Optimal. Leaf size=47 \[ \frac{b (c+d x)^{n+2}}{d^2 (n+2)}-\frac{(b c-a d) (c+d x)^{n+1}}{d^2 (n+1)} \]
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Rubi [A] time = 0.018252, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{b (c+d x)^{n+2}}{d^2 (n+2)}-\frac{(b c-a d) (c+d x)^{n+1}}{d^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (c+d x)^n \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^n}{d}+\frac{b (c+d x)^{1+n}}{d}\right ) \, dx\\ &=-\frac{(b c-a d) (c+d x)^{1+n}}{d^2 (1+n)}+\frac{b (c+d x)^{2+n}}{d^2 (2+n)}\\ \end{align*}
Mathematica [A] time = 0.028029, size = 41, normalized size = 0.87 \[ \frac{(c+d x)^{n+1} (a d (n+2)-b c+b d (n+1) x)}{d^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 46, normalized size = 1. \begin{align*}{\frac{ \left ( dx+c \right ) ^{1+n} \left ( bdnx+adn+bdx+2\,ad-bc \right ) }{{d}^{2} \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 = 1.88476, size = 171, normalized size = 3.64 \begin{align*} \frac{{\left (a c d n - b c^{2} + 2 \, a c d +{\left (b d^{2} n + b d^{2}\right )} x^{2} +{\left (2 \, a d^{2} +{\left (b c d + a d^{2}\right )} n\right )} x\right )}{\left (d x + c\right )}^{n}}{d^{2} n^{2} + 3 \, d^{2} n + 2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.841468, size = 377, normalized size = 8.02 \begin{align*} \begin{cases} c^{n} \left (a x + \frac{b x^{2}}{2}\right ) & \text{for}\: d = 0 \\- \frac{a d}{c d^{2} + d^{3} x} + \frac{b c \log{\left (\frac{c}{d} + x \right )}}{c d^{2} + d^{3} x} + \frac{b c}{c d^{2} + d^{3} x} + \frac{b d x \log{\left (\frac{c}{d} + x \right )}}{c d^{2} + d^{3} x} & \text{for}\: n = -2 \\\frac{a \log{\left (\frac{c}{d} + x \right )}}{d} - \frac{b c \log{\left (\frac{c}{d} + x \right )}}{d^{2}} + \frac{b x}{d} & \text{for}\: n = -1 \\\frac{a c d n \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{2 a c d \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{a d^{2} n x \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{2 a d^{2} x \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} - \frac{b c^{2} \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{b c d n x \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{b d^{2} n x^{2} \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{b d^{2} x^{2} \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06168, size = 178, normalized size = 3.79 \begin{align*} \frac{{\left (d x + c\right )}^{n} b d^{2} n x^{2} +{\left (d x + c\right )}^{n} b c d n x +{\left (d x + c\right )}^{n} a d^{2} n x +{\left (d x + c\right )}^{n} b d^{2} x^{2} +{\left (d x + c\right )}^{n} a c d n + 2 \,{\left (d x + c\right )}^{n} a d^{2} x -{\left (d x + c\right )}^{n} b c^{2} + 2 \,{\left (d x + c\right )}^{n} a c d}{d^{2} n^{2} + 3 \, d^{2} n + 2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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