Optimal. Leaf size=113 \[ \frac{(a+b x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b}+\frac{B n x (b c-a d)^2}{3 d^2}-\frac{B n (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac{B n (a+b x)^2 (b c-a d)}{6 b d} \]
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Rubi [A] time = 0.120999, antiderivative size = 125, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {6742, 2492, 43} \[ \frac{A (a+b x)^3}{3 b}+\frac{B n x (b c-a d)^2}{3 d^2}-\frac{B n (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac{B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac{B n (a+b x)^2 (b c-a d)}{6 b d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2492
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (a+b x)^2+B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A (a+b x)^3}{3 b}+B \int (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A (a+b x)^3}{3 b}+\frac{B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac{(B (b c-a d) n) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b}\\ &=\frac{A (a+b x)^3}{3 b}+\frac{B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac{(B (b c-a d) n) \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 n x}{3 d^2}-\frac{B (b c-a d) n (a+b x)^2}{6 b d}+\frac{A (a+b x)^3}{3 b}-\frac{B (b c-a d)^3 n \log (c+d x)}{3 b d^3}+\frac{B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}\\ \end{align*}
Mathematica [A] time = 0.281757, size = 194, normalized size = 1.72 \[ \frac{b d x \left (2 a^2 d^2 (3 A+2 B n)+a b d (6 A d x-6 B c n+B d n x)+b^2 \left (2 A d^2 x^2+B c n (2 c-d x)\right )\right )-2 B n \left (3 a^2 b c d^2-3 a^3 d^3-3 a b^2 c^2 d+b^3 c^3\right ) \log (c+d x)+2 B d^3 \left (3 a^2 b x+3 a^3+3 a b^2 x^2+b^3 x^3\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-4 a^3 B d^3 n \log (a+b x)}{6 b d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.527, size = 1325, normalized size = 11.7 \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 = 1.28175, size = 397, normalized size = 3.51 \begin{align*} \frac{1}{3} \, B b^{2} x^{3} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac{1}{3} \, A b^{2} x^{3} + B a b x^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a b x^{2} + B a^{2} x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a^{2} x + \frac{{\left (\frac{a e n \log \left (b x + a\right )}{b} - \frac{c e n \log \left (d x + c\right )}{d}\right )} B a^{2}}{e} - \frac{{\left (\frac{a^{2} e n \log \left (b x + a\right )}{b^{2}} - \frac{c^{2} e n \log \left (d x + c\right )}{d^{2}} + \frac{{\left (b c e n - a d e n\right )} x}{b d}\right )} B a b}{e} + \frac{{\left (\frac{2 \, a^{3} e n \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} e n \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d e n - a b d^{2} e n\right )} x^{2} - 2 \,{\left (b^{2} c^{2} e n - a^{2} d^{2} e n\right )} x}{b^{2} d^{2}}\right )} B b^{2}}{6 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.04023, size = 599, normalized size = 5.3 \begin{align*} \frac{2 \, A b^{3} d^{3} x^{3} +{\left (6 \, A a b^{2} d^{3} -{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} + 2 \,{\left (3 \, A a^{2} b d^{3} +{\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + 2 \, B a^{2} b d^{3}\right )} n\right )} x + 2 \,{\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + B a^{3} d^{3} n\right )} \log \left (b x + a\right ) - 2 \,{\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x +{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (d x + c\right ) + 2 \,{\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x\right )} \log \left (e\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 = 2.29535, size = 317, normalized size = 2.81 \begin{align*} \frac{B a^{3} n \log \left (b x + a\right )}{3 \, b} + \frac{1}{3} \,{\left (A b^{2} + B b^{2}\right )} x^{3} - \frac{{\left (B b^{2} c n - B a b d n - 6 \, A a b d - 6 \, B a b d\right )} x^{2}}{6 \, d} + \frac{1}{3} \,{\left (B b^{2} n x^{3} + 3 \, B a b n x^{2} + 3 \, B a^{2} n x\right )} \log \left (b x + a\right ) - \frac{1}{3} \,{\left (B b^{2} n x^{3} + 3 \, B a b n x^{2} + 3 \, B a^{2} n x\right )} \log \left (d x + c\right ) + \frac{{\left (B b^{2} c^{2} n - 3 \, B a b c d n + 2 \, B a^{2} d^{2} n + 3 \, A a^{2} d^{2} + 3 \, B a^{2} d^{2}\right )} x}{3 \, d^{2}} - \frac{{\left (B b^{2} c^{3} n - 3 \, B a b c^{2} d n + 3 \, B a^{2} c d^{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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