Integrand size = 26, antiderivative size = 47 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {8}{3}\right )}{4 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {507, 435, 430} \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {8}{3}\right )}{4 \sqrt {3}} \]
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Rule 430
Rule 435
Rule 507
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{4} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx\right )+\frac {1}{4} \int \frac {\sqrt {1+4 x^2}}{\sqrt {2-3 x^2}} \, dx \\ & = \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}}-\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\frac {i \left (E\left (i \text {arcsinh}(2 x)\left |-\frac {3}{8}\right .\right )-\operatorname {EllipticF}\left (i \text {arcsinh}(2 x),-\frac {3}{8}\right )\right )}{3 \sqrt {2}} \]
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Time = 3.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {\sqrt {3}\, \left (F\left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )-E\left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )\right )}{12}\) | \(35\) |
elliptic | \(-\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (4 x^{2}+1\right )}\, \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \left (F\left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )-E\left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )\right )}{24 \sqrt {-3 x^{2}+2}\, \sqrt {-12 x^{4}+5 x^{2}+2}}\) | \(85\) |
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (35) = 70\).
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.66 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=-\frac {4 \, \sqrt {3} \sqrt {2} \sqrt {-3} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,-\frac {3}{8}) - 4 \, \sqrt {3} \sqrt {2} \sqrt {-3} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,-\frac {3}{8}) + 9 \, \sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{108 \, x} \]
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\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\int \frac {x^{2}}{\sqrt {2 - 3 x^{2}} \sqrt {4 x^{2} + 1}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx=\int \frac {x^2}{\sqrt {2-3\,x^2}\,\sqrt {4\,x^2+1}} \,d x \]
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