Optimal. Leaf size=59 \[ \frac{5}{4} \sqrt{2 x^2-x+3} x+\frac{39}{16} \sqrt{2 x^2-x+3}+\frac{17 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
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Rubi [A] time = 0.032932, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1661, 640, 619, 215} \[ \frac{5}{4} \sqrt{2 x^2-x+3} x+\frac{39}{16} \sqrt{2 x^2-x+3}+\frac{17 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{2+3 x+5 x^2}{\sqrt{3-x+2 x^2}} \, dx &=\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{1}{4} \int \frac{-7+\frac{39 x}{2}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{39}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}-\frac{17}{32} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{39}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}-\frac{17 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{32 \sqrt{46}}\\ &=\frac{39}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{17 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0382121, size = 45, normalized size = 0.76 \[ \frac{1}{64} \left (4 \sqrt{2 x^2-x+3} (20 x+39)+17 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 45, normalized size = 0.8 \begin{align*}{\frac{5\,x}{4}\sqrt{2\,{x}^{2}-x+3}}+{\frac{39}{16}\sqrt{2\,{x}^{2}-x+3}}-{\frac{17\,\sqrt{2}}{64}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53221, size = 62, normalized size = 1.05 \begin{align*} \frac{5}{4} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{17}{64} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{39}{16} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32083, size = 163, normalized size = 2.76 \begin{align*} \frac{1}{16} \, \sqrt{2 \, x^{2} - x + 3}{\left (20 \, x + 39\right )} + \frac{17}{128} \, \sqrt{2} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 x^{2} + 3 x + 2}{\sqrt{2 x^{2} - x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15619, size = 72, normalized size = 1.22 \begin{align*} \frac{1}{16} \, \sqrt{2 \, x^{2} - x + 3}{\left (20 \, x + 39\right )} + \frac{17}{64} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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