Integrand size = 26, antiderivative size = 203 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {x \sqrt {a+b x^2}}{\sqrt {-c-d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Time = 0.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {433, 429, 506, 422} \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \]
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Rule 422
Rule 429
Rule 433
Rule 506
Rubi steps \begin{align*} \text {integral}& = a \int \frac {1}{\sqrt {a+b x^2} \sqrt {-c-d x^2}} \, dx+b \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {-c-d x^2}} \, dx \\ & = \frac {x \sqrt {a+b x^2}}{\sqrt {-c-d x^2}}+\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 )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+c \int \frac {\sqrt {a+b x^2}}{\left (-c-d x^2\right )^{3/2}} \, dx \\ & = \frac {x \sqrt {a+b x^2}}{\sqrt {-c-d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\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 {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {-c-d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {\frac {c+d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {\frac {a+b x^2}{a}} \sqrt {-c-d x^2}} \]
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Time = 2.47 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.53
method | result | size |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}\, a \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )}{\left (-b d \,x^{4}-a d \,x^{2}-c b \,x^{2}-a c \right ) \sqrt {-\frac {d}{c}}}\) | \(108\) |
elliptic | \(\frac {\sqrt {-\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}-c b \,x^{2}-a c}}-\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}-c b \,x^{2}-a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}}\) | \(266\) |
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none
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {-b d} b c^{2} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b x^{2} + a} \sqrt {-d x^{2} - c} b c d - {\left (b c^{2} + a d^{2}\right )} \sqrt {-b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c})}{b c d^{2} x} \]
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\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\sqrt {- c - d x^{2}}}\, dx \]
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\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-d x^{2} - c}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-d x^{2} - c}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {-c-d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\sqrt {-d\,x^2-c}} \,d x \]
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