Optimal. Leaf size=19 \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]
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Rubi [A] time = 0.0051805, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]
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 )^{\frac{1}{n}}\right ) \, dx &=a x+b \int \left (c x^n\right )^{\frac{1}{n}} \, dx\\ &=a x+\frac{\left (b \left (c x^n\right )^{\frac{1}{n}}\right ) \int x \, dx}{x}\\ &=a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}}\\ \end{align*}
Mathematica [A] time = 0.0052905, size = 19, normalized size = 1. \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 22, normalized size = 1.2 \begin{align*} ax+{\frac{bx}{2}{{\rm e}^{{\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^{\left (\frac{1}{n}\right )} \int{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}\,{d x} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27623, size = 34, normalized size = 1.79 \begin{align*} \frac{1}{2} \, b c^{\left (\frac{1}{n}\right )} x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.237278, size = 19, normalized size = 1. \begin{align*} a x + \frac{b c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26899, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{2} \, b c^{\left (\frac{1}{n}\right )} x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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