Optimal. Leaf size=54 \[ -\frac {1-a x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\sin ^{-1}(a x)}{a^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {811, 655, 222,
651} \begin {gather*} -\frac {\text {ArcSin}(a x)}{a^3}-\frac {1-a x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 651
Rule 655
Rule 811
Rubi steps
\begin {align*} \int \frac {x^2 (1-a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}-\frac {\int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {1-a x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {1-a x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\sin ^{-1}(a x)}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 71, normalized size = 1.31 \begin {gather*} \frac {(-2-a x) \sqrt {1-a^2 x^2}}{a^3 (1+a x)}-\frac {\log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{\left (-a^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 90, normalized size = 1.67
method | result | size |
default | \(-a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\) | \(90\) |
risch | \(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{4} \left (x +\frac {1}{a}\right )}\) | \(92\) |
meijerg | \(-\frac {-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}}{a^{3} \sqrt {\pi }}-\frac {\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 63, normalized size = 1.17 \begin {gather*} \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.04, size = 66, normalized size = 1.22 \begin {gather*} -\frac {2 \, a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 2\right )} + 2}{a^{4} x + a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.53, size = 119, normalized size = 2.20 \begin {gather*} - a \left (\begin {cases} \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{6} x^{2} - a^{4}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{a^{6} x^{2} - a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i x}{a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \operatorname {acosh}{\left (a x \right )}}{a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x}{a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\operatorname {asin}{\left (a x \right )}}{a^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 70, normalized size = 1.30 \begin {gather*} -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3}} + \frac {2}{a^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 84, normalized size = 1.56 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{\left (a\,\sqrt {-a^2}+a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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