Optimal. Leaf size=195 \[ -\frac{1930441 \left (2 x^2-x+3\right )^{5/2}}{13934592 (2 x+5)^5}+\frac{114335 \left (2 x^2-x+3\right )^{5/2}}{193536 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}-\frac{(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{2293235712 (2 x+5)^4}-\frac{(101679102454 x+146583836191) \sqrt{2 x^2-x+3}}{440301256704 (2 x+5)^2}+\frac{412760561351 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5283615080448 \sqrt{2}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}} \]
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Rubi [A] time = 0.261981, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1650, 810, 843, 619, 215, 724, 206} \[ -\frac{1930441 \left (2 x^2-x+3\right )^{5/2}}{13934592 (2 x+5)^5}+\frac{114335 \left (2 x^2-x+3\right )^{5/2}}{193536 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{4032 (2 x+5)^7}-\frac{(411822458 x+463558457) \left (2 x^2-x+3\right )^{3/2}}{2293235712 (2 x+5)^4}-\frac{(101679102454 x+146583836191) \sqrt{2 x^2-x+3}}{440301256704 (2 x+5)^2}+\frac{412760561351 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5283615080448 \sqrt{2}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 810
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (3-x+2 x^2\right )^{3/2} \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 )^{5/2}}{4032 (5+2 x)^7}-\frac{1}{504} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{76715}{16}-\frac{14855 x}{2}+3402 x^2-1260 x^3\right )}{(5+2 x)^7} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{13334715}{16}-\frac{4631913 x}{4}+272160 x^2\right )}{(5+2 x)^6} \, dx}{217728}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac{\int \frac{\left (\frac{516687885}{16}-48988800 x\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^5} \, dx}{78382080}\\ &=-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}+\frac{\int \frac{\left (-\frac{283730747265}{8}+56435097600 x\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^3} \, dx}{180592312320}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac{\int \frac{\frac{64992568300695}{4}-32506616217600 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{208042343792640}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}+\frac{5}{64} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx-\frac{412760561351 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{880602513408}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}+\frac{412760561351 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{440301256704}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{64 \sqrt{46}}\\ &=-\frac{(146583836191+101679102454 x) \sqrt{3-x+2 x^2}}{440301256704 (5+2 x)^2}-\frac{(463558457+411822458 x) \left (3-x+2 x^2\right )^{3/2}}{2293235712 (5+2 x)^4}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{4032 (5+2 x)^7}+\frac{114335 \left (3-x+2 x^2\right )^{5/2}}{193536 (5+2 x)^6}-\frac{1930441 \left (3-x+2 x^2\right )^{5/2}}{13934592 (5+2 x)^5}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}+\frac{412760561351 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{5283615080448 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.272285, size = 108, normalized size = 0.55 \[ \frac{-\frac{24 \sqrt{2 x^2-x+3} \left (38463671680832 x^6+402255822731712 x^5+2069947287085104 x^4+5966329646300704 x^3+9976065367498188 x^2+9065154700300572 x+3479517268702637\right )}{(2 x+5)^7}+2889323929457 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-2889476997120 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{73970611126272} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 267, normalized size = 1.4 \begin{align*} -{\frac{1930441}{445906944} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}}+{\frac{-17957520133+71830080532\,x}{31701690482688}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{769352975}{23776267862016} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{412760561351\,\sqrt{2}}{10567230160896}{\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{114335}{12386304} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}+{\frac{5\,\sqrt{2}}{128}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{412760561351}{1711891286065152} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{412760561351}{31701690482688}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{32967491}{330225942528} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{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{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-7}}+{\frac{-27452157541+109808630164\,x}{1711891286065152} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{27452157541}{855945643032576} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{7861079}{9172942848} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61073, size = 470, normalized size = 2.41 \begin{align*} -\frac{769352975}{11888133931008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{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{114335 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{193536 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac{1930441 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{13934592 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac{7861079 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{573308928 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac{32967491 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{41278242816 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{769352975 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5944066965504 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{17957520133}{7925422620672} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{5}{128} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{412760561351}{10567230160896} \, \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{35893173457}{2641807540224} \, \sqrt{2 \, x^{2} - x + 3} - \frac{27452157541 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{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 = 1.53481, size = 915, normalized size = 4.69 \begin{align*} \frac{2889476997120 \, \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 (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 2889323929457 \, \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 (38463671680832 \, x^{6} + 402255822731712 \, x^{5} + 2069947287085104 \, x^{4} + 5966329646300704 \, x^{3} + 9976065367498188 \, x^{2} + 9065154700300572 \, x + 3479517268702637\right )} \sqrt{2 \, x^{2} - x + 3}}{147941222252544 \,{\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{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \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 = 1.26089, size = 660, normalized size = 3.38 \begin{align*} -\frac{5}{128} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{412760561351}{10567230160896} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{412760561351}{10567230160896} \, \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 (1121897398412224 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{13} + 48260296303776704 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{12} + 444673458321712704 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} + 3996455936659982656 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} + 6725227967167489360 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} - 17132661028483948080 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} - 63713012094737246112 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} + 106515880136064432096 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} + 226947197958946260516 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} - 856601202771483308188 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 617998258357377713732 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 467121785339763351756 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 92292080735560562227 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 15161716093827501349\right )}}{6164217593856 \,{\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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