Optimal. Leaf size=20 \[ \frac{\left (a+b x+c x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.0054882, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {629} \[ \frac{\left (a+b x+c x^2\right )^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx &=\frac{\left (a+b x+c x^2\right )^{1+p}}{1+p}\\ \end{align*}
Mathematica [A] time = 0.0078453, size = 19, normalized size = 0.95 \[ \frac{(a+x (b+c x))^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 21, normalized size = 1.1 \begin{align*}{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+p}}{1+p}} \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 = 1.07846, size = 63, normalized size = 3.15 \begin{align*} \frac{{\left (c x^{2} + b x + a\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 51.315, size = 104, normalized size = 5.2 \begin{align*} \begin{cases} \frac{a \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac{b x \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac{c x^{2} \left (a + b x + c x^{2}\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (\frac{b}{2 c} + x - \frac{\sqrt{- 4 a c + b^{2}}}{2 c} \right )} + \log{\left (\frac{b}{2 c} + x + \frac{\sqrt{- 4 a c + b^{2}}}{2 c} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11774, size = 72, normalized size = 3.6 \begin{align*} \frac{{\left (c x^{2} + b x + a\right )}^{p} c x^{2} +{\left (c x^{2} + b x + a\right )}^{p} b x +{\left (c x^{2} + b x + a\right )}^{p} a}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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