Optimal. Leaf size=95 \[ -\frac{3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)}-\frac{\sin ^{-1}(a x)}{a^3} \]
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Rubi [A] time = 0.132497, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1637, 659, 651, 663, 216} \[ -\frac{3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)}-\frac{\sin ^{-1}(a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 1637
Rule 659
Rule 651
Rule 663
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{1-a^2 x^2}}{(1-a x)^4} \, dx &=\int \left (\frac{\sqrt{1-a^2 x^2}}{a^2 (-1+a x)^4}+\frac{2 \sqrt{1-a^2 x^2}}{a^2 (-1+a x)^3}+\frac{\sqrt{1-a^2 x^2}}{a^2 (-1+a x)^2}\right ) \, dx\\ &=\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^2}+\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^2}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^2}\\ &=\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}-\frac{2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3 (1-a x)^3}-\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^2}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}-\frac{3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}-\frac{\sin ^{-1}(a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.126385, size = 50, normalized size = 0.53 \[ \frac{\frac{\left (-13 a^2 x^2+19 a x-8\right ) \sqrt{1-a^2 x^2}}{(a x-1)^3}-5 \sin ^{-1}(a x)}{5 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 200, normalized size = 2.1 \begin{align*}{\frac{1}{{a}^{5}} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{1}{{a}^{3}}\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) }}-{\frac{1}{{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{3}{5\,{a}^{6}} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{1}{5\,{a}^{7}} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{{\left (a x - 1\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60036, size = 278, normalized size = 2.93 \begin{align*} \frac{8 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 24 \, a x + 10 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (13 \, a^{2} x^{2} - 19 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} - 8}{5 \,{\left (a^{6} x^{3} - 3 \, a^{5} x^{2} + 3 \, a^{4} x - a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x - 1\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11031, size = 217, normalized size = 2.28 \begin{align*} -\frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a^{2}{\left | a \right |}} - \frac{2 \,{\left (\frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{55 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{25 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 8\right )}}{5 \, a^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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