Optimal. Leaf size=62 \[ a^2 b x \left (c x^n\right )^{2/n}+a^3 x+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n} \]
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Rubi [A] time = 0.0232592, antiderivative size = 62, 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} \[ a^2 b x \left (c x^n\right )^{2/n}+a^3 x+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n} \]
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 )^3 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (a+b x^2\right )^3 \, 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^3+3 a^2 b x^2+3 a b^2 x^4+b^3 x^6\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=a^3 x+a^2 b x \left (c x^n\right )^{2/n}+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n}\\ \end{align*}
Mathematica [A] time = 0.0211043, size = 62, normalized size = 1. \[ a^2 b x \left (c x^n\right )^{2/n}+a^3 x+\frac{3}{5} a b^2 x \left (c x^n\right )^{4/n}+\frac{1}{7} b^3 x \left (c x^n\right )^{6/n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{3}\, 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^{3} c^{\frac{6}{n}} \int{\left (x^{n}\right )}^{\frac{6}{n}}\,{d x} + 3 \, a b^{2} c^{\frac{4}{n}} \int{\left (x^{n}\right )}^{\frac{4}{n}}\,{d x} + 3 \, a^{2} b c^{\frac{2}{n}} \int{\left (x^{n}\right )}^{\frac{2}{n}}\,{d x} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59242, size = 99, normalized size = 1.6 \begin{align*} \frac{1}{7} \, b^{3} c^{\frac{6}{n}} x^{7} + \frac{3}{5} \, a b^{2} c^{\frac{4}{n}} x^{5} + a^{2} b c^{\frac{2}{n}} x^{3} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.769372, size = 63, normalized size = 1.02 \begin{align*} a^{3} x + a^{2} b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}} + \frac{3 a b^{2} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} + \frac{b^{3} c^{\frac{6}{n}} x \left (x^{n}\right )^{\frac{6}{n}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13503, size = 70, normalized size = 1.13 \begin{align*} \frac{1}{7} \, b^{3} c^{\frac{6}{n}} x^{7} + \frac{3}{5} \, a b^{2} c^{\frac{4}{n}} x^{5} + a^{2} b c^{\frac{2}{n}} x^{3} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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