Optimal. Leaf size=85 \[ \frac{g (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}-\frac{B g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]
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Rubi [B] time = 0.292345, 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} \[ -\frac{b g \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (c+d x)}+\frac{g (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i^3 (c+d x)^2}+\frac{b^2 B g \log (a+b x)}{2 d^2 i^3 (b c-a d)}-\frac{b^2 B g \log (c+d x)}{2 d^2 i^3 (b c-a d)}-\frac{B g (b c-a d)}{4 d^2 i^3 (c+d x)^2}+\frac{b B g}{2 d^2 i^3 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{(a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(49 c+49 d x)^3} \, dx &=\int \left (\frac{(-b c+a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{117649 d (c+d x)^3}+\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{117649 d (c+d x)^2}\right ) \, dx\\ &=\frac{(b g) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{117649 d}-\frac{((b c-a d) g) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{117649 d}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}+\frac{(b B g) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{117649 d^2}-\frac{(B (b c-a d) g) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{235298 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}+\frac{(b B (b c-a d) g) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{117649 d^2}-\frac{\left (B (b c-a d)^2 g\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{235298 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}+\frac{(b B (b c-a d) g) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{117649 d^2}-\frac{\left (B (b c-a d)^2 g\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{235298 d^2}\\ &=-\frac{B (b c-a d) g}{470596 d^2 (c+d x)^2}+\frac{b B g}{235298 d^2 (c+d x)}+\frac{b^2 B g \log (a+b x)}{235298 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}-\frac{b^2 B g \log (c+d x)}{235298 d^2 (b c-a d)}\\ \end{align*}
Mathematica [B] time = 0.151509, size = 207, normalized size = 2.44 \[ \frac{g \left (-\frac{b \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 (c+d x)}+\frac{(b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 (c+d x)^2}-\frac{B \left (\frac{2 b^2 \log (a+b x)}{b c-a d}-\frac{2 b^2 \log (c+d x)}{b c-a d}+\frac{b c-a d}{(c+d x)^2}+\frac{2 b}{c+d x}\right )}{4 d^2}+\frac{b B \left (\frac{b \log (a+b x)}{b c-a d}-\frac{b \log (c+d x)}{b c-a d}+\frac{1}{c+d x}\right )}{d^2}\right )}{i^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 1049, normalized size = 12.3 \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 = 1.25232, size = 765, normalized size = 9. \begin{align*} -\frac{1}{4} \, B b g{\left (\frac{2 \,{\left (2 \, d x + c\right )} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{d^{4} i^{3} x^{2} + 2 \, c d^{3} i^{3} x + c^{2} d^{2} i^{3}} - \frac{b c^{2} - 3 \, a c d + 2 \,{\left (b c d - 2 \, a d^{2}\right )} x}{{\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}} - \frac{2 \,{\left (b^{2} c - 2 \, a b d\right )} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} i^{3}} + \frac{2 \,{\left (b^{2} c - 2 \, a b d\right )} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} i^{3}}\right )} + \frac{1}{4} \, B a g{\left (\frac{2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x +{\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} - \frac{2 \, \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}} + \frac{2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac{2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac{{\left (2 \, d x + c\right )} A b g}{2 \,{\left (d^{4} i^{3} x^{2} + 2 \, c d^{3} i^{3} x + c^{2} d^{2} i^{3}\right )}} - \frac{A a g}{2 \,{\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.476337, 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 )} g x +{\left ({\left (2 \, A - B\right )} b^{2} c^{2} -{\left (2 \, A - B\right )} a^{2} d^{2}\right )} g - 2 \,{\left (B b^{2} d^{2} g x^{2} + 2 \, B a b d^{2} g x + B a^{2} d^{2} g\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.02962, size = 382, normalized size = 4.49 \begin{align*} \frac{B b^{2} g \log{\left (x + \frac{- \frac{B a^{2} b^{2} d^{2} g}{a d - b c} + \frac{2 B a b^{3} c d g}{a d - b c} + B a b^{2} d g - \frac{B b^{4} c^{2} g}{a d - b c} + B b^{3} c g}{2 B b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac{B b^{2} g \log{\left (x + \frac{\frac{B a^{2} b^{2} d^{2} g}{a d - b c} - \frac{2 B a b^{3} c d g}{a d - b c} + B a b^{2} d g + \frac{B b^{4} c^{2} g}{a d - b c} + B b^{3} c g}{2 B b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac{2 A a d g + 2 A b c g - B a d g - B b c g + x \left (4 A b d g - 2 B b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac{\left (- B a d g - B b c g - 2 B b d g x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38173, size = 258, normalized size = 3.04 \begin{align*} -\frac{B b^{2} g \log \left (b x + a\right )}{2 \,{\left (b c d^{2} i - a d^{3} i\right )}} + \frac{B b^{2} g \log \left (d x + c\right )}{2 \,{\left (b c d^{2} i - a d^{3} i\right )}} - \frac{{\left (2 \, B b d g i x + B b c g i + B a d g i\right )} \log \left (\frac{b x + a}{d x + c}\right )}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} - \frac{4 \, A b d g i x + 2 \, B b d g i x + 2 \, A b c g i + B b c g i + 2 \, A a d g i + B a d g i}{4 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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