Integrand size = 23, antiderivative size = 235 \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\frac {(3 b+2 d) x \sqrt {2+b x^2}}{3 b \sqrt {3+d x^2}}+\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}-\frac {\sqrt {2} (3 b+2 d) \sqrt {2+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {2 \sqrt {2} \sqrt {2+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {428, 545, 429, 506, 422} \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\frac {2 \sqrt {2} \sqrt {b x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}-\frac {\sqrt {2} (3 b+2 d) \sqrt {b x^2+2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}+\frac {x (3 b+2 d) \sqrt {b x^2+2}}{3 b \sqrt {d x^2+3}} \]
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Rule 422
Rule 428
Rule 429
Rule 506
Rule 545
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}+\frac {2}{3} \int \frac {6+\frac {1}{2} (3 b+2 d) x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx \\ & = \frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}+4 \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx+\frac {1}{3} (3 b+2 d) \int \frac {x^2}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx \\ & = \frac {(3 b+2 d) x \sqrt {2+b x^2}}{3 b \sqrt {3+d x^2}}+\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}+\frac {2 \sqrt {2} \sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {(-3 b-2 d) \int \frac {\sqrt {2+b x^2}}{\left (3+d x^2\right )^{3/2}} \, dx}{b} \\ & = \frac {(3 b+2 d) x \sqrt {2+b x^2}}{3 b \sqrt {3+d x^2}}+\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}-\frac {\sqrt {2} (3 b+2 d) \sqrt {2+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {2 \sqrt {2} \sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\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 = 0.88 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.54 \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\frac {\sqrt {b} d x \sqrt {2+b x^2} \sqrt {3+d x^2}-i \sqrt {3} (3 b+2 d) E\left (i \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )+i \sqrt {3} (3 b-2 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right ),\frac {2 d}{3 b}\right )}{3 \sqrt {b} d} \]
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Time = 3.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {x \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}{3}+\frac {\left (\frac {2 \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}-\frac {\left (3 b +2 d \right ) \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-E\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{3 \sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\right ) \sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}}{\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}\) | \(252\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}\, \left (\frac {x \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}{3}+\frac {2 \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}-\frac {\left (b +\frac {2 d}{3}\right ) \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-E\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\right )}{\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}\) | \(253\) |
default | \(\frac {\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, \left (b^{2} d \,x^{5} \sqrt {-d}+3 b^{2} x^{3} \sqrt {-d}+2 b d \,x^{3} \sqrt {-d}+3 \sqrt {2}\, F\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) b \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}-2 \sqrt {2}\, F\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+3 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) b \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+2 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+6 b x \sqrt {-d}\right )}{3 \left (b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6\right ) \sqrt {-d}\, b}\) | \(303\) |
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Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.63 \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=-\frac {3 \, \sqrt {3} \sqrt {b d} {\left (3 \, b + 2 \, d\right )} x \sqrt {-\frac {1}{d}} E(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - \sqrt {3} \sqrt {b d} {\left (4 \, d^{2} + 9 \, b + 6 \, d\right )} x \sqrt {-\frac {1}{d}} F(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - {\left (b d^{2} x^{2} + 3 \, b d + 2 \, d^{2}\right )} \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}{3 \, b d^{2} x} \]
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\[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}\, dx \]
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\[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int { \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3} \,d x } \]
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\[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int { \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3} \,d x } \]
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Timed out. \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int \sqrt {b\,x^2+2}\,\sqrt {d\,x^2+3} \,d x \]
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