Optimal. Leaf size=126 \[ \frac{5}{32} \sqrt{2 x^2-x+3} (2 x+5)-\frac{243}{64} \sqrt{2 x^2-x+3}-\frac{3667 \sqrt{2 x^2-x+3}}{576 (2 x+5)}+\frac{158527 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{6912 \sqrt{2}}-\frac{2943 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
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Rubi [A] time = 0.202046, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1650, 1653, 843, 619, 215, 724, 206} \[ \frac{5}{32} \sqrt{2 x^2-x+3} (2 x+5)-\frac{243}{64} \sqrt{2 x^2-x+3}-\frac{3667 \sqrt{2 x^2-x+3}}{576 (2 x+5)}+\frac{158527 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{6912 \sqrt{2}}-\frac{2943 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 1653
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx &=-\frac{3667 \sqrt{3-x+2 x^2}}{576 (5+2 x)}-\frac{1}{72} \int \frac{\frac{12007}{16}-1323 x+486 x^2-180 x^3}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{576 (5+2 x)}+\frac{5}{32} (5+2 x) \sqrt{3-x+2 x^2}-\frac{\int \frac{30314-27216 x+34992 x^2}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{2304}\\ &=-\frac{243}{64} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{576 (5+2 x)}+\frac{5}{32} (5+2 x) \sqrt{3-x+2 x^2}-\frac{\int \frac{417472-847584 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{18432}\\ &=-\frac{243}{64} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{576 (5+2 x)}+\frac{5}{32} (5+2 x) \sqrt{3-x+2 x^2}+\frac{2943}{128} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx-\frac{158527 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{1152}\\ &=-\frac{243}{64} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{576 (5+2 x)}+\frac{5}{32} (5+2 x) \sqrt{3-x+2 x^2}+\frac{158527}{576} \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )+\frac{2943 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt{46}}\\ &=-\frac{243}{64} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{576 (5+2 x)}+\frac{5}{32} (5+2 x) \sqrt{3-x+2 x^2}-\frac{2943 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}+\frac{158527 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{6912 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.114362, size = 88, normalized size = 0.7 \[ \frac{\frac{48 \sqrt{2 x^2-x+3} \left (180 x^2-1287 x-6176\right )}{2 x+5}+158527 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-158922 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{13824} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 96, normalized size = 0.8 \begin{align*}{\frac{5\,x}{16}\sqrt{2\,{x}^{2}-x+3}}-{\frac{193}{64}\sqrt{2\,{x}^{2}-x+3}}+{\frac{2943\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{158527\,\sqrt{2}}{13824}{\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{3667}{1152}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52076, size = 139, normalized size = 1.1 \begin{align*} \frac{5}{16} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{2943}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{158527}{13824} \, \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{193}{64} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \, \sqrt{2 \, x^{2} - x + 3}}{576 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44426, size = 389, normalized size = 3.09 \begin{align*} \frac{158922 \, \sqrt{2}{\left (2 \, x + 5\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 158527 \, \sqrt{2}{\left (2 \, x + 5\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 ) + 96 \,{\left (180 \, x^{2} - 1287 \, x - 6176\right )} \sqrt{2 \, x^{2} - x + 3}}{27648 \,{\left (2 \, x + 5\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{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{2} \sqrt{2 x^{2} - x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{4} - x^{3} + 3 \, x^{2} + x + 2}{\sqrt{2 \, x^{2} - x + 3}{\left (2 \, x + 5\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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