Optimal. Leaf size=91 \[ \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}-\frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}} \]
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Rubi [A] time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {423, 426, 424, 421, 419} \[ \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}-\frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rule 423
Rule 424
Rule 426
Rubi steps
\begin {align*} \int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx &=-\frac {\int \frac {\sqrt {c+d x^2}}{\sqrt {4-x^2}} \, dx}{d}-\frac {(-c-4 d) \int \frac {1}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx}{d}\\ &=-\frac {\sqrt {c+d x^2} \int \frac {\sqrt {1+\frac {d x^2}{c}}}{\sqrt {4-x^2}} \, dx}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {\left ((-c-4 d) \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {4-x^2} \sqrt {1+\frac {d x^2}{c}}} \, dx}{d \sqrt {c+d x^2}}\\ &=-\frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}+\frac {(c+4 d) \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.66 \[ \frac {2 \sqrt {\frac {c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c}{4 d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 4}}{\sqrt {d x^{2} + c}}, x\right ) \]
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 \[ \int \frac {\sqrt {-x^{2} + 4}}{\sqrt {d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 78, normalized size = 0.86 \[ \frac {\left (-c \EllipticE \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )+c \EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )+4 d \EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) \sqrt {\frac {d \,x^{2}+c}{c}}}{\sqrt {d \,x^{2}+c}\, d} \]
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 \[ \int \frac {\sqrt {-x^{2} + 4}}{\sqrt {d x^{2} + c}}\,{d x} \]
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 \[ \int \frac {\sqrt {4-x^2}}{\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {\sqrt {- \left (x - 2\right ) \left (x + 2\right )}}{\sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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