Optimal. Leaf size=151 \[ \frac{357391 \left (2 x^2-x+3\right )^{3/2}}{82944 (2 x+5)}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}+\frac{5}{48} \left (2 x^2-x+3\right )^{3/2}+\frac{5 (661065-110099 x) \sqrt{2 x^2-x+3}}{82944}-\frac{12670805 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{55296 \sqrt{2}}+\frac{117315 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}} \]
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Rubi [A] time = 0.228359, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1650, 1653, 814, 843, 619, 215, 724, 206} \[ \frac{357391 \left (2 x^2-x+3\right )^{3/2}}{82944 (2 x+5)}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}+\frac{5}{48} \left (2 x^2-x+3\right )^{3/2}+\frac{5 (661065-110099 x) \sqrt{2 x^2-x+3}}{82944}-\frac{12670805 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{55296 \sqrt{2}}+\frac{117315 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 1653
Rule 814
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}-\frac{1}{144} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{27681}{16}-\frac{14251 x}{4}+972 x^2-360 x^3\right )}{(5+2 x)^2} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac{357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{1531305}{16}-\frac{492175 x}{2}+12960 x^2\right )}{5+2 x} \, dx}{10368}\\ &=\frac{5}{48} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac{357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}+\frac{\int \frac{\left (\frac{4982715}{2}-6605940 x\right ) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{248832}\\ &=\frac{5 (661065-110099 x) \sqrt{3-x+2 x^2}}{82944}+\frac{5}{48} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac{357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}-\frac{\int \frac{-1825161120+3648965760 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{7962624}\\ &=\frac{5 (661065-110099 x) \sqrt{3-x+2 x^2}}{82944}+\frac{5}{48} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac{357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}-\frac{117315}{512} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{12670805 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{9216}\\ &=\frac{5 (661065-110099 x) \sqrt{3-x+2 x^2}}{82944}+\frac{5}{48} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac{357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}-\frac{12670805 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{4608}-\frac{117315 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{512 \sqrt{46}}\\ &=\frac{5 (661065-110099 x) \sqrt{3-x+2 x^2}}{82944}+\frac{5}{48} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac{357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}+\frac{117315 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}}-\frac{12670805 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{55296 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.155759, size = 98, normalized size = 0.65 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (3840 x^4-25632 x^3+272520 x^2+2959330 x+4880551\right )}{(2 x+5)^2}-12670805 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+12670020 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{110592} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 158, normalized size = 1.1 \begin{align*}{\frac{5}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{-149+596\,x}{256}\sqrt{2\,{x}^{2}-x+3}}-{\frac{117315\,\sqrt{2}}{1024}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{3667}{4608} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{357391}{165888} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{12670805}{331776}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{12670805\,\sqrt{2}}{110592}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }-{\frac{-357391+1429564\,x}{331776}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50259, size = 193, normalized size = 1.28 \begin{align*} \frac{5}{48} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{149}{64} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{117315}{1024} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{12670805}{110592} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) + \frac{3877}{144} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1152 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{357391 \, \sqrt{2 \, x^{2} - x + 3}}{4608 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41095, size = 479, normalized size = 3.17 \begin{align*} \frac{12670020 \, \sqrt{2}{\left (4 \, x^{2} + 20 \, x + 25\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 12670805 \, \sqrt{2}{\left (4 \, x^{2} + 20 \, x + 25\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \,{\left (3840 \, x^{4} - 25632 \, x^{3} + 272520 \, x^{2} + 2959330 \, x + 4880551\right )} \sqrt{2 \, x^{2} - x + 3}}{221184 \,{\left (4 \, x^{2} + 20 \, 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{\sqrt{2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27493, size = 348, normalized size = 2.3 \begin{align*} \frac{1}{768} \,{\left (4 \,{\left (40 \, x - 467\right )} x + 19695\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{117315}{1024} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{12670805}{110592} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{12670805}{110592} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{\sqrt{2}{\left (10693526 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 79895946 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 124044603 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 80334011\right )}}{9216 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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