Optimal. Leaf size=43 \[ a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n} \]
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Rubi [A] time = 0.0171888, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {254, 194} \[ a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n} \]
Antiderivative was successfully verified.
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Rule 254
Rule 194
Rubi steps
\begin{align*} \int \left (a+b \left (c x^n\right )^{2/n}\right )^2 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (a+b x^2\right )^2 \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n}\\ \end{align*}
Mathematica [A] time = 0.0123648, size = 43, normalized size = 1. \[ a^2 x+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{5} b^2 x \left (c x^n\right )^{4/n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.161, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \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{4}{n}} \int{\left (x^{n}\right )}^{\frac{4}{n}}\,{d x} + 2 \, a b c^{\frac{2}{n}} \int{\left (x^{n}\right )}^{\frac{2}{n}}\,{d x} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3387, size = 69, normalized size = 1.6 \begin{align*} \frac{1}{5} \, b^{2} c^{\frac{4}{n}} x^{5} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.418959, size = 42, normalized size = 0.98 \begin{align*} a^{2} x + \frac{2 a b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} + \frac{b^{2} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11505, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{5} \, b^{2} c^{\frac{4}{n}} x^{5} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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