Optimal. Leaf size=38 \[ -\frac {1}{3} \sqrt {-3 x^2+5 x+2}-\frac {5 \sin ^{-1}\left (\frac {1}{7} (5-6 x)\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {640, 619, 216} \begin {gather*} -\frac {1}{3} \sqrt {-3 x^2+5 x+2}-\frac {5 \sin ^{-1}\left (\frac {1}{7} (5-6 x)\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {2+5 x-3 x^2}} \, dx &=-\frac {1}{3} \sqrt {2+5 x-3 x^2}+\frac {5}{6} \int \frac {1}{\sqrt {2+5 x-3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2+5 x-3 x^2}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{49}}} \, dx,x,5-6 x\right )}{42 \sqrt {3}}\\ &=-\frac {1}{3} \sqrt {2+5 x-3 x^2}-\frac {5 \sin ^{-1}\left (\frac {1}{7} (5-6 x)\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \sqrt {-3 x^2+5 x+2}-\frac {5 \sin ^{-1}\left (\frac {1}{7} (5-6 x)\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 56, normalized size = 1.47 \begin {gather*} -\frac {1}{3} \sqrt {-3 x^2+5 x+2}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {-3 x^2+5 x+2}}{3 x+1}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 55, normalized size = 1.45 \begin {gather*} -\frac {5}{18} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 5\right )}}{6 \, {\left (3 \, x^{2} - 5 \, x - 2\right )}}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 26, normalized size = 0.68 \begin {gather*} \frac {5}{18} \, \sqrt {3} \arcsin \left (\frac {6}{7} \, x - \frac {5}{7}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.71 \begin {gather*} \frac {5 \sqrt {3}\, \arcsin \left (\frac {6 x}{7}-\frac {5}{7}\right )}{18}-\frac {\sqrt {-3 x^{2}+5 x +2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 26, normalized size = 0.68 \begin {gather*} -\frac {5}{18} \, \sqrt {3} \arcsin \left (-\frac {6}{7} \, x + \frac {5}{7}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 46, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {-3\,x^2+5\,x+2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x-\frac {5}{2}\right )\,1{}\mathrm {i}}{3}\right )\,5{}\mathrm {i}}{18} \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 {x}{\sqrt {- \left (x - 2\right ) \left (3 x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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