Optimal. Leaf size=124 \[ \frac{g^2 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b}+\frac{B g^2 n x (b c-a d)^2}{3 d^2}-\frac{B g^2 n (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac{B g^2 n (a+b x)^2 (b c-a d)}{6 b d} \]
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Rubi [A] time = 0.089504, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2525, 12, 43} \[ \frac{g^2 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b}+\frac{B g^2 n x (b c-a d)^2}{3 d^2}-\frac{B g^2 n (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac{B g^2 n (a+b x)^2 (b c-a d)}{6 b d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac{(B n) \int \frac{(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{3 b g}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac{\left (B (b c-a d) g^2 n\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac{\left (B (b c-a d) g^2 n\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b}\\ &=\frac{B (b c-a d)^2 g^2 n x}{3 d^2}-\frac{B (b c-a d) g^2 n (a+b x)^2}{6 b d}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac{B (b c-a d)^3 g^2 n \log (c+d x)}{3 b d^3}\\ \end{align*}
Mathematica [A] time = 0.0621068, size = 103, normalized size = 0.83 \[ \frac{g^2 \left (\frac{B n (a d-b c) \left (d \left (a^2 d+4 a b d x+b^2 x (d x-2 c)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}+(a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.402, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21147, size = 417, normalized size = 3.36 \begin{align*} \frac{1}{3} \, B b^{2} g^{2} x^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{1}{3} \, A b^{2} g^{2} x^{3} + B a b g^{2} x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A a b g^{2} x^{2} + \frac{1}{6} \, B b^{2} g^{2} n{\left (\frac{2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - B a b g^{2} n{\left (\frac{a^{2} \log \left (b x + a\right )}{b^{2}} - \frac{c^{2} \log \left (d x + c\right )}{d^{2}} + \frac{{\left (b c - a d\right )} x}{b d}\right )} + B a^{2} g^{2} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B a^{2} g^{2} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A a^{2} g^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.906362, size = 622, normalized size = 5.02 \begin{align*} \frac{2 \, A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} n \log \left (b x + a\right ) - 2 \,{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} n \log \left (d x + c\right ) +{\left (6 \, A a b^{2} d^{3} g^{2} -{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} g^{2} n\right )} x^{2} + 2 \,{\left (3 \, A a^{2} b d^{3} g^{2} +{\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + 2 \, B a^{2} b d^{3}\right )} g^{2} n\right )} x + 2 \,{\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (e\right ) + 2 \,{\left (B b^{3} d^{3} g^{2} n x^{3} + 3 \, B a b^{2} d^{3} g^{2} n x^{2} + 3 \, B a^{2} b d^{3} g^{2} n x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{6 \, b d^{3}} \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 [B] time = 26.7209, size = 354, normalized size = 2.85 \begin{align*} \frac{B a^{3} g^{2} n \log \left (b x + a\right )}{3 \, b} + \frac{1}{3} \,{\left (A b^{2} g^{2} + B b^{2} g^{2}\right )} x^{3} - \frac{{\left (B b^{2} c g^{2} n - B a b d g^{2} n - 6 \, A a b d g^{2} - 6 \, B a b d g^{2}\right )} x^{2}}{6 \, d} + \frac{1}{3} \,{\left (B b^{2} g^{2} n x^{3} + 3 \, B a b g^{2} n x^{2} + 3 \, B a^{2} g^{2} n x\right )} \log \left (\frac{b x + a}{d x + c}\right ) + \frac{{\left (B b^{2} c^{2} g^{2} n - 3 \, B a b c d g^{2} n + 2 \, B a^{2} d^{2} g^{2} n + 3 \, A a^{2} d^{2} g^{2} + 3 \, B a^{2} d^{2} g^{2}\right )} x}{3 \, d^{2}} - \frac{{\left (B b^{2} c^{3} g^{2} n - 3 \, B a b c^{2} d g^{2} n + 3 \, B a^{2} c d^{2} g^{2} n\right )} \log \left (-d x - c\right )}{3 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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