Optimal. Leaf size=120 \[ \frac{1}{16} \sqrt{2 x^2-x+3} (2 x+5)^4-\frac{105}{128} \sqrt{2 x^2-x+3} (2 x+5)^3+\frac{761}{256} \sqrt{2 x^2-x+3} (2 x+5)^2-\frac{(4676 x+19227) \sqrt{2 x^2-x+3}}{2048}-\frac{85429 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]
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Rubi [A] time = 0.135734, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1653, 779, 619, 215} \[ \frac{1}{16} \sqrt{2 x^2-x+3} (2 x+5)^4-\frac{105}{128} \sqrt{2 x^2-x+3} (2 x+5)^3+\frac{761}{256} \sqrt{2 x^2-x+3} (2 x+5)^2-\frac{(4676 x+19227) \sqrt{2 x^2-x+3}}{2048}-\frac{85429 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 779
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 )}{\sqrt{3-x+2 x^2}} \, dx &=\frac{1}{16} (5+2 x)^4 \sqrt{3-x+2 x^2}+\frac{1}{160} \int \frac{(5+2 x) \left (-5055-4390 x-5580 x^2-4200 x^3\right )}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{105}{128} (5+2 x)^3 \sqrt{3-x+2 x^2}+\frac{1}{16} (5+2 x)^4 \sqrt{3-x+2 x^2}+\frac{\int \frac{(5+2 x) \left (327480+105440 x+365280 x^2\right )}{\sqrt{3-x+2 x^2}} \, dx}{10240}\\ &=\frac{761}{256} (5+2 x)^2 \sqrt{3-x+2 x^2}-\frac{105}{128} (5+2 x)^3 \sqrt{3-x+2 x^2}+\frac{1}{16} (5+2 x)^4 \sqrt{3-x+2 x^2}+\frac{\int \frac{(919200-1122240 x) (5+2 x)}{\sqrt{3-x+2 x^2}} \, dx}{245760}\\ &=\frac{761}{256} (5+2 x)^2 \sqrt{3-x+2 x^2}-\frac{105}{128} (5+2 x)^3 \sqrt{3-x+2 x^2}+\frac{1}{16} (5+2 x)^4 \sqrt{3-x+2 x^2}-\frac{(19227+4676 x) \sqrt{3-x+2 x^2}}{2048}+\frac{85429 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4096}\\ &=\frac{761}{256} (5+2 x)^2 \sqrt{3-x+2 x^2}-\frac{105}{128} (5+2 x)^3 \sqrt{3-x+2 x^2}+\frac{1}{16} (5+2 x)^4 \sqrt{3-x+2 x^2}-\frac{(19227+4676 x) \sqrt{3-x+2 x^2}}{2048}+\frac{85429 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4096 \sqrt{46}}\\ &=\frac{761}{256} (5+2 x)^2 \sqrt{3-x+2 x^2}-\frac{105}{128} (5+2 x)^3 \sqrt{3-x+2 x^2}+\frac{1}{16} (5+2 x)^4 \sqrt{3-x+2 x^2}-\frac{(19227+4676 x) \sqrt{3-x+2 x^2}}{2048}-\frac{85429 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.115793, size = 60, normalized size = 0.5 \[ \frac{4 \sqrt{2 x^2-x+3} \left (2048 x^4+7040 x^3+352 x^2-6916 x+2973\right )-85429 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 95, normalized size = 0.8 \begin{align*}{x}^{4}\sqrt{2\,{x}^{2}-x+3}+{\frac{55\,{x}^{3}}{16}\sqrt{2\,{x}^{2}-x+3}}+{\frac{11\,{x}^{2}}{64}\sqrt{2\,{x}^{2}-x+3}}-{\frac{1729\,x}{512}\sqrt{2\,{x}^{2}-x+3}}+{\frac{2973}{2048}\sqrt{2\,{x}^{2}-x+3}}+{\frac{85429\,\sqrt{2}}{8192}{\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 = 1.47478, size = 130, normalized size = 1.08 \begin{align*} \sqrt{2 \, x^{2} - x + 3} x^{4} + \frac{55}{16} \, \sqrt{2 \, x^{2} - x + 3} x^{3} + \frac{11}{64} \, \sqrt{2 \, x^{2} - x + 3} x^{2} - \frac{1729}{512} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{85429}{8192} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{2973}{2048} \, \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 = 1.29906, size = 223, normalized size = 1.86 \begin{align*} \frac{1}{2048} \,{\left (2048 \, x^{4} + 7040 \, x^{3} + 352 \, x^{2} - 6916 \, x + 2973\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{85429}{16384} \, \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{\left (2 x + 5\right ) \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\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 = 1.20803, size = 92, normalized size = 0.77 \begin{align*} \frac{1}{2048} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \, x + 55\right )} x + 11\right )} x - 1729\right )} x + 2973\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{85429}{8192} \, \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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