Optimal. Leaf size=86 \[ \frac{6055-28981 x}{3174 \sqrt{2 x^2-x+3}}+\frac{5}{4} \sqrt{2 x^2-x+3}-\frac{53-373 x}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac{71 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0822592, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1660, 640, 619, 215} \[ \frac{6055-28981 x}{3174 \sqrt{2 x^2-x+3}}+\frac{5}{4} \sqrt{2 x^2-x+3}-\frac{53-373 x}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac{71 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1660
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 )^{5/2}} \, dx &=-\frac{53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-\frac{233}{4}+483 x^2+\frac{345 x^3}{2}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{6055-28981 x}{3174 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{\frac{52371}{16}+\frac{7935 x}{8}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{6055-28981 x}{3174 \sqrt{3-x+2 x^2}}+\frac{5}{4} \sqrt{3-x+2 x^2}+\frac{71}{8} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{6055-28981 x}{3174 \sqrt{3-x+2 x^2}}+\frac{5}{4} \sqrt{3-x+2 x^2}+\frac{71 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{8 \sqrt{46}}\\ &=-\frac{53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{6055-28981 x}{3174 \sqrt{3-x+2 x^2}}+\frac{5}{4} \sqrt{3-x+2 x^2}-\frac{71 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.259808, size = 60, normalized size = 0.7 \[ \frac{31740 x^4-147664 x^3+185337 x^2-199290 x+102869}{6348 \left (2 x^2-x+3\right )^{3/2}}+\frac{71 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 163, normalized size = 1.9 \begin{align*} -{\frac{71\,{x}^{3}}{12} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{401\,{x}^{2}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{71\,\sqrt{2}}{16}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-643+2572\,x}{12696}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-2327+9308\,x}{35328} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{945\,x}{128} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{{x}^{4}}{ \left ( 2\,{x}^{2}-x+3 \right ) ^{3/2}}}-{\frac{71\,x}{8}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{71}{32}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{11749}{512} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.50182, size = 273, normalized size = 3.17 \begin{align*} \frac{5 \, x^{4}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{71}{12696} \, x{\left (\frac{284 \, x}{\sqrt{2 \, x^{2} - x + 3}} - \frac{3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{2 \, x^{2} - x + 3}} + \frac{805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} + \frac{71}{16} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{5041}{6348} \, \sqrt{2 \, x^{2} - x + 3} - \frac{10007 \, x}{3174 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{59 \, x^{2}}{2 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{2959}{2116 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{807 \, x}{92 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{7603}{276 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34327, size = 331, normalized size = 3.85 \begin{align*} \frac{112677 \, \sqrt{2}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\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 (31740 \, x^{4} - 147664 \, x^{3} + 185337 \, x^{2} - 199290 \, x + 102869\right )} \sqrt{2 \, x^{2} - x + 3}}{50784 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\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{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17698, size = 89, normalized size = 1.03 \begin{align*} -\frac{71}{16} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left ({\left (4 \,{\left (7935 \, x - 36916\right )} x + 185337\right )} x - 199290\right )} x + 102869}{6348 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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