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 = 0.22, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 206
Rule 724
Rule 806
Rule 1646
Rule 1650
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*}
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Mathematica [A] time = 0.31, size = 97, normalized size = 0.72 \[ \frac {774079 \log \left (12 \sqrt {4 x^2-2 x+6}-22 x+17\right )+\frac {12 \sqrt {2} \left (217883368 x^5+107028732 x^4-1503926130 x^3-5919924791 x^2+2280511668 x-8953831359\right )}{529 (2 x+5)^2 \left (2 x^2-x+3\right )^{3/2}}-774079 \log (2 x+5)}{322486272 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 155, normalized size = 1.15 \[ \frac {409487791 \, \sqrt {2} {\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (217883368 \, x^{5} + 107028732 \, x^{4} - 1503926130 \, x^{3} - 5919924791 \, x^{2} + 2280511668 \, x - 8953831359\right )} \sqrt {2 \, x^{2} - x + 3}}{682380951552 \, {\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 228, normalized size = 1.69 \[ \frac {774079}{644972544} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {774079}{644972544} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {\sqrt {2} {\left (44558 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 10136238 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 16812201 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 10182217\right )}}{53747712 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} + \frac {{\left ({\left (4296526 \, x - 11507857\right )} x + 10720752\right )} x - 11003805}{592344576 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 200, normalized size = 1.48 \[ \frac {774079 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{644972544}-\frac {5}{48 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {149 \left (4 x -1\right )}{1104 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {149 \left (4 x -1\right )}{1587 \sqrt {2 x^{2}-x +3}}+\frac {115369}{165888 \left (x +\frac {5}{2}\right ) \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {774079}{17915904 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {57937675 x}{103016448}-\frac {57937675}{412065792}}{\left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {5366174813 x}{14216269824}-\frac {5366174813}{56865079296}}{\sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}-\frac {774079}{107495424 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}-\frac {3667}{4608 \left (x +\frac {5}{2}\right )^{2} \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 178, normalized size = 1.32 \[ -\frac {774079}{644972544} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {27235421 \, x}{14216269824 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {36393601}{4738756608 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {2323723 \, x}{103016448 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3667}{1152 \, {\left (4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 20 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 25 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {115369}{82944 \, {\left (2 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 5 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {5254255}{34338816 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {5\,x^4-x^3+3\,x^2+x+2}{{\left (2\,x+5\right )}^3\,{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{3} \left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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