Optimal. Leaf size=87 \[ \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {421, 419} \[ \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx &=\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}} \, dx}{\sqrt {c-d x^2}}\\ &=\frac {\left (\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}{\sqrt {a+b x^2} \sqrt {c-d x^2}}\\ &=\frac {\sqrt {c} \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 )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 89, normalized size = 1.02 \[ \frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {\frac {c-d x^2}{c}} \operatorname {EllipticF}\left (\sin ^{-1}\left (x \sqrt {-\frac {b}{a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}}{b d x^{4} - {\left (b c - a d\right )} x^{2} - a c}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 106, normalized size = 1.22 \[ -\frac {\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {-\frac {d \,x^{2}-c}{c}}\, \sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}\, \EllipticF \left (\sqrt {\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \left (b d \,x^{4}+a d \,x^{2}-b c \,x^{2}-a c \right )} \]
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 \[ \int \frac {1}{\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {c-d\,x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b x^{2}} \sqrt {c - d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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