Optimal. Leaf size=22 \[ -3 x^3-\frac {1}{9} \sqrt {1-9 x^2}+x \]
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Rubi [A] time = 0.08, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 261
Rule 6742
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*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ -3 x^3-\frac {1}{9} \sqrt {1-9 x^2}+x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 0.82 \[ -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 18, normalized size = 0.82 \[ -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.86 \[ -3 x^{3}+x -\frac {\sqrt {-9 x^{2}+1}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 18, normalized size = 0.82 \[ -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 18, normalized size = 0.82 \[ x-3\,x^3-\frac {\sqrt {\frac {1}{9}-x^2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.17, size = 17, normalized size = 0.77 \[ - 3 x^{3} + x - \frac {\sqrt {1 - 9 x^{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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