Optimal. Leaf size=125 \[ \frac{B g (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 i}+\frac{g (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A+B\right )}{d^2 i}+\frac{g (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i} \]
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Rubi [A] time = 0.354912, antiderivative size = 213, normalized size of antiderivative = 1.7, number of steps used = 14, number of rules used = 11, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2528, 2486, 31, 2524, 12, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B g (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac{g (b c-a d) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 i}-\frac{B g (b c-a d) \log ^2(c+d x)}{2 d^2 i}+\frac{B g (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i}-\frac{B g (b c-a d) \log (c+d x)}{d^2 i}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d i}+\frac{A b g x}{d i} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 12
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
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 )}{33 c+33 d x} \, dx &=\int \left (\frac{b g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{33 d}+\frac{(-b c+a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{33 d (c+d x)}\right ) \, dx\\ &=\frac{(b g) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{33 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} \, dx}{33 d}\\ &=\frac{A b g x}{33 d}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac{(b B g) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{33 d}+\frac{(B (b c-a d) g) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{33 d^2}\\ &=\frac{A b g x}{33 d}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{33 d}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac{(B (b c-a d) g) \int \frac{1}{c+d x} \, dx}{33 d}+\frac{(B (b c-a d) g) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{33 d^2 e}\\ &=\frac{A b g x}{33 d}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{33 d}-\frac{B (b c-a d) g \log (c+d x)}{33 d^2}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac{(B (b c-a d) g) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{33 d^2 e}\\ &=\frac{A b g x}{33 d}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{33 d}-\frac{B (b c-a d) g \log (c+d x)}{33 d^2}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac{(b B (b c-a d) g) \int \frac{\log (c+d x)}{a+b x} \, dx}{33 d^2}-\frac{(B (b c-a d) g) \int \frac{\log (c+d x)}{c+d x} \, dx}{33 d}\\ &=\frac{A b g x}{33 d}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{33 d}-\frac{B (b c-a d) g \log (c+d x)}{33 d^2}+\frac{B (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac{(B (b c-a d) g) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{33 d^2}-\frac{(B (b c-a d) g) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{33 d}\\ &=\frac{A b g x}{33 d}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{33 d}-\frac{B (b c-a d) g \log (c+d x)}{33 d^2}+\frac{B (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac{B (b c-a d) g \log ^2(c+d x)}{66 d^2}-\frac{(B (b c-a d) g) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{33 d^2}\\ &=\frac{A b g x}{33 d}+\frac{B g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{33 d}-\frac{B (b c-a d) g \log (c+d x)}{33 d^2}+\frac{B (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac{(b c-a d) g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac{B (b c-a d) g \log ^2(c+d x)}{66 d^2}+\frac{B (b c-a d) g \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{33 d^2}\\ \end{align*}
Mathematica [A] time = 0.107321, size = 162, normalized size = 1.3 \[ \frac{g \left (B (b c-a d) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 (b c-a d) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )-2 B (b c-a d) \log (c+d x)+2 A b d x\right )}{2 d^2 i} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.158, size = 895, normalized size = 7.2 \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 [A] time = 1.49421, size = 298, normalized size = 2.38 \begin{align*} A b g{\left (\frac{x}{d i} - \frac{c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac{A a g \log \left (d i x + c i\right )}{d i} - \frac{{\left (b c g - a d g\right )}{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )} B}{d^{2} i} + \frac{{\left (a d g \log \left (e\right ) -{\left (g \log \left (e\right ) + g\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} - \frac{2 \, B b d g x \log \left (d x + c\right ) - 2 \, B b d g x \log \left (e\right ) -{\left (b c g - a d g\right )} B \log \left (d x + c\right )^{2} - 2 \,{\left (B b d g x + B a d g\right )} \log \left (b x + a\right )}{2 \, d^{2} i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A b g x + A a g +{\left (B b g x + B a g\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{d i x + c i}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b g x + a g\right )}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{d i x + c i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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