Optimal. Leaf size=85 \[ -\frac{i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac{B i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]
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Rubi [B] time = 0.281116, antiderivative size = 191, normalized size of antiderivative = 2.25, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 44} \[ -\frac{d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac{i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac{B d^2 i \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac{B d^2 i \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac{B i (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{B d i}{2 b^2 g^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{(7 c+7 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac{7 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g^3 (a+b x)^3}+\frac{7 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac{(7 d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac{(7 (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac{7 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{7 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac{(7 B d) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{(7 B (b c-a d)) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac{7 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{7 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac{(7 B d (b c-a d)) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (7 B (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac{7 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{7 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac{(7 B d (b c-a d)) \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^2 g^3}+\frac{\left (7 B (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}{2 b^2 g^3}\\ &=-\frac{7 B (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{7 B d}{2 b^2 g^3 (a+b x)}-\frac{7 B d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac{7 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{7 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac{7 B d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}\\ \end{align*}
Mathematica [B] time = 0.162672, size = 208, normalized size = 2.45 \[ \frac{i \left (-\frac{d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 (a+b x)}-\frac{(b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 (a+b x)^2}-\frac{B \left (-\frac{2 d^2 \log (a+b x)}{b c-a d}+\frac{2 d^2 \log (c+d x)}{b c-a d}+\frac{b c-a d}{(a+b x)^2}-\frac{2 d}{a+b x}\right )}{4 b^2}-\frac{B d \left (\frac{d \log (a+b x)}{b c-a d}-\frac{d \log (c+d x)}{b c-a d}+\frac{1}{a+b x}\right )}{b^2}\right )}{g^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 394, normalized size = 4.6 \begin{align*}{\frac{d{e}^{2}iAa}{2\, \left ( ad-bc \right ) ^{2}{g}^{3}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-2}}-{\frac{{e}^{2}iAbc}{2\, \left ( ad-bc \right ) ^{2}{g}^{3}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-2}}+{\frac{d{e}^{2}iBa}{2\, \left ( ad-bc \right ) ^{2}{g}^{3}}\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 ) ^{-2}}-{\frac{{e}^{2}iBbc}{2\, \left ( ad-bc \right ) ^{2}{g}^{3}}\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 ) ^{-2}}+{\frac{d{e}^{2}iBa}{4\, \left ( ad-bc \right ) ^{2}{g}^{3}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-2}}-{\frac{{e}^{2}iBbc}{4\, \left ( ad-bc \right ) ^{2}{g}^{3}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.37955, size = 770, normalized size = 9.06 \begin{align*} -\frac{1}{4} \, B d i{\left (\frac{2 \,{\left (2 \, b x + a\right )} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} + \frac{3 \, a b c - a^{2} d + 2 \,{\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac{2 \,{\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac{2 \,{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac{1}{4} \, B c i{\left (\frac{2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x +{\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac{2 \, \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac{2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac{2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac{{\left (2 \, b x + a\right )} A d i}{2 \,{\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac{A c i}{2 \,{\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.08231, size = 366, normalized size = 4.31 \begin{align*} -\frac{2 \,{\left ({\left (2 \, A + B\right )} b^{2} c d -{\left (2 \, A + B\right )} a b d^{2}\right )} i x +{\left ({\left (2 \, A + B\right )} b^{2} c^{2} -{\left (2 \, A + B\right )} a^{2} d^{2}\right )} i + 2 \,{\left (B b^{2} d^{2} i x^{2} + 2 \, B b^{2} c d i x + B b^{2} c^{2} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.21001, size = 382, normalized size = 4.49 \begin{align*} - \frac{B d^{2} i \log{\left (x + \frac{- \frac{B a^{2} d^{4} i}{a d - b c} + \frac{2 B a b c d^{3} i}{a d - b c} + B a d^{3} i - \frac{B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac{B d^{2} i \log{\left (x + \frac{\frac{B a^{2} d^{4} i}{a d - b c} - \frac{2 B a b c d^{3} i}{a d - b c} + B a d^{3} i + \frac{B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} - \frac{2 A a d i + 2 A b c i + B a d i + B b c i + x \left (4 A b d i + 2 B b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac{\left (- B a d i - B b c i - 2 B b d i x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35108, size = 286, normalized size = 3.36 \begin{align*} \frac{B d^{2} \log \left (b x + a\right )}{2 \,{\left (b^{3} c g^{3} i - a b^{2} d g^{3} i\right )}} - \frac{B d^{2} \log \left (d x + c\right )}{2 \,{\left (b^{3} c g^{3} i - a b^{2} d g^{3} i\right )}} - \frac{{\left (2 \, B b d i x + B b c i + B a d i\right )} \log \left (\frac{b x + a}{d x + c}\right )}{2 \,{\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac{4 \, A b d i x + 6 \, B b d i x + 2 \, A b c i + 3 \, B b c i + 2 \, A a d i + 3 \, B a d i}{4 \,{\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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