Optimal. Leaf size=90 \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {427, 426, 424} \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 424
Rule 426
Rule 427
Rubi steps
\begin {align*} \int \frac {\sqrt {-c-d x^2}}{\sqrt {a-b x^2}} \, dx &=\frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {-c-d x^2}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\left (\sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2}\right ) \int \frac {\sqrt {1+\frac {d x^2}{c}}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}\\ &=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 1.00 \[ \frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {a-b x^2} \sqrt {\frac {c+d x^2}{c}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{2} + a} \sqrt {-d x^{2} - c}}{b x^{2} - a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-d x^{2} - c}}{\sqrt {-b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 171, normalized size = 1.90 \[ \frac {\left (a d \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )-a d \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )-b c \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )\right ) \sqrt {-d \,x^{2}-c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {-\frac {b \,x^{2}-a}{a}}}{\left (b d \,x^{4}-a d \,x^{2}+b c \,x^{2}-a c \right ) \sqrt {-\frac {d}{c}}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-d x^{2} - c}}{\sqrt {-b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {-d\,x^2-c}}{\sqrt {a-b\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c - d x^{2}}}{\sqrt {a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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