Integrand size = 32, antiderivative size = 102 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\frac {c^{3/2} \sqrt {e+f x^2} \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{a \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]
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Time = 0.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {553} \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\frac {c^{3/2} \sqrt {e+f x^2} \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{a \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]
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Rule 553
Rubi steps \begin{align*} \text {integral}& = \frac {c^{3/2} \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.67 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=-\frac {i \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (a d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+(b c-a d) \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{a b \sqrt {\frac {d}{c}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
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Time = 3.34 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d -\Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a d +\Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b c \right ) \sqrt {\frac {f \,x^{2}+e}{e}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{b a \sqrt {-\frac {d}{c}}\, \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(191\) |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {d \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c}{a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(325\) |
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Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}} \,d x } \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}} \,d x \]
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