Optimal. Leaf size=253 \[ \frac{2 B^2 g^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{3 b d^3}+\frac{B g^2 (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{3 b d^3}+\frac{B g^2 (a+b x) (b c-a d)^2 \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )+2 A+B\right )}{3 b d^2}-\frac{B g^2 (a+b x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 b} \]
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Rubi [A] time = 0.552817, antiderivative size = 389, normalized size of antiderivative = 1.54, number of steps used = 20, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {2525, 12, 2528, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{2 B^2 g^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b d^3}-\frac{2 B g^2 (b c-a d)^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b d^3}+\frac{2 A B g^2 x (b c-a d)^2}{3 d^2}-\frac{B g^2 (a+b x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 b}+\frac{2 B^2 g^2 (a+b x) (b c-a d)^2 \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}+\frac{B^2 g^2 x (b c-a d)^2}{3 d^2}-\frac{B^2 g^2 (b c-a d)^3 \log ^2(c+d x)}{3 b d^3}-\frac{B^2 g^2 (b c-a d)^3 \log (c+d x)}{b d^3}+\frac{2 B^2 g^2 (b c-a d)^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b d^3} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 43
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{(2 B) \int \frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b g}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{\left (2 B (b c-a d) g^2\right ) \int \frac{(a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{\left (2 B (b c-a d) g^2\right ) \int \left (-\frac{b (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{b (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{(-b c+a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2 (c+d x)}\right ) \, dx}{3 b}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{\left (2 B (b c-a d) g^2\right ) \int (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 d}+\frac{\left (2 B (b c-a d)^2 g^2\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{3 d^2}-\frac{\left (2 B (b c-a d)^3 g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b d^2}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}+\frac{\left (B^2 (b c-a d) g^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{3 b d}+\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{3 d^2}+\frac{\left (2 B^2 (b c-a d)^3 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}{3 b d^3}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \int \frac{a+b x}{c+d x} \, dx}{3 b d}-\frac{\left (2 B^2 (b c-a d)^3 g^2\right ) \int \frac{1}{c+d x} \, dx}{3 b d^2}+\frac{\left (2 B^2 (b c-a d)^3 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}{3 b d^3 e}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{2 B^2 (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{3 b d}+\frac{\left (2 B^2 (b c-a d)^3 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}{3 b d^3 e}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac{B^2 (b c-a d)^2 g^2 x}{3 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}+\frac{\left (2 B^2 (b c-a d)^3 g^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 d^3}-\frac{\left (2 B^2 (b c-a d)^3 g^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b d^2}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac{B^2 (b c-a d)^2 g^2 x}{3 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}+\frac{2 B^2 (b c-a d)^3 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}-\frac{\left (2 B^2 (b c-a d)^3 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b d^3}-\frac{\left (2 B^2 (b c-a d)^3 g^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b d^2}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac{B^2 (b c-a d)^2 g^2 x}{3 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}+\frac{2 B^2 (b c-a d)^3 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}-\frac{B^2 (b c-a d)^3 g^2 \log ^2(c+d x)}{3 b d^3}-\frac{\left (2 B^2 (b c-a d)^3 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 )}{3 b d^3}\\ &=\frac{2 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac{B^2 (b c-a d)^2 g^2 x}{3 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{3 b d^2}-\frac{B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b}-\frac{B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}+\frac{2 B^2 (b c-a d)^3 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac{2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b d^3}-\frac{B^2 (b c-a d)^3 g^2 \log ^2(c+d x)}{3 b d^3}+\frac{2 B^2 (b c-a d)^3 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b d^3}\\ \end{align*}
Mathematica [A] time = 0.220747, size = 287, normalized size = 1.13 \[ \frac{g^2 \left (\frac{B (b c-a d) \left (B (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-d^2 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 (b c-a d)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )-2 B (b c-a d)^2 \log (c+d x)+B (b c-a d) ((a d-b c) \log (c+d x)+b d x)\right )}{d^3}+(a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.991, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.03334, size = 1573, normalized size = 6.22 \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 (A^{2} b^{2} g^{2} x^{2} + 2 \, A^{2} a b g^{2} x + A^{2} a^{2} g^{2} +{\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B b^{2} g^{2} x^{2} + 2 \, A B a b g^{2} x + A B a^{2} g^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right ), 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{\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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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