Optimal. Leaf size=46 \[ \frac{(b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)} \]
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Rubi [A] time = 0.017749, antiderivative size = 46, 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} \[ \frac{(b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^n (c+d x) \, dx &=\int \left (\frac{(b c-a d) (a+b x)^n}{b}+\frac{d (a+b x)^{1+n}}{b}\right ) \, dx\\ &=\frac{(b c-a d) (a+b x)^{1+n}}{b^2 (1+n)}+\frac{d (a+b x)^{2+n}}{b^2 (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0299921, size = 41, normalized size = 0.89 \[ \frac{(a+b x)^{n+1} (-a d+b c (n+2)+b d (n+1) x)}{b^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 49, normalized size = 1.1 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -bdnx-bcn-bdx+ad-2\,bc \right ) }{{b}^{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 = 1.634, size = 171, normalized size = 3.72 \begin{align*} \frac{{\left (a b c n + 2 \, a b c - a^{2} d +{\left (b^{2} d n + b^{2} d\right )} x^{2} +{\left (2 \, b^{2} c +{\left (b^{2} c + a b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.877254, size = 377, normalized size = 8.2 \begin{align*} \begin{cases} a^{n} \left (c x + \frac{d x^{2}}{2}\right ) & \text{for}\: b = 0 \\\frac{a d \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a d}{a b^{2} + b^{3} x} - \frac{b c}{a b^{2} + b^{3} x} + \frac{b d x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: n = -2 \\- \frac{a d \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{c \log{\left (\frac{a}{b} + x \right )}}{b} + \frac{d x}{b} & \text{for}\: n = -1 \\- \frac{a^{2} d \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b c n \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{2 a b c \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b d n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} c n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{2 b^{2} c x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} d n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} d x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.51389, size = 178, normalized size = 3.87 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{2} d n x^{2} +{\left (b x + a\right )}^{n} b^{2} c n x +{\left (b x + a\right )}^{n} a b d n x +{\left (b x + a\right )}^{n} b^{2} d x^{2} +{\left (b x + a\right )}^{n} a b c n + 2 \,{\left (b x + a\right )}^{n} b^{2} c x + 2 \,{\left (b x + a\right )}^{n} a b c -{\left (b x + a\right )}^{n} a^{2} d}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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