Optimal. Leaf size=160 \[ -\frac{27754539-31190998 x}{31986607104 \sqrt{2 x^2-x+3}}+\frac{475357 \sqrt{2 x^2-x+3}}{1934917632 (2 x+5)}-\frac{89137 \sqrt{2 x^2-x+3}}{80621568 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{559872 (2 x+5)^3}+\frac{369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}+\frac{4778789 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{7739670528 \sqrt{2}} \]
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Rubi [A] time = 0.282999, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1646, 1650, 806, 724, 206} \[ -\frac{27754539-31190998 x}{31986607104 \sqrt{2 x^2-x+3}}+\frac{475357 \sqrt{2 x^2-x+3}}{1934917632 (2 x+5)}-\frac{89137 \sqrt{2 x^2-x+3}}{80621568 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{559872 (2 x+5)^3}+\frac{369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}+\frac{4778789 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{7739670528 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1646
Rule 1650
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{606939313}{26873856}+\frac{727085495 x}{13436928}+\frac{186705485 x^2}{2239488}-\frac{10162483 x^3}{3359232}-\frac{175877 x^4}{419904}}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac{27754539-31190998 x}{31986607104 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{-\frac{4811736919}{40310784}-\frac{3560904781 x}{13436928}-\frac{87176555 x^2}{1679616}-\frac{39913579 x^3}{10077696}}{(5+2 x)^4 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac{27754539-31190998 x}{31986607104 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{559872 (5+2 x)^3}-\frac{\int \frac{\frac{86989289}{11664}+\frac{1265556853 x}{186624}+\frac{39913579 x^2}{93312}}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx}{85698}\\ &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac{27754539-31190998 x}{31986607104 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{559872 (5+2 x)^3}-\frac{89137 \sqrt{3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac{\int \frac{-\frac{5274322027}{20736}-\frac{301114735 x}{5184}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{12340512}\\ &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac{27754539-31190998 x}{31986607104 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{559872 (5+2 x)^3}-\frac{89137 \sqrt{3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac{475357 \sqrt{3-x+2 x^2}}{1934917632 (5+2 x)}-\frac{4778789 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{1289945088}\\ &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac{27754539-31190998 x}{31986607104 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{559872 (5+2 x)^3}-\frac{89137 \sqrt{3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac{475357 \sqrt{3-x+2 x^2}}{1934917632 (5+2 x)}+\frac{4778789 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{644972544}\\ &=\frac{369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac{27754539-31190998 x}{31986607104 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{559872 (5+2 x)^3}-\frac{89137 \sqrt{3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac{475357 \sqrt{3-x+2 x^2}}{1934917632 (5+2 x)}+\frac{4778789 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{7739670528 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.232494, size = 89, normalized size = 0.56 \[ \frac{\frac{24 \left (6664404208 x^6+34872810880 x^5+46210466520 x^4+27484986184 x^3-6702882569 x^2+73621973154 x-95241881529\right )}{(2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}}+2527979381 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{8188571418624} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 207, normalized size = 1.3 \begin{align*} -{\frac{-72646615+290586460\,x}{9889579008} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{25951}{110592} \left ( x+{\frac{5}{2}} \right ) ^{-2} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{4778789\,\sqrt{2}}{15479341056}{\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}{13824} \left ( x+{\frac{5}{2}} \right ) ^{-3} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-8183108657+32732434628\,x}{1364761903104}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{4778789}{429981696} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-10+40\,x}{1587}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{-5+20\,x}{552} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{4778789}{2579890176}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{34861}{3981312} \left ( x+{\frac{5}{2}} \right ) ^{-1} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59192, size = 332, normalized size = 2.08 \begin{align*} -\frac{4778789}{15479341056} \, \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{416525263 \, x}{341190475776 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{245375387}{113730158592 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{16932905 \, x}{2472394752 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3667}{1728 \,{\left (8 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 60 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 150 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 125 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{25951}{27648 \,{\left (4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 20 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 25 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{34861}{1990656 \,{\left (2 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 5 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{10570421}{824131584 \,{\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 = 1.39946, size = 582, normalized size = 3.64 \begin{align*} \frac{2527979381 \, \sqrt{2}{\left (32 \, x^{7} + 208 \, x^{6} + 464 \, x^{5} + 632 \, x^{4} + 1162 \, x^{3} + 1265 \, x^{2} + 600 \, x + 1125\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 (6664404208 \, x^{6} + 34872810880 \, x^{5} + 46210466520 \, x^{4} + 27484986184 \, x^{3} - 6702882569 \, x^{2} + 73621973154 \, x - 95241881529\right )} \sqrt{2 \, x^{2} - x + 3}}{16377142837248 \,{\left (32 \, x^{7} + 208 \, x^{6} + 464 \, x^{5} + 632 \, x^{4} + 1162 \, x^{3} + 1265 \, x^{2} + 600 \, x + 1125\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24861, size = 377, normalized size = 2.36 \begin{align*} \frac{4778789}{15479341056} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{4778789}{15479341056} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{{\left ({\left (15595499 \, x - 21675019\right )} x + 27298005\right )} x - 14440149}{7996651776 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{\sqrt{2}{\left (38030012 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 734231900 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 122834956 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 2154595396 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 1659431083 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 760577429\right )}}{3869835264 \,{\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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