Optimal. Leaf size=14 \[ -2 F\left (\left .\sin ^{-1}\left (\sqrt {4-x}\right )\right |-1\right ) \]
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Rubi [A]
time = 0.02, 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 = {1976, 703, 227}
\begin {gather*} -2 F\left (\left .\text {ArcSin}\left (\sqrt {4-x}\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 703
Rule 1976
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 \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.01, size = 28, normalized size = 2.00 \begin {gather*} -2 \sqrt {4-x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(-4+x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs.
\(2(12)=24\).
time = 0.12, size = 39, normalized size = 2.79
method | result | size |
default | \(\frac {\EllipticF \left (\frac {\sqrt {-6+2 x}}{2}, \sqrt {2}\right ) \sqrt {10-2 x}\, \sqrt {-6+2 x}}{\sqrt {-\left (-5+x \right ) \left (-3+x \right )}}\) | \(39\) |
elliptic | \(\frac {\sqrt {\left (-4+x \right ) \left (-5+x \right ) \left (-3+x \right )}\, \sqrt {-6+2 x}\, \sqrt {10-2 x}\, \EllipticF \left (\frac {\sqrt {-6+2 x}}{2}, \sqrt {2}\right )}{\sqrt {-\left (-5+x \right ) \left (-3+x \right )}\, \sqrt {x^{3}-12 x^{2}+47 x -60}}\) | \(66\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.20, size = 8, normalized size = 0.57 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (4, 0, x - 4\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- \left (x - 5\right ) \left (x - 3\right )} \sqrt {4 - x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {-\left (x-3\right )\,\left (x-5\right )}\,\sqrt {4-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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