Optimal. Leaf size=84 \[ \frac{(a+b x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{2 b}+\frac{B n (b c-a d)^2 \log (c+d x)}{2 b d^2}-\frac{B n x (b c-a d)}{2 d} \]
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Rubi [A] time = 0.0908031, antiderivative size = 96, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6742, 2492, 43} \[ \frac{A (a+b x)^2}{2 b}+\frac{B n (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac{B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B n x (b c-a d)}{2 d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2492
Rule 43
Rubi steps
\begin{align*} \int (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (a+b x)+B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A (a+b x)^2}{2 b}+B \int (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A (a+b x)^2}{2 b}+\frac{B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{(B (b c-a d) n) \int \frac{a+b x}{c+d x} \, dx}{2 b}\\ &=\frac{A (a+b x)^2}{2 b}+\frac{B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{(B (b c-a d) n) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac{B (b c-a d) n x}{2 d}+\frac{A (a+b x)^2}{2 b}+\frac{B (b c-a d)^2 n \log (c+d x)}{2 b d^2}+\frac{B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.146117, size = 126, normalized size = 1.5 \[ \frac{d \left (B d \left (2 a^2+2 a b x+b^2 x^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+b x (2 a A d+a B d n+A b d x-b B c n)\right )+B n \left (2 a^2 d^2-2 a b c d+b^2 c^2\right ) \log (c+d x)-a^2 B d^2 n \log (a+b x)}{2 b d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.51, size = 817, normalized size = 9.7 \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 [A] time = 1.15689, size = 208, normalized size = 2.48 \begin{align*} \frac{1}{2} \, B b x^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac{1}{2} \, A b x^{2} + B a x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a 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}{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 b}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.07428, size = 354, normalized size = 4.21 \begin{align*} \frac{A b^{2} d^{2} x^{2} +{\left (2 \, A a b d^{2} -{\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x +{\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x + B a^{2} d^{2} n\right )} \log \left (b x + a\right ) -{\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x -{\left (B b^{2} c^{2} - 2 \, B a b c d\right )} n\right )} \log \left (d x + c\right ) +{\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x\right )} \log \left (e\right )}{2 \, b d^{2}} \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 [A] time = 1.70434, size = 171, normalized size = 2.04 \begin{align*} \frac{B a^{2} n \log \left (b x + a\right )}{2 \, b} + \frac{1}{2} \,{\left (A b + B b\right )} x^{2} + \frac{1}{2} \,{\left (B b n x^{2} + 2 \, B a n x\right )} \log \left (b x + a\right ) - \frac{1}{2} \,{\left (B b n x^{2} + 2 \, B a n x\right )} \log \left (d x + c\right ) - \frac{{\left (B b c n - B a d n - 2 \, A a d - 2 \, B a d\right )} x}{2 \, d} + \frac{{\left (B b c^{2} n - 2 \, B a c d n\right )} \log \left (d x + c\right )}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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