Integrand size = 25, antiderivative size = 66 \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}+\frac {(4-3 a x) \sqrt {1-a^2 x^2}}{2 a^4}+\frac {3 \arcsin (a x)}{2 a^4} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {864, 833, 794, 222} \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {3 \arcsin (a x)}{2 a^4}+\frac {x^2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}+\frac {(4-3 a x) \sqrt {1-a^2 x^2}}{2 a^4} \]
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Rule 222
Rule 794
Rule 833
Rule 864
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (4+a x-a^2 x^2\right )}{2 a^4 (1+a x)}+\frac {3 \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )}{a^4} \]
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Time = 0.37 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.47
method | result | size |
risch | \(\frac {\left (a x -2\right ) \left (a^{2} x^{2}-1\right )}{2 a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{3} \sqrt {a^{2}}}+\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{5} \left (x +\frac {1}{a}\right )}\) | \(97\) |
default | \(\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a}+\frac {\sqrt {-a^{2} x^{2}+1}}{a^{4}}+\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{5} \left (x +\frac {1}{a}\right )}\) | \(134\) |
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {4 \, a x - 6 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (a^{2} x^{2} - a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} + 4}{2 \, {\left (a^{5} x + a^{4}\right )}} \]
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\[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{3}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} x + a^{4}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{3}} + \frac {3 \, \arcsin \left (a x\right )}{2 \, a^{4}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{4}} \]
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Exception generated. \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.76 \[ \int \frac {x^3}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^3\,\sqrt {-a^2}}-\frac {\left (\frac {1}{a^2\,\sqrt {-a^2}}+\frac {x\,\sqrt {-a^2}}{2\,a^3}\right )\,\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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