Optimal. Leaf size=14 \[ -2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {4-x}\right ),-1\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1982, 689, 221} \[ -2 F\left (\left .\sin ^{-1}\left (\sqrt {4-x}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 689
Rule 1982
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {4-x} \sqrt {(5-x) (-3+x)}} \, dx &=\int \frac {1}{\sqrt {4-x} \sqrt {-15+8 x-x^2}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\sqrt {4-x}\right )\right )\\ &=-2 F\left (\left .\sin ^{-1}\left (\sqrt {4-x}\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 28, normalized size = 2.00 \[ -2 \sqrt {4-x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(x-4)^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 8 \, x - 15} \sqrt {-x + 4}}{x^{3} - 12 \, x^{2} + 47 \, x - 60}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-{\left (x - 3\right )} {\left (x - 5\right )}} \sqrt {-x + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 35, normalized size = 2.50 \[ -\frac {2 \sqrt {-x +5}\, \sqrt {x -3}\, \EllipticF \left (\sqrt {-x +4}, i\right )}{\sqrt {-\left (x -5\right ) \left (x -3\right )}} \]
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 \[ \int \frac {1}{\sqrt {-{\left (x - 3\right )} {\left (x - 5\right )}} \sqrt {-x + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {-\left (x-3\right )\,\left (x-5\right )}\,\sqrt {4-x}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \left (x - 5\right ) \left (x - 3\right )} \sqrt {4 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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