Optimal. Leaf size=43 \[ \frac{a^2 x^3}{3}+\frac{2 a b x^{n+3}}{n+3}+\frac{b^2 x^{2 n+3}}{2 n+3} \]
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Rubi [A] time = 0.0199481, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^3}{3}+\frac{2 a b x^{n+3}}{n+3}+\frac{b^2 x^{2 n+3}}{2 n+3} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin{align*} \int x^2 \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x^2+b^2 x^{2 (1+n)}+2 a b x^{2+n}\right ) \, dx\\ &=\frac{a^2 x^3}{3}+\frac{2 a b x^{3+n}}{3+n}+\frac{b^2 x^{3+2 n}}{3+2 n}\\ \end{align*}
Mathematica [A] time = 0.0350172, size = 40, normalized size = 0.93 \[ \frac{1}{3} x^3 \left (a^2+\frac{6 a b x^n}{n+3}+\frac{3 b^2 x^{2 n}}{2 n+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{x}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3+2\,n}}+{\frac{{x}^{3}{a}^{2}}{3}}+2\,{\frac{{x}^{3}ab{{\rm e}^{n\ln \left ( x \right ) }}}{3+n}} \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 = 1.08461, size = 163, normalized size = 3.79 \begin{align*} \frac{3 \,{\left (b^{2} n + 3 \, b^{2}\right )} x^{3} x^{2 \, n} + 6 \,{\left (2 \, a b n + 3 \, a b\right )} x^{3} x^{n} +{\left (2 \, a^{2} n^{2} + 9 \, a^{2} n + 9 \, a^{2}\right )} x^{3}}{3 \,{\left (2 \, n^{2} + 9 \, n + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.72169, size = 211, normalized size = 4.91 \begin{align*} \begin{cases} \frac{a^{2} x^{3}}{3} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{3 x^{3}} & \text{for}\: n = -3 \\\frac{a^{2} x^{3}}{3} + \frac{4 a b x^{\frac{3}{2}}}{3} + b^{2} \log{\left (x \right )} & \text{for}\: n = - \frac{3}{2} \\\frac{2 a^{2} n^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac{9 a^{2} n x^{3}}{6 n^{2} + 27 n + 27} + \frac{9 a^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac{12 a b n x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac{18 a b x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac{3 b^{2} n x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} + \frac{9 b^{2} x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21544, size = 123, normalized size = 2.86 \begin{align*} \frac{3 \, b^{2} n x^{3} x^{2 \, n} + 12 \, a b n x^{3} x^{n} + 2 \, a^{2} n^{2} x^{3} + 9 \, b^{2} x^{3} x^{2 \, n} + 18 \, a b x^{3} x^{n} + 9 \, a^{2} n x^{3} + 9 \, a^{2} x^{3}}{3 \,{\left (2 \, n^{2} + 9 \, n + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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