Optimal. Leaf size=34 \[ \frac{\left (a x+\frac{b x^2}{2}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
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Rubi [A] time = 0.0084399, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1591} \[ \frac{\left (a x+\frac{b x^2}{2}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int (a+b x) \left (1+\left (a x+\frac{b x^2}{2}\right )^n\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^n\right ) \, dx,x,a x+\frac{b x^2}{2}\right )\\ &=a x+\frac{b x^2}{2}+\frac{\left (a x+\frac{b x^2}{2}\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0547979, size = 34, normalized size = 1. \[ \frac{x (2 a+b x) \left (\left (a x+\frac{b x^2}{2}\right )^n+n+1\right )}{2 (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{1}{1+n} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) ^{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69603, size = 70, normalized size = 2.06 \begin{align*} \frac{1}{2} \, b x^{2} + a x + \frac{{\left (b x^{2} + 2 \, a x\right )} e^{\left (n \log \left (b x + 2 \, a\right ) + n \log \left (x\right )\right )}}{2^{n + 1} n + 2^{n + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39302, size = 112, normalized size = 3.29 \begin{align*} \frac{{\left (b n + b\right )} x^{2} +{\left (b x^{2} + 2 \, a x\right )}{\left (\frac{1}{2} \, b x^{2} + a x\right )}^{n} + 2 \,{\left (a n + a\right )} x}{2 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 87.1078, size = 230, normalized size = 6.76 \begin{align*} \begin{cases} a \left (x + \frac{\log{\left (x \right )}}{a}\right ) & \text{for}\: b = 0 \wedge n = -1 \\a \left (\frac{a^{n} x x^{n}}{n + 1} + \frac{n x}{n + 1} + \frac{x}{n + 1}\right ) & \text{for}\: b = 0 \\a x + \frac{b x^{2}}{2} + \log{\left (x \right )} + \log{\left (\frac{2 a}{b} + x \right )} & \text{for}\: n = -1 \\\frac{2 \cdot 2^{n} a b n x}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac{2 \cdot 2^{n} a b x}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac{2^{n} b^{2} n x^{2}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac{2^{n} b^{2} x^{2}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac{2 a b x \left (2 a x + b x^{2}\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac{b^{2} x^{2} \left (2 a x + b x^{2}\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12755, size = 41, normalized size = 1.21 \begin{align*} \frac{1}{2} \, b x^{2} + a x + \frac{{\left (\frac{1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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