Optimal. Leaf size=22 \[ x-3 x^3-\frac {1}{9} \sqrt {1-9 x^2} \]
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Rubi [A]
time = 0.05, 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 = {6874, 267}
\begin {gather*} -3 x^3-\frac {1}{9} \sqrt {1-9 x^2}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6874
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 \begin {gather*} x-3 x^3-\frac {1}{9} \sqrt {1-9 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 19, normalized size = 0.86
method | result | size |
default | \(x -3 x^{3}-\frac {\sqrt {-9 x^{2}+1}}{9}\) | \(19\) |
trager | \(-\left (3 x^{2}-1\right ) x -\frac {\sqrt {-9 x^{2}+1}}{9}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.82 \begin {gather*} -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 18, normalized size = 0.82 \begin {gather*} -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 17, normalized size = 0.77 \begin {gather*} - 3 x^{3} + x - \frac {\sqrt {1 - 9 x^{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.32, size = 18, normalized size = 0.82 \begin {gather*} -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \end {gather*}
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 \begin {gather*} x-3\,x^3-\frac {\sqrt {\frac {1}{9}-x^2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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