Optimal. Leaf size=53 \[ \frac{(a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac{2 a (a c-b c x)^{n+1}}{b c (n+1)} \]
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Rubi [A] time = 0.0170643, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{(a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac{2 a (a c-b c x)^{n+1}}{b c (n+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (a c-b c x)^n \, dx &=\int \left (2 a (a c-b c x)^n-\frac{(a c-b c x)^{1+n}}{c}\right ) \, dx\\ &=-\frac{2 a (a c-b c x)^{1+n}}{b c (1+n)}+\frac{(a c-b c x)^{2+n}}{b c^2 (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0210716, size = 43, normalized size = 0.81 \[ \frac{(b x-a) (a (n+3)+b (n+1) x) (c (a-b x))^n}{b (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 47, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -bcx+ac \right ) ^{n} \left ( bnx+an+bx+3\,a \right ) \left ( -bx+a \right ) }{b \left ({n}^{2}+3\,n+2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13825, size = 117, normalized size = 2.21 \begin{align*} -\frac{{\left (a^{2} n - 2 \, a b x -{\left (b^{2} n + b^{2}\right )} x^{2} + 3 \, a^{2}\right )}{\left (-b c x + a c\right )}^{n}}{b n^{2} + 3 \, b n + 2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.615571, size = 245, normalized size = 4.62 \begin{align*} \begin{cases} a x \left (a c\right )^{n} & \text{for}\: b = 0 \\- \frac{a \log{\left (- \frac{a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} - \frac{2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac{b x \log{\left (- \frac{a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} & \text{for}\: n = -2 \\- \frac{2 a \log{\left (- \frac{a}{b} + x \right )}}{b c} - \frac{x}{c} & \text{for}\: n = -1 \\- \frac{a^{2} n \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} - \frac{3 a^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac{2 a b x \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac{b^{2} n x^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac{b^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05295, size = 139, normalized size = 2.62 \begin{align*} \frac{{\left (-b c x + a c\right )}^{n} b^{2} n x^{2} +{\left (-b c x + a c\right )}^{n} b^{2} x^{2} -{\left (-b c x + a c\right )}^{n} a^{2} n + 2 \,{\left (-b c x + a c\right )}^{n} a b x - 3 \,{\left (-b c x + a c\right )}^{n} a^{2}}{b n^{2} + 3 \, b n + 2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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