Optimal. Leaf size=260 \[ \frac{2 b B g^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac{b g^2 (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )+2 A+B\right )}{d^3 i^2}+\frac{g^2 (2 A+B) (a+b x) (b c-a d)}{d^2 i^2 (c+d x)}+\frac{g^2 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (c+d x)}+\frac{2 B g^2 (a+b x) (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}-\frac{2 B g^2 (a+b x) (b c-a d)}{d^2 i^2 (c+d x)} \]
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Rubi [A] time = 0.532412, antiderivative size = 336, normalized size of antiderivative = 1.29, number of steps used = 18, number of rules used = 13, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.325, Rules used = {2528, 2486, 31, 2525, 12, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{2 b B g^2 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^3 i^2}-\frac{g^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^2 (c+d x)}-\frac{2 b g^2 (b c-a d) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^2}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^2 i^2}+\frac{B g^2 (b c-a d)^2}{d^3 i^2 (c+d x)}-\frac{b B g^2 (b c-a d) \log ^2(c+d x)}{d^3 i^2}+\frac{b B g^2 (b c-a d) \log (a+b x)}{d^3 i^2}+\frac{2 b B g^2 (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^3 i^2}-\frac{2 b B g^2 (b c-a d) \log (c+d x)}{d^3 i^2}+\frac{A b^2 g^2 x}{d^2 i^2} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int \frac{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(40 c+40 d x)^2} \, dx &=\int \left (\frac{b^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^2}+\frac{(-b c+a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^2 (c+d x)^2}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{800 d^2 (c+d x)}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{1600 d^2}-\frac{\left (b (b c-a d) g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{800 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1600 d^2}\\ &=\frac{A b^2 g^2 x}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}+\frac{\left (b^2 B g^2\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{1600 d^2}+\frac{\left (b B (b c-a d) g^2\right ) \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}{800 d^3}+\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1600 d^3}\\ &=\frac{A b^2 g^2 x}{1600 d^2}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \int \frac{1}{c+d x} \, dx}{1600 d^2}+\frac{\left (B (b c-a d)^3 g^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1600 d^3}+\frac{\left (b B (b c-a d) g^2\right ) \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}{800 d^3 e}\\ &=\frac{A b^2 g^2 x}{1600 d^2}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b B (b c-a d) g^2 \log (c+d x)}{1600 d^3}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}+\frac{\left (B (b c-a d)^3 g^2\right ) \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}{1600 d^3}+\frac{\left (b B (b c-a d) g^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{800 d^3 e}\\ &=\frac{A b^2 g^2 x}{1600 d^2}+\frac{B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac{b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b B (b c-a d) g^2 \log (c+d x)}{800 d^3}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}+\frac{\left (b^2 B (b c-a d) g^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{800 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{800 d^2}\\ &=\frac{A b^2 g^2 x}{1600 d^2}+\frac{B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac{b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b B (b c-a d) g^2 \log (c+d x)}{800 d^3}+\frac{b B (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{800 d^3}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{800 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{800 d^2}\\ &=\frac{A b^2 g^2 x}{1600 d^2}+\frac{B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac{b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b B (b c-a d) g^2 \log (c+d x)}{800 d^3}+\frac{b B (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{800 d^3}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac{b B (b c-a d) g^2 \log ^2(c+d x)}{1600 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{800 d^3}\\ &=\frac{A b^2 g^2 x}{1600 d^2}+\frac{B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac{b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac{b B g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1600 d^2}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1600 d^3 (c+d x)}-\frac{b B (b c-a d) g^2 \log (c+d x)}{800 d^3}+\frac{b B (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{800 d^3}-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac{b B (b c-a d) g^2 \log ^2(c+d x)}{1600 d^3}+\frac{b B (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{800 d^3}\\ \end{align*}
Mathematica [A] time = 0.244676, size = 239, normalized size = 0.92 \[ \frac{g^2 \left (b B (b c-a d) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 b (b c-a d) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{(b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+b B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )+\frac{B (b c-a d)^2}{c+d x}+b B (b c-a d) \log (a+b x)-2 b B (b c-a d) \log (c+d x)+A b^2 d x\right )}{d^3 i^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 1382, normalized size = 5.3 \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 = 1.53016, size = 1196, normalized size = 4.6 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} +{\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{d^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, 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 )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (d i x + c i\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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