Integrand size = 23, antiderivative size = 87 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {429} \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rule 429
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {\frac {c+d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 2.94 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right )}\) | \(100\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\) | \(122\) |
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none
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c})}{b c} \]
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\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \]
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