Optimal. Leaf size=105 \[ \frac{4 (18982-20383 x)}{1587 \sqrt{2 x^2-x+3}}+\frac{5}{4} x \sqrt{2 x^2-x+3}+\frac{247}{16} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
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Rubi [A] time = 0.130773, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{4 (18982-20383 x)}{1587 \sqrt{2 x^2-x+3}}+\frac{5}{4} x \sqrt{2 x^2-x+3}+\frac{247}{16} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
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)^2 \left (2+x+3 x^2-x^3+5 x^4\right )}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-145-\frac{1725 x}{2}+2415 x^2+\frac{3657 x^3}{2}+345 x^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{\frac{33327}{2}+\frac{46023 x}{4}+\frac{7935 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{242811}{4}+\frac{391989 x}{8}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{247}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{1471}{32} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{247}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{1471 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{32 \sqrt{46}}\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{247}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.711421, size = 65, normalized size = 0.62 \[ \frac{126960 x^5+1440996 x^4-3764360 x^3+8639625 x^2-6410082 x+6663133}{25392 \left (2 x^2-x+3\right )^{3/2}}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 180, normalized size = 1.7 \begin{align*} 5\,{\frac{{x}^{5}}{ \left ( 2\,{x}^{2}-x+3 \right ) ^{3/2}}}-{\frac{1471\,{x}^{3}}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{19073\,{x}^{2}}{64} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{1471\,\sqrt{2}}{64}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{-162931+651724\,x}{50784}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-753223+3012892\,x}{141312} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{32257\,x}{512} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{227\,{x}^{4}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{1471\,x}{32}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1471}{128}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{577397}{2048} \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 = 1.55226, size = 296, normalized size = 2.82 \begin{align*} \frac{5 \, x^{5}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{227 \, x^{4}}{4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1471}{50784} \, 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{1471}{64} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{104441}{25392} \, \sqrt{2 \, x^{2} - x + 3} - \frac{383581 \, x}{12696 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{321 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{15965}{4232 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{4147 \, x}{46 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{42883}{138 \,{\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 = 1.33287, size = 359, normalized size = 3.42 \begin{align*} \frac{2334477 \, \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 (126960 \, x^{5} + 1440996 \, x^{4} - 3764360 \, x^{3} + 8639625 \, x^{2} - 6410082 \, x + 6663133\right )} \sqrt{2 \, x^{2} - x + 3}}{203136 \,{\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 )^{2} \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 = 1.16211, size = 96, normalized size = 0.91 \begin{align*} -\frac{1471}{64} \, \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 (1587 \,{\left (20 \, x + 227\right )} x - 941090\right )} x + 8639625\right )} x - 6410082\right )} x + 6663133}{25392 \,{\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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