Optimal. Leaf size=22 \[ \frac{x^{2 (n+1)} (c+d x)^{n+1}}{n+1} \]
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Rubi [A] time = 0.009202, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {845} \[ \frac{x^{2 (n+1)} (c+d x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 845
Rubi steps
\begin{align*} \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx &=\frac{x^{2 (1+n)} (c+d x)^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0103197, size = 22, normalized size = 1. \[ \frac{x^{2 n+2} (c+d x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{{x}^{2+2\,n} \left ( dx+c \right ) ^{1+n}}{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16639, size = 43, normalized size = 1.95 \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 = 1.4309, size = 61, normalized size = 2.77 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )}{\left (d x + c\right )}^{n} x^{2 \, n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.10322, size = 53, normalized size = 2.41 \begin{align*} \begin{cases} \frac{c x^{2} x^{2 n} \left (c + d x\right )^{n}}{n + 1} + \frac{d x^{3} x^{2 n} \left (c + d x\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 [A] time = 1.15633, size = 55, normalized size = 2.5 \begin{align*} \frac{{\left (d x + c\right )}^{n} d x^{3} x^{2 \, n} +{\left (d x + c\right )}^{n} c x^{2} x^{2 \, n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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