Optimal. Leaf size=21 \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]
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Rubi [A] time = 0.0056403, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 30} \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (a+b \left (c x^n\right )^{2/n}\right ) \, dx &=a x+b \int \left (c x^n\right )^{2/n} \, dx\\ &=a x+\frac{\left (b \left (c x^n\right )^{2/n}\right ) \int x^2 \, dx}{x^2}\\ &=a x+\frac{1}{3} b x \left (c x^n\right )^{2/n}\\ \end{align*}
Mathematica [A] time = 0.0020764, size = 21, normalized size = 1. \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 23, normalized size = 1.1 \begin{align*} ax+{\frac{bx}{3}{{\rm e}^{2\,{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b c^{\frac{2}{n}} \int{\left (x^{n}\right )}^{\frac{2}{n}}\,{d x} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3099, size = 34, normalized size = 1.62 \begin{align*} \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.230187, size = 19, normalized size = 0.9 \begin{align*} a x + \frac{b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0945, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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