Optimal. Leaf size=35 \[ \frac{\left (a x+\frac{b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
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Rubi [A] time = 0.0090755, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1591} \[ \frac{\left (a x+\frac{b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int (a+b x) \left (1+\left (c+a x+\frac{b x^2}{2}\right )^n\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^n\right ) \, dx,x,c+a x+\frac{b x^2}{2}\right )\\ &=a x+\frac{b x^2}{2}+\frac{\left (c+a x+\frac{b x^2}{2}\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0540426, size = 35, normalized size = 1. \[ \frac{\left (a x+\frac{b x^2}{2}+c\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 0.9 \begin{align*} c+ax+{\frac{b{x}^{2}}{2}}+{\frac{1}{1+n} \left ( c+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 = 1.66477, size = 73, normalized size = 2.09 \begin{align*} \frac{1}{2} \, b x^{2} + a x + \frac{{\left (b x^{2} + 2 \, a x + 2 \, c\right )}{\left (b x^{2} + 2 \, a x + 2 \, c\right )}^{n}}{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 = 1.37815, size = 126, normalized size = 3.6 \begin{align*} \frac{{\left (b n + b\right )} x^{2} +{\left (b x^{2} + 2 \, a x + 2 \, c\right )}{\left (\frac{1}{2} \, b x^{2} + a x + c\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 [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 [B] time = 1.21129, size = 107, normalized size = 3.06 \begin{align*} \frac{{\left (\frac{1}{2} \, b x^{2} + a x + c\right )}^{n} b x^{2} + b n x^{2} + 2 \,{\left (\frac{1}{2} \, b x^{2} + a x + c\right )}^{n} a x + 2 \, a n x + b x^{2} + 2 \,{\left (\frac{1}{2} \, b x^{2} + a x + c\right )}^{n} c + 2 \, a x}{2 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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