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 = 0.27, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 206
Rule 720
Rule 724
Rule 806
Rule 834
Rule 1650
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*}
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Mathematica [A] time = 0.20, 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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fricas [A] time = 0.71, size = 171, normalized size = 0.88 \[ \frac {2023498715 \, \sqrt {2} {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (1574342277056 \, x^{6} + 27976951397184 \, x^{5} + 4982916071952 \, x^{4} + 41058010262368 \, x^{3} + 14716683780036 \, x^{2} + 590492177460 \, x - 20465234808721\right )} \sqrt {2 \, x^{2} - x + 3}}{7988826001637376 \, {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 456, normalized size = 2.35 \[ -\frac {289071245}{570630428688384} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {289071245}{570630428688384} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (129503917760 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{13} - 3320259746027840 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{12} - 23966708071916736 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} - 186055342532355520 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} - 274256644494948976 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 796135370176031760 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 2531523139171005408 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} - 4610393811900786336 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 7997126854300052364 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 30842713619423538868 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 21873571601855032556 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 16204706960604668100 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 3196254593191113265 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 536799032216117911\right )}}{332867750068224 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 216, normalized size = 1.11 \[ -\frac {289071245 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{570630428688384}+\frac {289071245 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{1711891286065152}-\frac {3667 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{516096 \left (x +\frac {5}{2}\right )^{7}}+\frac {948341 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{111476736 \left (x +\frac {5}{2}\right )^{6}}-\frac {1464037 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{445906944 \left (x +\frac {5}{2}\right )^{5}}+\frac {246159769 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{6934744793088 \left (x +\frac {5}{2}\right )^{3}}+\frac {19414831 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{64210599936 \left (x +\frac {5}{2}\right )^{4}}-\frac {12568315 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{23776267862016 \left (x +\frac {5}{2}\right )^{2}}+\frac {138251465 \left (4 x -1\right ) \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{1711891286065152}-\frac {138251465 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{855945643032576 \left (x +\frac {5}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 301, normalized size = 1.55 \[ \frac {289071245}{570630428688384} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {12568315}{11888133931008} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4032 \, {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} + \frac {948341 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1741824 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac {1464037 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{13934592 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac {19414831 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4013162496 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {246159769 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{866843099136 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {12568315 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5944066965504 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {138251465 \, \sqrt {2 \, x^{2} - x + 3}}{23776267862016 \, {\left (2 \, x + 5\right )}} \]
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 \[ \int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^8} \,d x \]
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 \[ \int \frac {\sqrt {2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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