^{2} - 18 \, a^{3} b^{5} c^{2} d g^{5} x^{2} + 18 \, a^{4} b^{4} c d^{2} g^{5} x^{2} - 6 \, a^{5} b^{3} d^{3} g^{5} x^{2} + 4 \, a^{3} b^{5} c^{3} g^{5} x - 12 \, a^{4} b^{4} c^{2} d g^{5} x + 12 \, a^{5} b^{3} c d^{2} g^{5} x - 4 \, a^{6} b^{2} d^{3} g^{5} x + a^{4} b^{4} c^{3} g^{5} - 3 \, a^{5} b^{3} c^{2} d g^{5} + 3 \, a^{6} b^{2} c d^{2} g^{5} - a^{7} b d^{3} g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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