Optimal. Leaf size=45 \[ -\frac {11 (3 x+5)}{23 \sqrt {2 x^2-x+3}}-\frac {5 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1660, 12, 619, 215} \[ -\frac {11 (3 x+5)}{23 \sqrt {2 x^2-x+3}}-\frac {5 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 619
Rule 1660
Rubi steps
\begin {align*} \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=-\frac {11 (5+3 x)}{23 \sqrt {3-x+2 x^2}}+\frac {2}{23} \int \frac {115}{4 \sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {11 (5+3 x)}{23 \sqrt {3-x+2 x^2}}+\frac {5}{2} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {11 (5+3 x)}{23 \sqrt {3-x+2 x^2}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2 \sqrt {46}}\\ &=-\frac {11 (5+3 x)}{23 \sqrt {3-x+2 x^2}}-\frac {5 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 45, normalized size = 1.00 \[ \frac {5 \sinh ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{2 \sqrt {2}}-\frac {11 (3 x+5)}{23 \sqrt {2 x^2-x+3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 82, normalized size = 1.82 \[ \frac {115 \, \sqrt {2} {\left (2 \, x^{2} - x + 3\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 88 \, \sqrt {2 \, x^{2} - x + 3} {\left (3 \, x + 5\right )}}{184 \, {\left (2 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 53, normalized size = 1.18 \[ -\frac {5}{4} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {11 \, {\left (3 \, x + 5\right )}}{23 \, \sqrt {2 \, x^{2} - x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 1.42 \[ -\frac {5 x}{2 \sqrt {2 x^{2}-x +3}}+\frac {5 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4}-\frac {17}{8 \sqrt {2 x^{2}-x +3}}+\frac {\frac {49 x}{46}-\frac {49}{184}}{\sqrt {2 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 46, normalized size = 1.02 \[ \frac {5}{4} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {33 \, x}{23 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {55}{23 \, \sqrt {2 \, x^{2} - x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 87, normalized size = 1.93 \[ \frac {5\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{4}+\frac {3\,\left (2\,x-12\right )}{23\,\sqrt {2\,x^2-x+3}}-\frac {10\,\left (\frac {11\,x}{2}+\frac {3}{2}\right )}{23\,\sqrt {2\,x^2-x+3}}+\frac {16\,x-4}{23\,\sqrt {2\,x^2-x+3}} \]
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 \[ \int \frac {5 x^{2} + 3 x + 2}{\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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