Optimal. Leaf size=21 \[ a x+\frac{1}{4} b x \left (c x^n\right )^{3/n} \]
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Rubi [A] time = 0.0063415, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 30} \[ a x+\frac{1}{4} b x \left (c x^n\right )^{3/n} \]
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 )^{3/n}\right ) \, dx &=a x+b \int \left (c x^n\right )^{3/n} \, dx\\ &=a x+\frac{\left (b \left (c x^n\right )^{3/n}\right ) \int x^3 \, dx}{x^3}\\ &=a x+\frac{1}{4} b x \left (c x^n\right )^{3/n}\\ \end{align*}
Mathematica [A] time = 0.0018409, size = 21, normalized size = 1. \[ a x+\frac{1}{4} b x \left (c x^n\right )^{3/n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 23, normalized size = 1.1 \begin{align*} ax+{\frac{bx}{4}{{\rm e}^{3\,{\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^{\frac{3}{n}} \int{\left (x^{n}\right )}^{\frac{3}{n}}\,{d x} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25145, size = 34, normalized size = 1.62 \begin{align*} \frac{1}{4} \, b c^{\frac{3}{n}} x^{4} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.223396, size = 19, normalized size = 0.9 \begin{align*} a x + \frac{b 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 = 1.14367, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{4} \, b c^{\frac{3}{n}} x^{4} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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