Optimal. Leaf size=19 \[ \frac{\left (b x+c x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.0054707, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {629} \[ \frac{\left (b x+c x^2\right )^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b+2 c x) \left (b x+c x^2\right )^p \, dx &=\frac{\left (b x+c x^2\right )^{1+p}}{1+p}\\ \end{align*}
Mathematica [A] time = 0.0090834, size = 17, normalized size = 0.89 \[ \frac{(x (b+c x))^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 24, normalized size = 1.3 \begin{align*}{\frac{x \left ( cx+b \right ) \left ( c{x}^{2}+bx \right ) ^{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 = 1.33617, size = 53, normalized size = 2.79 \begin{align*} \frac{{\left (c x^{2} + b x\right )}{\left (c x^{2} + b x\right )}^{p}}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.560271, size = 46, normalized size = 2.42 \begin{align*} \begin{cases} \frac{b x \left (b x + c x^{2}\right )^{p}}{p + 1} + \frac{c x^{2} \left (b x + c x^{2}\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (x \right )} + \log{\left (\frac{b}{c} + x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25932, size = 50, normalized size = 2.63 \begin{align*} \frac{{\left (c x^{2} + b x\right )}^{p} c x^{2} +{\left (c x^{2} + b x\right )}^{p} b x}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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