Optimal. Leaf size=158 \[ -\frac{6467659 \left (2 x^2-x+3\right )^{3/2}}{5971968 (2 x+5)}+\frac{158527 \left (2 x^2-x+3\right )^{3/2}}{82944 (2 x+5)^2}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{1728 (2 x+5)^3}-\frac{(44378877-7400779 x) \sqrt{2 x^2-x+3}}{5971968}+\frac{170114729 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{3981312 \sqrt{2}}-\frac{10939 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{256 \sqrt{2}} \]
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Rubi [A] time = 0.22581, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1650, 814, 843, 619, 215, 724, 206} \[ -\frac{6467659 \left (2 x^2-x+3\right )^{3/2}}{5971968 (2 x+5)}+\frac{158527 \left (2 x^2-x+3\right )^{3/2}}{82944 (2 x+5)^2}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{1728 (2 x+5)^3}-\frac{(44378877-7400779 x) \sqrt{2 x^2-x+3}}{5971968}+\frac{170114729 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{3981312 \sqrt{2}}-\frac{10939 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{256 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
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)^4} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}-\frac{1}{216} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{36021}{16}-3969 x+1458 x^2-540 x^3\right )}{(5+2 x)^3} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}+\frac{158527 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)^2}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{2672127}{16}-\frac{1284285 x}{4}+38880 x^2\right )}{(5+2 x)^2} \, dx}{31104}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}+\frac{158527 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)^2}-\frac{6467659 \left (3-x+2 x^2\right )^{3/2}}{5971968 (5+2 x)}-\frac{\int \frac{\left (\frac{66297447}{16}-\frac{22202337 x}{2}\right ) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{2239488}\\ &=-\frac{(44378877-7400779 x) \sqrt{3-x+2 x^2}}{5971968}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}+\frac{158527 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)^2}-\frac{6467659 \left (3-x+2 x^2\right )^{3/2}}{5971968 (5+2 x)}+\frac{\int \frac{-3061291212+6124439808 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{71663616}\\ &=-\frac{(44378877-7400779 x) \sqrt{3-x+2 x^2}}{5971968}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}+\frac{158527 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)^2}-\frac{6467659 \left (3-x+2 x^2\right )^{3/2}}{5971968 (5+2 x)}+\frac{10939}{256} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx-\frac{170114729 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{663552}\\ &=-\frac{(44378877-7400779 x) \sqrt{3-x+2 x^2}}{5971968}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}+\frac{158527 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)^2}-\frac{6467659 \left (3-x+2 x^2\right )^{3/2}}{5971968 (5+2 x)}+\frac{170114729 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{331776}+\frac{10939 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{256 \sqrt{46}}\\ &=-\frac{(44378877-7400779 x) \sqrt{3-x+2 x^2}}{5971968}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{1728 (5+2 x)^3}+\frac{158527 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)^2}-\frac{6467659 \left (3-x+2 x^2\right )^{3/2}}{5971968 (5+2 x)}-\frac{10939 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{256 \sqrt{2}}+\frac{170114729 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{3981312 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.158555, size = 98, normalized size = 0.62 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (414720 x^4-5453568 x^3-97682900 x^2-329667508 x-327735797\right )}{(2 x+5)^3}+170114729 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-170123328 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{7962624} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 165, normalized size = 1. \begin{align*}{\frac{-5+20\,x}{128}\sqrt{2\,{x}^{2}-x+3}}+{\frac{10939\,\sqrt{2}}{512}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{158527}{331776} \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{6467659}{11943936} \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{170114729}{23887872}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{170114729\,\sqrt{2}}{7962624}{\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{-6467659+25870636\,x}{23887872}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3667}{13824} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56178, size = 216, normalized size = 1.37 \begin{align*} \frac{5}{32} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{10939}{512} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{170114729}{7962624} \, \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{693775}{165888} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1728 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{158527 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{82944 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{6467659 \, \sqrt{2 \, x^{2} - x + 3}}{331776 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42033, size = 543, normalized size = 3.44 \begin{align*} \frac{170123328 \, \sqrt{2}{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 170114729 \, \sqrt{2}{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\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 (414720 \, x^{4} - 5453568 \, x^{3} - 97682900 \, x^{2} - 329667508 \, x - 327735797\right )} \sqrt{2 \, x^{2} - x + 3}}{15925248 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\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 )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3983, size = 410, normalized size = 2.59 \begin{align*} \frac{1}{128} \, \sqrt{2 \, x^{2} - x + 3}{\left (20 \, x - 413\right )} - \frac{10939}{512} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{170114729}{7962624} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{170114729}{7962624} \, \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 (575810908 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 9206213116 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 9688786604 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 73157325092 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 49481952947 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 20269228621\right )}}{663552 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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