Optimal. Leaf size=89 \[ -\frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)}-\frac{B i^3 (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)} \]
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Rubi [B] time = 0.72228, antiderivative size = 373, normalized size of antiderivative = 4.19, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^3 i^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^5 (a+b x)}-\frac{3 d^2 i^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 g^5 (a+b x)^2}-\frac{d i^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^5 (a+b x)^3}-\frac{i^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^4 g^5 (a+b x)^4}-\frac{3 B d^2 i^3 (b c-a d)}{8 b^4 g^5 (a+b x)^2}-\frac{B d^4 i^3 \log (a+b x)}{4 b^4 g^5 (b c-a d)}+\frac{B d^4 i^3 \log (c+d x)}{4 b^4 g^5 (b c-a d)}-\frac{B d i^3 (b c-a d)^2}{4 b^4 g^5 (a+b x)^3}-\frac{B i^3 (b c-a d)^3}{16 b^4 g^5 (a+b x)^4}-\frac{B d^3 i^3}{4 b^4 g^5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{(28 c+28 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac{21952 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^5}+\frac{65856 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}+\frac{65856 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}+\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}\right ) \, dx\\ &=\frac{\left (21952 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^5}+\frac{\left (65856 d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^5}+\frac{\left (65856 d (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^5}+\frac{\left (21952 (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^3 g^5}\\ &=-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (21952 B d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (32928 B d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (21952 B d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (5488 B (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^5}\\ &=-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (21952 B d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (32928 B d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (21952 B d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (5488 B (b c-a d)^4\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^5}\\ &=-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (21952 B d^3 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^5}+\frac{\left (32928 B d^2 (b c-a d)^2\right ) \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^4 g^5}+\frac{\left (21952 B d (b c-a d)^3\right ) \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}{b^4 g^5}+\frac{\left (5488 B (b c-a d)^4\right ) \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}{b^4 g^5}\\ &=-\frac{1372 B (b c-a d)^3}{b^4 g^5 (a+b x)^4}-\frac{5488 B d (b c-a d)^2}{b^4 g^5 (a+b x)^3}-\frac{8232 B d^2 (b c-a d)}{b^4 g^5 (a+b x)^2}-\frac{5488 B d^3}{b^4 g^5 (a+b x)}-\frac{5488 B d^4 \log (a+b x)}{b^4 (b c-a d) g^5}-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{5488 B d^4 \log (c+d x)}{b^4 (b c-a d) g^5}\\ \end{align*}
Mathematica [B] time = 0.495376, size = 427, normalized size = 4.8 \[ -\frac{i^3 \left (-24 a^2 A b^2 d^4 x^2-16 a^3 A b d^4 x-4 a^4 A d^4+4 B \left (-6 a^2 b^2 d^4 x^2-4 a^3 b d^4 x-a^4 d^4-4 a b^3 d^4 x^3+b^4 c \left (4 c^2 d x+c^3+6 c d^2 x^2+4 d^3 x^3\right )\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )-24 a^2 b^2 B d^4 x^2 \log (c+d x)-6 a^2 b^2 B d^4 x^2-16 a^3 b B d^4 x \log (c+d x)-4 a^3 b B d^4 x-4 a^4 B d^4 \log (c+d x)-a^4 B d^4-16 a A b^3 d^4 x^3-16 a b^3 B d^4 x^3 \log (c+d x)-4 a b^3 B d^4 x^3+4 B d^4 (a+b x)^4 \log (a+b x)+24 A b^4 c^2 d^2 x^2+16 A b^4 c^3 d x+4 A b^4 c^4+16 A b^4 c d^3 x^3+6 b^4 B c^2 d^2 x^2+4 b^4 B c^3 d x+b^4 B c^4+4 b^4 B c d^3 x^3-4 b^4 B d^4 x^4 \log (c+d x)\right )}{16 b^4 g^5 (a+b x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 406, normalized size = 4.6 \begin{align*}{\frac{{e}^{4}d{i}^{3}Aa}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}-{\frac{{e}^{4}{i}^{3}Abc}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}+{\frac{{e}^{4}d{i}^{3}Ba}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}-{\frac{{e}^{4}{i}^{3}Bbc}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}+{\frac{{e}^{4}d{i}^{3}Ba}{16\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}-{\frac{{e}^{4}{i}^{3}Bbc}{16\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.99735, size = 4194, normalized size = 47.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.542133, size = 710, normalized size = 7.98 \begin{align*} -\frac{4 \,{\left ({\left (4 \, A + B\right )} b^{4} c d^{3} -{\left (4 \, A + B\right )} a b^{3} d^{4}\right )} i^{3} x^{3} + 6 \,{\left ({\left (4 \, A + B\right )} b^{4} c^{2} d^{2} -{\left (4 \, A + B\right )} a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 4 \,{\left ({\left (4 \, A + B\right )} b^{4} c^{3} d -{\left (4 \, A + B\right )} a^{3} b d^{4}\right )} i^{3} x +{\left ({\left (4 \, A + B\right )} b^{4} c^{4} -{\left (4 \, A + B\right )} a^{4} d^{4}\right )} i^{3} + 4 \,{\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x + B b^{4} c^{4} i^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{16 \,{\left ({\left (b^{9} c - a b^{8} d\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c - a^{2} b^{7} d\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c - a^{3} b^{6} d\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c - a^{4} b^{5} d\right )} g^{5} x +{\left (a^{4} b^{5} c - a^{5} b^{4} d\right )} g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 157.452, size = 864, normalized size = 9.71 \begin{align*} - \frac{B d^{4} i^{3} \log{\left (x + \frac{- \frac{B a^{2} d^{6} i^{3}}{a d - b c} + \frac{2 B a b c d^{5} i^{3}}{a d - b c} + B a d^{5} i^{3} - \frac{B b^{2} c^{2} d^{4} i^{3}}{a d - b c} + B b c d^{4} i^{3}}{2 B b d^{5} i^{3}} \right )}}{4 b^{4} g^{5} \left (a d - b c\right )} + \frac{B d^{4} i^{3} \log{\left (x + \frac{\frac{B a^{2} d^{6} i^{3}}{a d - b c} - \frac{2 B a b c d^{5} i^{3}}{a d - b c} + B a d^{5} i^{3} + \frac{B b^{2} c^{2} d^{4} i^{3}}{a d - b c} + B b c d^{4} i^{3}}{2 B b d^{5} i^{3}} \right )}}{4 b^{4} g^{5} \left (a d - b c\right )} - \frac{4 A a^{3} d^{3} i^{3} + 4 A a^{2} b c d^{2} i^{3} + 4 A a b^{2} c^{2} d i^{3} + 4 A b^{3} c^{3} i^{3} + B a^{3} d^{3} i^{3} + B a^{2} b c d^{2} i^{3} + B a b^{2} c^{2} d i^{3} + B b^{3} c^{3} i^{3} + x^{3} \left (16 A b^{3} d^{3} i^{3} + 4 B b^{3} d^{3} i^{3}\right ) + x^{2} \left (24 A a b^{2} d^{3} i^{3} + 24 A b^{3} c d^{2} i^{3} + 6 B a b^{2} d^{3} i^{3} + 6 B b^{3} c d^{2} i^{3}\right ) + x \left (16 A a^{2} b d^{3} i^{3} + 16 A a b^{2} c d^{2} i^{3} + 16 A b^{3} c^{2} d i^{3} + 4 B a^{2} b d^{3} i^{3} + 4 B a b^{2} c d^{2} i^{3} + 4 B b^{3} c^{2} d i^{3}\right )}{16 a^{4} b^{4} g^{5} + 64 a^{3} b^{5} g^{5} x + 96 a^{2} b^{6} g^{5} x^{2} + 64 a b^{7} g^{5} x^{3} + 16 b^{8} g^{5} x^{4}} + \frac{\left (- B a^{3} d^{3} i^{3} - B a^{2} b c d^{2} i^{3} - 4 B a^{2} b d^{3} i^{3} x - B a b^{2} c^{2} d i^{3} - 4 B a b^{2} c d^{2} i^{3} x - 6 B a b^{2} d^{3} i^{3} x^{2} - B b^{3} c^{3} i^{3} - 4 B b^{3} c^{2} d i^{3} x - 6 B b^{3} c d^{2} i^{3} x^{2} - 4 B b^{3} d^{3} i^{3} x^{3}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{4 a^{4} b^{4} g^{5} + 16 a^{3} b^{5} g^{5} x + 24 a^{2} b^{6} g^{5} x^{2} + 16 a b^{7} g^{5} x^{3} + 4 b^{8} g^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32788, size = 776, normalized size = 8.72 \begin{align*} -\frac{B d^{4} \log \left (b x + a\right )}{4 \,{\left (b^{5} c g^{5} i - a b^{4} d g^{5} i\right )}} + \frac{B d^{4} \log \left (d x + c\right )}{4 \,{\left (b^{5} c g^{5} i - a b^{4} d g^{5} i\right )}} + \frac{{\left (4 \, B b^{3} d^{3} i x^{3} + 6 \, B b^{3} c d^{2} i x^{2} + 6 \, B a b^{2} d^{3} i x^{2} + 4 \, B b^{3} c^{2} d i x + 4 \, B a b^{2} c d^{2} i x + 4 \, B a^{2} b d^{3} i x + B b^{3} c^{3} i + B a b^{2} c^{2} d i + B a^{2} b c d^{2} i + B a^{3} d^{3} i\right )} \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left (b^{8} g^{5} x^{4} + 4 \, a b^{7} g^{5} x^{3} + 6 \, a^{2} b^{6} g^{5} x^{2} + 4 \, a^{3} b^{5} g^{5} x + a^{4} b^{4} g^{5}\right )}} + \frac{16 \, A b^{3} d^{3} i x^{3} + 20 \, B b^{3} d^{3} i x^{3} + 24 \, A b^{3} c d^{2} i x^{2} + 30 \, B b^{3} c d^{2} i x^{2} + 24 \, A a b^{2} d^{3} i x^{2} + 30 \, B a b^{2} d^{3} i x^{2} + 16 \, A b^{3} c^{2} d i x + 20 \, B b^{3} c^{2} d i x + 16 \, A a b^{2} c d^{2} i x + 20 \, B a b^{2} c d^{2} i x + 16 \, A a^{2} b d^{3} i x + 20 \, B a^{2} b d^{3} i x + 4 \, A b^{3} c^{3} i + 5 \, B b^{3} c^{3} i + 4 \, A a b^{2} c^{2} d i + 5 \, B a b^{2} c^{2} d i + 4 \, A a^{2} b c d^{2} i + 5 \, B a^{2} b c d^{2} i + 4 \, A a^{3} d^{3} i + 5 \, B a^{3} d^{3} i}{16 \,{\left (b^{8} g^{5} x^{4} + 4 \, a b^{7} g^{5} x^{3} + 6 \, a^{2} b^{6} g^{5} x^{2} + 4 \, a^{3} b^{5} g^{5} x + a^{4} b^{4} g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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