Optimal. Leaf size=128 \[ -\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}+\frac {2 \sqrt {-1+3 x-x^2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}}-\frac {2 \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {707, 705, 704,
313, 227, 1195, 21, 435} \begin {gather*} -\frac {2 \sqrt {-x^2+3 x-1} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {x^2-3 x+1}}+\frac {2 \sqrt {-x^2+3 x-1} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{5 \sqrt {3-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 227
Rule 313
Rule 435
Rule 704
Rule 705
Rule 707
Rule 1195
Rubi steps
\begin {align*} \int \frac {1}{(3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \, dx &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}-\frac {1}{5} \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}-\frac {\sqrt {-1+3 x-x^2} \int \frac {\sqrt {3-2 x}}{\sqrt {-\frac {1}{5}+\frac {3 x}{5}-\frac {x^2}{5}}} \, dx}{5 \sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}+\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{5 \sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}-\frac {\left (2 \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 )}{5 \sqrt {1-3 x+x^2}}+\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {5}}}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{5 \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}-\frac {2 \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}}+\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {5}}}{\sqrt {\frac {1}{\sqrt {5}}-\frac {x^2}{5}} \sqrt {\frac {1}{\sqrt {5}}+\frac {x^2}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{5 \sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}-\frac {2 \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}}+\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {1}{\sqrt {5}}+\frac {x^2}{5}}}{\sqrt {\frac {1}{\sqrt {5}}-\frac {x^2}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{5 \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{5 \sqrt {3-2 x}}+\frac {2 \sqrt {-1+3 x-x^2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt {1-3 x+x^2}}-\frac {2 \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \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 = 10.03, size = 63, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {-1+3 x-x^2} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};\frac {1}{5} (3-2 x)^2\right )}{\sqrt {5} \sqrt {3-2 x} \sqrt {1-3 x+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 116, normalized size = 0.91
method | result | size |
default | \(\frac {\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}\, \left (\sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {5}\, \sqrt {\left (-3+2 x \right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \EllipticE \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right )+20 x^{2}-60 x +20\right )}{50 x^{3}-225 x^{2}+275 x -75}\) | \(116\) |
elliptic | \(\frac {\sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \left (\frac {-\frac {4}{5} x^{2}+\frac {12}{5} x -\frac {4}{5}}{\sqrt {\left (x -\frac {3}{2}\right ) \left (-2 x^{2}+6 x -2\right )}}+\frac {6 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{125 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}-\frac {4 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \left (\frac {\sqrt {5}\, \EllipticE \left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}+\frac {3 \EllipticF \left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}\right )}{125 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right )}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(256\) |
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 = 0.27, size = 26, normalized size = 0.20 \begin {gather*} \frac {4 \, \sqrt {x^{2} - 3 \, x + 1} \sqrt {-2 \, x + 3}}{5 \, {\left (2 \, x - 3\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}{\left (3 - 2 x\right )^{\frac {3}{2}} \sqrt {x^{2} - 3 x + 1}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{{\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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