Optimal. Leaf size=124 \[ \frac{5}{12} \left (2 x^2-x+3\right )^{3/2} x^3+\frac{7}{80} \left (2 x^2-x+3\right )^{3/2} x^2-\frac{71 \left (2 x^2-x+3\right )^{3/2} x}{1280}+\frac{287 \left (2 x^2-x+3\right )^{3/2}}{5120}-\frac{4609 (1-4 x) \sqrt{2 x^2-x+3}}{16384}-\frac{106007 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}} \]
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Rubi [A] time = 0.0921227, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1661, 640, 612, 619, 215} \[ \frac{5}{12} \left (2 x^2-x+3\right )^{3/2} x^3+\frac{7}{80} \left (2 x^2-x+3\right )^{3/2} x^2-\frac{71 \left (2 x^2-x+3\right )^{3/2} x}{1280}+\frac{287 \left (2 x^2-x+3\right )^{3/2}}{5120}-\frac{4609 (1-4 x) \sqrt{2 x^2-x+3}}{16384}-\frac{106007 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \sqrt{3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{12} \int \sqrt{3-x+2 x^2} \left (24+12 x-9 x^2+\frac{21 x^3}{2}\right ) \, dx\\ &=\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{120} \int \left (240+57 x-\frac{213 x^2}{4}\right ) \sqrt{3-x+2 x^2} \, dx\\ &=-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{960} \int \left (\frac{8319}{4}+\frac{2583 x}{8}\right ) \sqrt{3-x+2 x^2} \, dx\\ &=\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{4609 \int \sqrt{3-x+2 x^2} \, dx}{2048}\\ &=-\frac{4609 (1-4 x) \sqrt{3-x+2 x^2}}{16384}+\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{106007 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{32768}\\ &=-\frac{4609 (1-4 x) \sqrt{3-x+2 x^2}}{16384}+\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{\left (4609 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{32768}\\ &=-\frac{4609 (1-4 x) \sqrt{3-x+2 x^2}}{16384}+\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}-\frac{106007 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.100818, size = 65, normalized size = 0.52 \[ \frac{4 \sqrt{2 x^2-x+3} \left (204800 x^5-59392 x^4+258432 x^3+105696 x^2+221868 x-27807\right )-1590105 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{983040} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 98, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{12} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{x}^{2}}{80} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{71\,x}{1280} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{287}{5120} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-4609+18436\,x}{16384}\sqrt{2\,{x}^{2}-x+3}}+{\frac{106007\,\sqrt{2}}{65536}{\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 = 1.57703, size = 147, normalized size = 1.19 \begin{align*} \frac{5}{12} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{7}{80} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{71}{1280} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{287}{5120} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{4609}{4096} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{106007}{65536} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4609}{16384} \, \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 = 1.35483, size = 258, normalized size = 2.08 \begin{align*} \frac{1}{245760} \,{\left (204800 \, x^{5} - 59392 \, x^{4} + 258432 \, x^{3} + 105696 \, x^{2} + 221868 \, x - 27807\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{106007}{131072} \, \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 \sqrt{2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17368, size = 99, normalized size = 0.8 \begin{align*} \frac{1}{245760} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 29\right )} x + 2019\right )} x + 3303\right )} x + 55467\right )} x - 27807\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{106007}{65536} \, \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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