Integrand size = 26, antiderivative size = 95 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}-\frac {\arcsin (a x)}{a^3} \]
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Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1651, 673, 665, 677, 222} \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=-\frac {\arcsin (a x)}{a^3}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)} \]
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Rule 222
Rule 665
Rule 673
Rule 677
Rule 1651
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\frac {\left (-8+19 a x-13 a^2 x^2\right ) \sqrt {1-a^2 x^2}}{5 a^3 (-1+a x)^3}-\frac {2 \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )}{a^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(85)=170\).
Time = 0.38 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.58
method | result | size |
default | \(\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{a^{4}}+\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}}{a^{6}}+\frac {2 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{6} \left (x -\frac {1}{a}\right )^{3}}\) | \(245\) |
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none
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.33 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\frac {8 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 24 \, a x + 10 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (13 \, a^{2} x^{2} - 19 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{5 \, {\left (a^{6} x^{3} - 3 \, a^{5} x^{2} + 3 \, a^{4} x - a^{3}\right )}} \]
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\[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\int \frac {x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x - 1\right )^{4}}\, dx \]
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\[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{{\left (a x - 1\right )}^{4}} \,d x } \]
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Exception generated. \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\text {Exception raised: TypeError} \]
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Time = 11.46 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.32 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx=\frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,x^2-2\,a^6\,x+a^5\right )}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,\sqrt {-a^2}}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (a\,\sqrt {-a^2}-3\,a^2\,x\,\sqrt {-a^2}+3\,a^3\,x^2\,\sqrt {-a^2}-a^4\,x^3\,\sqrt {-a^2}\right )}-\frac {13\,\sqrt {1-a^2\,x^2}}{5\,\left (a\,\sqrt {-a^2}-a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,x^2-2\,a^4\,x+a^3\right )} \]
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