Optimal. Leaf size=21 \[ \frac{\left (c x^2+d x^3\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.012103, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1588} \[ \frac{\left (c x^2+d x^3\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int x (2 c+3 d x) \left (c x^2+d x^3\right )^n \, dx &=\frac{\left (c x^2+d x^3\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0046984, size = 19, normalized size = 0.9 \[ \frac{\left (x^2 (c+d x)\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 28, normalized size = 1.3 \begin{align*}{\frac{ \left ( d{x}^{3}+c{x}^{2} \right ) ^{n}{x}^{2} \left ( dx+c \right ) }{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17194, size = 43, normalized size = 2.05 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41657, size = 58, normalized size = 2.76 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )}{\left (d x^{3} + c x^{2}\right )}^{n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.773654, size = 53, normalized size = 2.52 \begin{align*} \begin{cases} \frac{c x^{2} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} + \frac{d x^{3} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\2 \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29148, size = 58, normalized size = 2.76 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )}^{n} d x^{3} +{\left (d x^{3} + c x^{2}\right )}^{n} c x^{2}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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