Optimal. Leaf size=55 \[ -\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\sqrt {1-a^2 x^2}}{a^3 (1+a x)}-\frac {\sin ^{-1}(a x)}{a^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1653, 12, 807,
222} \begin {gather*} -\frac {\text {ArcSin}(a x)}{a^3}-\frac {\sqrt {1-a^2 x^2}}{a^3 (a x+1)}-\frac {\sqrt {1-a^2 x^2}}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 807
Rule 1653
Rubi steps
\begin {align*} \int \frac {x^2}{(1+a x) \sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\int \frac {a^3 x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\sqrt {1-a^2 x^2}}{a^3 (1+a x)}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\sqrt {1-a^2 x^2}}{a^3 (1+a x)}-\frac {\sin ^{-1}(a x)}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 71, normalized size = 1.29 \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 = 0.06, size = 84, normalized size = 1.53
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{3}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{4} \left (x +\frac {1}{a}\right )}\) | \(84\) |
risch | \(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{4} \left (x +\frac {1}{a}\right )}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 52, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1}}{a^{4} x + a^{3}} - \frac {\arcsin \left (a x\right )}{a^{3}} - \frac {\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 = 2.54, size = 66, normalized size = 1.20 \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 70, normalized size = 1.27 \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 = 0.07, size = 84, normalized size = 1.53 \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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