Integrand size = 24, antiderivative size = 189 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx=-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {434, 438, 437, 435, 432, 430} \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}} \]
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Rule 430
Rule 432
Rule 434
Rule 435
Rule 437
Rule 438
Rubi steps \begin{align*} \text {integral}& = -\frac {d \int \frac {\sqrt {a+b x^2}}{\sqrt {c-d x^2}} \, dx}{b}+\frac {(b c+a d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx}{b} \\ & = -\frac {\left (d \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {a+b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{b \sqrt {c-d x^2}}+\frac {\left ((b c+a d) \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}} \, dx}{b \sqrt {c-d x^2}} \\ & = -\frac {\left (d \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\left ((b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}} \, dx}{b \sqrt {a+b x^2} \sqrt {c-d x^2}} \\ & = -\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \\ \end{align*}
Time = 0.85 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {c-d x^2} E\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {\frac {c-d x^2}{c}}} \]
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Time = 2.43 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \left (a d F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )+c F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b -a d E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )\right )}{\left (-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {\frac {d}{c}}\, b}\) | \(161\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (\frac {c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}}+\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}\, b}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\) | \(257\) |
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Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {-b d} c x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - \sqrt {-b d} {\left (c - d\right )} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + \sqrt {b x^{2} + a} \sqrt {-d x^{2} + c} d}{b d x} \]
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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. \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {c-d\,x^2}}{\sqrt {b\,x^2+a}} \,d x \]
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