Optimal. Leaf size=55 \[ \frac{1}{2} a^2 x \left (c x^n\right )^{\frac{1}{n}}+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{4} b^2 x \left (c x^n\right )^{3/n} \]
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Rubi [A] time = 0.0252137, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {15, 368, 43} \[ \frac{1}{2} a^2 x \left (c x^n\right )^{\frac{1}{n}}+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{4} b^2 x \left (c x^n\right )^{3/n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 368
Rule 43
Rubi steps
\begin{align*} \int \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2 \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \int x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2 \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x (a+b x)^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 x+2 a b x^2+b^2 x^3\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{1}{2} a^2 x \left (c x^n\right )^{\frac{1}{n}}+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{4} b^2 x \left (c x^n\right )^{3/n}\\ \end{align*}
Mathematica [A] time = 0.0178952, size = 49, normalized size = 0.89 \[ \frac{1}{12} x \left (c x^n\right )^{\frac{1}{n}} \left (6 a^2+8 a b \left (c x^n\right )^{\frac{1}{n}}+3 b^2 \left (c x^n\right )^{2/n}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.232, size = 0, normalized size = 0. \begin{align*} \int \sqrt [n]{c{x}^{n}} \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*} \int{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2} \left (c x^{n}\right )^{\left (\frac{1}{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64861, size = 88, normalized size = 1.6 \begin{align*} \frac{1}{4} \, b^{2} c^{\frac{3}{n}} x^{4} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{2} c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.755513, size = 56, normalized size = 1.02 \begin{align*} \frac{a^{2} c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + \frac{2 a b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} + \frac{b^{2} c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2783, size = 58, normalized size = 1.05 \begin{align*} \frac{1}{4} \, b^{2} c^{\frac{3}{n}} x^{4} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{2} c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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