Optimal. Leaf size=194 \[ \frac{246159769 \left (2 x^2-x+3\right )^{3/2}}{866843099136 (2 x+5)^3}+\frac{19414831 \left (2 x^2-x+3\right )^{3/2}}{4013162496 (2 x+5)^4}-\frac{1464037 \left (2 x^2-x+3\right )^{3/2}}{13934592 (2 x+5)^5}+\frac{948341 \left (2 x^2-x+3\right )^{3/2}}{1741824 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}-\frac{12568315 (17-22 x) \sqrt{2 x^2-x+3}}{23776267862016 (2 x+5)^2}-\frac{289071245 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{285315214344192 \sqrt{2}} \]
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Rubi [A] time = 0.267503, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1650, 834, 806, 720, 724, 206} \[ \frac{246159769 \left (2 x^2-x+3\right )^{3/2}}{866843099136 (2 x+5)^3}+\frac{19414831 \left (2 x^2-x+3\right )^{3/2}}{4013162496 (2 x+5)^4}-\frac{1464037 \left (2 x^2-x+3\right )^{3/2}}{13934592 (2 x+5)^5}+\frac{948341 \left (2 x^2-x+3\right )^{3/2}}{1741824 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}-\frac{12568315 (17-22 x) \sqrt{2 x^2-x+3}}{23776267862016 (2 x+5)^2}-\frac{289071245 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{285315214344192 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 834
Rule 806
Rule 720
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)^8} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}-\frac{1}{504} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{69381}{16}-5594 x+3402 x^2-1260 x^3\right )}{(5+2 x)^7} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{10506615}{16}-\frac{2815905 x}{4}+272160 x^2\right )}{(5+2 x)^6} \, dx}{217728}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}-\frac{\int \frac{\left (\frac{231748695}{16}-\frac{32095935 x}{2}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^5} \, dx}{78382080}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{\int \frac{\left (-\frac{2340515655}{16}+\frac{873667395 x}{4}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^4} \, dx}{22574039040}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}+\frac{12568315 \int \frac{\sqrt{3-x+2 x^2}}{(5+2 x)^3} \, dx}{82556485632}\\ &=-\frac{12568315 (17-22 x) \sqrt{3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}+\frac{289071245 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{47552535724032}\\ &=-\frac{12568315 (17-22 x) \sqrt{3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}-\frac{289071245 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{23776267862016}\\ &=-\frac{12568315 (17-22 x) \sqrt{3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}-\frac{289071245 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{285315214344192 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.206567, size = 96, normalized size = 0.49 \[ \frac{24 \sqrt{2 x^2-x+3} \left (1574342277056 x^6+27976951397184 x^5+4982916071952 x^4+41058010262368 x^3+14716683780036 x^2+590492177460 x-20465234808721\right )-2023498715 \sqrt{2} (2 x+5)^7 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{3994413000818688 (2 x+5)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 216, normalized size = 1.1 \begin{align*}{\frac{948341}{111476736} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}-{\frac{138251465}{855945643032576} \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{-138251465+553005860\,x}{1711891286065152}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{12568315}{23776267862016} \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{289071245\,\sqrt{2}}{570630428688384}{\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{289071245}{1711891286065152}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{19414831}{64210599936} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}}+{\frac{246159769}{6934744793088} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{3667}{516096} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-7}}-{\frac{1464037}{445906944} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54707, size = 406, normalized size = 2.09 \begin{align*} \frac{289071245}{570630428688384} \, \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{12568315}{11888133931008} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4032 \,{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} + \frac{948341 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1741824 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac{1464037 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{13934592 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac{19414831 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4013162496 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{246159769 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{866843099136 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{12568315 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{5944066965504 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{138251465 \, \sqrt{2 \, x^{2} - x + 3}}{23776267862016 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43963, size = 649, normalized size = 3.35 \begin{align*} \frac{2023498715 \, \sqrt{2}{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\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 (1574342277056 \, x^{6} + 27976951397184 \, x^{5} + 4982916071952 \, x^{4} + 41058010262368 \, x^{3} + 14716683780036 \, x^{2} + 590492177460 \, x - 20465234808721\right )} \sqrt{2 \, x^{2} - x + 3}}{7988826001637376 \,{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\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 )^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34148, size = 616, normalized size = 3.18 \begin{align*} -\frac{289071245}{570630428688384} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{289071245}{570630428688384} \, \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 (129503917760 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{13} - 3320259746027840 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{12} - 23966708071916736 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} - 186055342532355520 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} - 274256644494948976 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 796135370176031760 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 2531523139171005408 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} - 4610393811900786336 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 7997126854300052364 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 30842713619423538868 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 21873571601855032556 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 16204706960604668100 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 3196254593191113265 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 536799032216117911\right )}}{332867750068224 \,{\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 )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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