Integrand size = 26, antiderivative size = 110 \[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=\frac {x \sqrt {2+b x^2}}{b \sqrt {3+d x^2}}-\frac {\sqrt {2} \sqrt {2+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{b \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {506, 422} \[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=\frac {x \sqrt {b x^2+2}}{b \sqrt {d x^2+3}}-\frac {\sqrt {2} \sqrt {b x^2+2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{b \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
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Rule 422
Rule 506
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {2+b x^2}}{b \sqrt {3+d x^2}}-\frac {3 \int \frac {\sqrt {2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx}{b} \\ & = \frac {x \sqrt {2+b x^2}}{b \sqrt {3+d x^2}}-\frac {\sqrt {2} \sqrt {2+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{b \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=-\frac {i \sqrt {3} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right ),\frac {2 d}{3 b}\right )\right )}{\sqrt {b} d} \]
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Time = 3.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\left (-F\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )+E\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )\right ) \sqrt {2}}{b \sqrt {-d}}\) | \(70\) |
elliptic | \(-\frac {\sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}\, \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-E\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, \sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\) | \(145\) |
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none
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=-\frac {3 \, \sqrt {3} \sqrt {b d} x \sqrt {-\frac {1}{d}} E(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - 3 \, \sqrt {3} \sqrt {b d} x \sqrt {-\frac {1}{d}} F(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3} d}{b d^{2} x} \]
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\[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=\int \frac {x^{2}}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx=\int \frac {x^2}{\sqrt {b\,x^2+2}\,\sqrt {d\,x^2+3}} \,d x \]
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