Integrand size = 26, antiderivative size = 89 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {438, 437, 435} \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {c} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2-c}} \]
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Rule 435
Rule 437
Rule 438
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {-c+d x^2}} \\ & = \frac {\left (\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}{\sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2}} \\ & = \frac {\sqrt {c} \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 )}{\sqrt {d} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {a-b x^2} \sqrt {\frac {c-d x^2}{c}} E\left (\arcsin \left (\sqrt {\frac {d}{c}} x\right )|\frac {b c}{a d}\right )}{\sqrt {\frac {d}{c}} \sqrt {\frac {a-b x^2}{a}} \sqrt {-c+d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(161\) vs. \(2(74)=148\).
Time = 2.56 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.82
method | result | size |
default | \(\frac {\left (-a F\left (x \sqrt {\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d +b c F\left (x \sqrt {\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )-b c E\left (x \sqrt {\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}-c}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {-d \,x^{2}+c}{c}}}{\left (b d \,x^{4}-a d \,x^{2}-c b \,x^{2}+a c \right ) \sqrt {\frac {b}{a}}\, d}\) | \(162\) |
elliptic | \(\frac {\sqrt {-\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (\frac {a \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}+c b \,x^{2}-a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1-\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}+c b \,x^{2}-a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}-c}}\) | \(258\) |
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none
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\frac {\sqrt {-b d} a x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) - \sqrt {-b d} {\left (a - b\right )} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) + \sqrt {-b x^{2} + a} \sqrt {d x^{2} - c} b}{b d x} \]
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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. \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c+d x^2}} \, dx=\int \frac {\sqrt {a-b\,x^2}}{\sqrt {d\,x^2-c}} \,d x \]
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