Optimal. Leaf size=79 \[ -\frac {4}{3} \sqrt {3-2 x} \sqrt {1-3 x+x^2}-\frac {2\ 5^{3/4} \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt {1-3 x+x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {706, 705, 703,
227} \begin {gather*} -\frac {2\ 5^{3/4} \sqrt {-x^2+3 x-1} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt {x^2-3 x+1}}-\frac {4}{3} \sqrt {3-2 x} \sqrt {x^2-3 x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 703
Rule 705
Rule 706
Rubi steps
\begin {align*} \int \frac {(3-2 x)^{3/2}}{\sqrt {1-3 x+x^2}} \, dx &=-\frac {4}{3} \sqrt {3-2 x} \sqrt {1-3 x+x^2}+\frac {5}{3} \int \frac {1}{\sqrt {3-2 x} \sqrt {1-3 x+x^2}} \, dx\\ &=-\frac {4}{3} \sqrt {3-2 x} \sqrt {1-3 x+x^2}+\frac {\left (\sqrt {5} \sqrt {-1+3 x-x^2}\right ) \int \frac {1}{\sqrt {3-2 x} \sqrt {-\frac {1}{5}+\frac {3 x}{5}-\frac {x^2}{5}}} \, dx}{3 \sqrt {1-3 x+x^2}}\\ &=-\frac {4}{3} \sqrt {3-2 x} \sqrt {1-3 x+x^2}-\frac {\left (2 \sqrt {5} \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{3 \sqrt {1-3 x+x^2}}\\ &=-\frac {4}{3} \sqrt {3-2 x} \sqrt {1-3 x+x^2}-\frac {2\ 5^{3/4} \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt {1-3 x+x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 76, normalized size = 0.96 \begin {gather*} -\frac {2 \sqrt {3-2 x} \left (2-6 x+2 x^2+\sqrt {5} \sqrt {-1+3 x-x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {1}{5} (3-2 x)^2\right )\right )}{3 \sqrt {1-3 x+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.90, size = 118, normalized size = 1.49
method | result | size |
default | \(\frac {\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}\, \left (\sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {\left (-3+2 x \right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right )-8 x^{3}+36 x^{2}-44 x +12\right )}{6 x^{3}-27 x^{2}+33 x -9}\) | \(118\) |
elliptic | \(\frac {\sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \left (-\frac {4 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}{3}-\frac {2 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{15 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right )}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(137\) |
risch | \(\frac {4 \left (-3+2 x \right ) \sqrt {x^{2}-3 x +1}\, \sqrt {\left (3-2 x \right ) \left (x^{2}-3 x +1\right )}}{3 \sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \sqrt {3-2 x}}-\frac {2 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right ) \sqrt {\left (3-2 x \right ) \left (x^{2}-3 x +1\right )}}{15 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}\, \sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(173\) |
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 [A]
time = 0.42, size = 19, normalized size = 0.24 \begin {gather*} -\frac {4}{3} \, \sqrt {x^{2} - 3 \, x + 1} \sqrt {-2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.42, size = 41, normalized size = 0.52 \begin {gather*} \frac {\sqrt {5} i \left (3 - 2 x\right )^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {\left (3 - 2 x\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac {9}{4}\right )} \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 = -0.01 \begin {gather*} \int \frac {{\left (3-2\,x\right )}^{3/2}}{\sqrt {x^2-3\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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