Optimal. Leaf size=194 \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {-c-d x^2}}+\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}} \]
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Rubi [A] time = 0.13, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {423, 427, 426, 424, 421, 419} \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {-c-d x^2}}+\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rule 423
Rule 424
Rule 426
Rule 427
Rubi steps
\begin {align*} \int \frac {\sqrt {a-b x^2}}{\sqrt {-c-d x^2}} \, dx &=\frac {b \int \frac {\sqrt {-c-d x^2}}{\sqrt {a-b x^2}} \, dx}{d}+\frac {(b c+a d) \int \frac {1}{\sqrt {a-b x^2} \sqrt {-c-d x^2}} \, dx}{d}\\ &=\frac {\left (b \sqrt {1-\frac {b x^2}{a}}\right ) \int \frac {\sqrt {-c-d x^2}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{d \sqrt {a-b 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}{d \sqrt {-c-d x^2}}\\ &=\frac {\left (b \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}{d \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\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}{d \sqrt {a-b x^2} \sqrt {-c-d x^2}}\\ &=\frac {\sqrt {a} \sqrt {b} \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 )}{d \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} (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 {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {-c-d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 0.47 \[ \frac {\sqrt {a-b x^2} \sqrt {\frac {c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {\frac {a-b x^2}{a}} \sqrt {-c-d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{2} + a} \sqrt {-d x^{2} - c}}{d x^{2} + c}, x\right ) \]
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 \[ \int \frac {\sqrt {-b x^{2} + a}}{\sqrt {-d x^{2} - c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 111, normalized size = 0.57 \[ \frac {\sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {-\frac {b \,x^{2}-a}{a}}\, a \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )}{\left (b d \,x^{4}-a d \,x^{2}+b c \,x^{2}-a c \right ) \sqrt {-\frac {d}{c}}} \]
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 \[ \int \frac {\sqrt {-b x^{2} + a}}{\sqrt {-d x^{2} - c}}\,{d x} \]
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 \[ \int \frac {\sqrt {a-b\,x^2}}{\sqrt {-d\,x^2-c}} \,d x \]
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 \[ \int \frac {\sqrt {a - b x^{2}}}{\sqrt {- c - d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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