Optimal. Leaf size=135 \[ -\frac{4679797-2148263 x}{592344576 \sqrt{2 x^2-x+3}}-\frac{45979 \sqrt{2 x^2-x+3}}{26873856 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{373248 (2 x+5)^2}+\frac{65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}+\frac{774079 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{322486272 \sqrt{2}} \]
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Rubi [A] time = 0.220896, antiderivative size = 135, 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 = {1646, 1650, 806, 724, 206} \[ -\frac{4679797-2148263 x}{592344576 \sqrt{2 x^2-x+3}}-\frac{45979 \sqrt{2 x^2-x+3}}{26873856 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{373248 (2 x+5)^2}+\frac{65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}+\frac{774079 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{322486272 \sqrt{2}} \]
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)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{11115283}{746496}+\frac{3198845 x}{62208}+\frac{605005 x^2}{6912}-\frac{8779 x^3}{23328}}{(5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{-\frac{171639869}{2985984}-\frac{142392517 x}{746496}-\frac{16570925 x^2}{746496}}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{\int \frac{\frac{34040621}{10368}+\frac{28209983 x}{10368}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{57132}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{45979 \sqrt{3-x+2 x^2}}{26873856 (5+2 x)}-\frac{774079 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{53747712}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{45979 \sqrt{3-x+2 x^2}}{26873856 (5+2 x)}+\frac{774079 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{26873856}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{45979 \sqrt{3-x+2 x^2}}{26873856 (5+2 x)}+\frac{774079 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{322486272 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.306919, size = 97, normalized size = 0.72 \[ \frac{\frac{12 \sqrt{2} \left (217883368 x^5+107028732 x^4-1503926130 x^3-5919924791 x^2+2280511668 x-8953831359\right )}{529 (2 x+5)^2 \left (2 x^2-x+3\right )^{3/2}}+774079 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )-774079 \log (2 x+5)}{322486272 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 200, normalized size = 1.5 \begin{align*} -{\frac{5}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{-149+596\,x}{1104} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{-149+596\,x}{1587}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{3667}{4608} \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{115369}{165888} \left ( x+{\frac{5}{2}} \right ) ^{-1} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{774079}{17915904} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-57937675+231750700\,x}{412065792} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-5366174813+21464699252\,x}{56865079296}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{774079}{107495424}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{774079\,\sqrt{2}}{644972544}{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61319, size = 240, normalized size = 1.78 \begin{align*} -\frac{774079}{644972544} \, \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{27235421 \, x}{14216269824 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{36393601}{4738756608 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{2323723 \, x}{103016448 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3667}{1152 \,{\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{115369}{82944 \,{\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{5254255}{34338816 \,{\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 = 1.40361, size = 502, normalized size = 3.72 \begin{align*} \frac{409487791 \, \sqrt{2}{\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\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 (217883368 \, x^{5} + 107028732 \, x^{4} - 1503926130 \, x^{3} - 5919924791 \, x^{2} + 2280511668 \, x - 8953831359\right )} \sqrt{2 \, x^{2} - x + 3}}{682380951552 \,{\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\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 )^{3} \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 [B] time = 1.21714, size = 308, normalized size = 2.28 \begin{align*} \frac{774079}{644972544} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{774079}{644972544} \, \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 (44558 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 10136238 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 16812201 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 10182217\right )}}{53747712 \,{\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}} + \frac{{\left ({\left (4296526 \, x - 11507857\right )} x + 10720752\right )} x - 11003805}{592344576 \,{\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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