Optimal. Leaf size=34 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}{3 b} \]
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Rubi [A] time = 0.0087369, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {254, 32} \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 254
Rule 32
Rubi steps
\begin{align*} \int \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0133179, size = 38, normalized size = 1.12 \[ a^2 x+a b x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{3} b^2 x \left (c x^n\right )^{2/n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{2}\, dx \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^{2} c^{\frac{2}{n}} \int{\left (x^{n}\right )}^{\frac{2}{n}}\,{d x} + 2 \, a b c^{\left (\frac{1}{n}\right )} \int{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}\,{d x} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25701, size = 63, normalized size = 1.85 \begin{align*} \frac{1}{3} \, b^{2} c^{\frac{2}{n}} x^{3} + a b c^{\left (\frac{1}{n}\right )} x^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.441378, size = 39, normalized size = 1.15 \begin{align*} a^{2} x + a b c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}} + \frac{b^{2} 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 = 1.10464, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{2} c^{\frac{2}{n}} x^{3} + a b c^{\left (\frac{1}{n}\right )} x^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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