Optimal. Leaf size=22 \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]
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Rubi [A] time = 0.0820809, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6742, 261} \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]
Antiderivative was successfully verified.
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Rule 6742
Rule 261
Rubi steps
\begin{align*} \int \frac{x+\left (1-9 x^2\right )^{3/2}}{\sqrt{1-9 x^2}} \, dx &=\int \left (1-9 x^2+\frac{x}{\sqrt{1-9 x^2}}\right ) \, dx\\ &=x-3 x^3+\int \frac{x}{\sqrt{1-9 x^2}} \, dx\\ &=x-3 x^3-\frac{1}{9} \sqrt{1-9 x^2}\\ \end{align*}
Mathematica [A] time = 0.0018521, size = 22, normalized size = 1. \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 0.9 \begin{align*} x-3\,{x}^{3}-{\frac{1}{9}\sqrt{-9\,{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10861, size = 24, normalized size = 1.09 \begin{align*} -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80464, size = 47, normalized size = 2.14 \begin{align*} -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.15003, size = 17, normalized size = 0.77 \begin{align*} - 3 x^{3} + x - \frac{\sqrt{1 - 9 x^{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13827, size = 24, normalized size = 1.09 \begin{align*} -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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