Optimal. Leaf size=66 \[ \frac {3 \sin ^{-1}(a x)}{2 a^4}+\frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}+\frac {(4-3 a x) \sqrt {1-a^2 x^2}}{2 a^4} \]
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Rubi [A] time = 0.05, antiderivative size = 66, 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 = {850, 819, 780, 216} \[ \frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}+\frac {(4-3 a x) \sqrt {1-a^2 x^2}}{2 a^4}+\frac {3 \sin ^{-1}(a x)}{2 a^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 819
Rule 850
Rubi steps
\begin {align*} \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx &=\int \frac {x^3 (1-a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {\int \frac {x (2-3 a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}+\frac {(4-3 a x) \sqrt {1-a^2 x^2}}{2 a^4}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^3}\\ &=\frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}+\frac {(4-3 a x) \sqrt {1-a^2 x^2}}{2 a^4}+\frac {3 \sin ^{-1}(a x)}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 0.82 \[ \frac {\sqrt {1-a^2 x^2} \left (-a^2 x^2+a x+4\right )+3 (a x+1) \sin ^{-1}(a x)}{2 a^4 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 75, normalized size = 1.14 \[ \frac {4 \, a x - 6 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (a^{2} x^{2} - a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} + 4}{2 \, {\left (a^{5} x + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 100, normalized size = 1.52 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, x}{2 a^{3}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}\, a^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{a^{4}}+\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{\left (x +\frac {1}{a}\right ) a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 68, normalized size = 1.03 \[ \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} x + a^{4}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{3}} + \frac {3 \, \arcsin \left (a x\right )}{2 \, a^{4}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 116, normalized size = 1.76 \[ \frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^3\,\sqrt {-a^2}}-\frac {\left (\frac {1}{a^2\,\sqrt {-a^2}}+\frac {x\,\sqrt {-a^2}}{2\,a^3}\right )\,\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \int \frac {x^{3}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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