Optimal. Leaf size=54 \[ -\frac {\sin ^{-1}(a x)}{a^3}-\frac {1-a x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3} \]
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Rubi [A] time = 0.03, 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 = {797, 641, 216, 637} \[ -\frac {1-a x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\sin ^{-1}(a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 637
Rule 641
Rule 797
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 = 0.03, size = 50, normalized size = 0.93 \[ \frac {a^2 x^2-\sqrt {1-a^2 x^2} \sin ^{-1}(a x)+a x-2}{a^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 66, normalized size = 1.22 \[ -\frac {2 \, a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 2\right )} + 2}{a^{4} x + a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 70, normalized size = 1.30 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3}} + \frac {2}{a^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 85, normalized size = 1.57 \[ \frac {x^{2}}{\sqrt {-a^{2} x^{2}+1}\, a}+\frac {x}{\sqrt {-a^{2} x^{2}+1}\, a^{2}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}\, a^{2}}-\frac {2}{\sqrt {-a^{2} x^{2}+1}\, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 63, normalized size = 1.17 \[ \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{3}} \]
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 \[ \frac {\sqrt {1-a^2\,x^2}}{\left (a\,\sqrt {-a^2}+a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.33, size = 102, normalized size = 1.89 \[ - a \left (\begin {cases} - \frac {x^{2}}{a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {2}{a^{4} \sqrt {- a^{2} x^{2} + 1}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i x}{a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \operatorname {acosh}{\left (a x \right )}}{a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x}{a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\operatorname {asin}{\left (a x \right )}}{a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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