Optimal. Leaf size=79 \[ -\frac {4 \sqrt {1-3 x+x^2}}{15 (3-2 x)^{3/2}}-\frac {2 \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{15 \sqrt [4]{5} \sqrt {1-3 x+x^2}} \]
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Rubi [A]
time = 0.02, 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 = {707, 705, 703,
227} \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 )}{15 \sqrt [4]{5} \sqrt {x^2-3 x+1}}-\frac {4 \sqrt {x^2-3 x+1}}{15 (3-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 703
Rule 705
Rule 707
Rubi steps
\begin {align*} \int \frac {1}{(3-2 x)^{5/2} \sqrt {1-3 x+x^2}} \, dx &=-\frac {4 \sqrt {1-3 x+x^2}}{15 (3-2 x)^{3/2}}+\frac {1}{15} \int \frac {1}{\sqrt {3-2 x} \sqrt {1-3 x+x^2}} \, dx\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{15 (3-2 x)^{3/2}}+\frac {\sqrt {-1+3 x-x^2} \int \frac {1}{\sqrt {3-2 x} \sqrt {-\frac {1}{5}+\frac {3 x}{5}-\frac {x^2}{5}}} \, dx}{15 \sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{15 (3-2 x)^{3/2}}-\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 )}{15 \sqrt {5} \sqrt {1-3 x+x^2}}\\ &=-\frac {4 \sqrt {1-3 x+x^2}}{15 (3-2 x)^{3/2}}-\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 )}{15 \sqrt [4]{5} \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 = 10.02, size = 65, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {-1+3 x-x^2} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};\frac {1}{5} (3-2 x)^2\right )}{3 \sqrt {5} (3-2 x)^{3/2} \sqrt {1-3 x+x^2}} \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. \(171\) vs.
\(2(62)=124\).
time = 0.91, size = 172, normalized size = 2.18
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \left (-\frac {\sqrt {-2 x^{3}+9 x^{2}-11 x +3}}{15 \left (x -\frac {3}{2}\right )^{2}}-\frac {2 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{375 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right )}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(142\) |
default | \(\frac {\left (2 \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {\left (-3+2 x \right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right ) x -3 \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {\left (-3+2 x \right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right )-20 x^{2}+60 x -20\right ) \sqrt {3-2 x}}{75 \sqrt {x^{2}-3 x +1}\, \left (-3+2 x \right )^{2}}\) | \(172\) |
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 = 1.13, size = 31, normalized size = 0.39 \begin {gather*} -\frac {4 \, \sqrt {x^{2} - 3 \, x + 1} \sqrt {-2 \, x + 3}}{15 \, {\left (4 \, x^{2} - 12 \, x + 9\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 {5}{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 )}^{5/2}\,\sqrt {x^2-3\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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