Integrand size = 23, antiderivative size = 204 \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\frac {d x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {433, 429, 506, 422} \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}} \]
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Rule 422
Rule 429
Rule 433
Rule 506
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx+d \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx \\ & = \frac {d x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}+\frac {c^{3/2} \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}}-\frac {(c d) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{b} \\ & = \frac {d x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \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 = 0.79 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {\frac {c+d x^2}{c}}} \]
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Time = 2.36 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.50
method | result | size |
default | \(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )}{\left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}}\) | \(101\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(245\) |
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Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=-\frac {\sqrt {b d} c x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (c + d\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} d}{b d x} \]
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\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\sqrt {a + b x^{2}}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a}} \,d x } \]
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\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{\sqrt {b\,x^2+a}} \,d x \]
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