Optimal. Leaf size=103 \[ \frac{5}{6} \sqrt{2 x^2-x+3} x^2+\frac{193}{48} \sqrt{2 x^2-x+3} x+\frac{33}{64} \sqrt{2 x^2-x+3}-\frac{53-373 x}{23 \sqrt{2 x^2-x+3}}+\frac{3111 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
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Rubi [A] time = 0.101666, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{5}{6} \sqrt{2 x^2-x+3} x^2+\frac{193}{48} \sqrt{2 x^2-x+3} x+\frac{33}{64} \sqrt{2 x^2-x+3}-\frac{53-373 x}{23 \sqrt{2 x^2-x+3}}+\frac{3111 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1661
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{(5+2 x) \left (2+x+3 x^2-x^3+5 x^4\right )}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{-\frac{575}{4}+161 x^2+\frac{115 x^3}{2}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}+\frac{1}{69} \int \frac{-\frac{1725}{2}-345 x+\frac{4439 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}+\frac{1}{276} \int \frac{-\frac{27117}{4}+\frac{2277 x}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{33}{64} \sqrt{3-x+2 x^2}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}-\frac{3111}{128} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{33}{64} \sqrt{3-x+2 x^2}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}-\frac{3111 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt{46}}\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{33}{64} \sqrt{3-x+2 x^2}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}+\frac{3111 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.18094, size = 60, normalized size = 0.58 \[ \frac{7360 x^4+31832 x^3-2162 x^2+122607 x-3345}{4416 \sqrt{2 x^2-x+3}}-\frac{3111 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 115, normalized size = 1.1 \begin{align*}{\frac{5\,{x}^{4}}{3}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{3111\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-10185+40740\,x}{11776}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{3111\,x}{128}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{173\,{x}^{3}}{24}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{47\,{x}^{2}}{96}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{55}{512}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4559, size = 131, normalized size = 1.27 \begin{align*} \frac{5 \, x^{4}}{3 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{173 \, x^{3}}{24 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{47 \, x^{2}}{96 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3111}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{40869 \, x}{1472 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1115}{1472 \, \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 = 1.32192, size = 270, normalized size = 2.62 \begin{align*} \frac{214659 \, \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 ) + 8 \,{\left (7360 \, x^{4} + 31832 \, x^{3} - 2162 \, x^{2} + 122607 \, x - 3345\right )} \sqrt{2 \, x^{2} - x + 3}}{35328 \,{\left (2 \, x^{2} - x + 3\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{\left (2 x + 5\right ) \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14613, size = 90, normalized size = 0.87 \begin{align*} \frac{3111}{256} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (4 \,{\left (40 \, x + 173\right )} x - 47\right )} x + 122607\right )} x - 3345}{4416 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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