Optimal. Leaf size=165 \[ -\frac{9363383 \left (2 x^2-x+3\right )^{3/2}}{23887872 (2 x+5)^2}+\frac{593771 \left (2 x^2-x+3\right )^{3/2}}{497664 (2 x+5)^3}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{2304 (2 x+5)^4}+\frac{7 (9616196 x+52836655) \sqrt{2 x^2-x+3}}{95551488 (2 x+5)}-\frac{4640586097 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1146617856 \sqrt{2}}+\frac{259 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}} \]
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Rubi [A] time = 0.234099, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1650, 812, 843, 619, 215, 724, 206} \[ -\frac{9363383 \left (2 x^2-x+3\right )^{3/2}}{23887872 (2 x+5)^2}+\frac{593771 \left (2 x^2-x+3\right )^{3/2}}{497664 (2 x+5)^3}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{2304 (2 x+5)^4}+\frac{7 (9616196 x+52836655) \sqrt{2 x^2-x+3}}{95551488 (2 x+5)}-\frac{4640586097 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1146617856 \sqrt{2}}+\frac{259 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 812
Rule 843
Rule 619
Rule 215
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)^5} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}-\frac{1}{288} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{44361}{16}-\frac{17501 x}{4}+1944 x^2-720 x^3\right )}{(5+2 x)^4} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{4140069}{16}-404352 x+77760 x^2\right )}{(5+2 x)^3} \, dx}{62208}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}-\frac{\int \frac{\left (\frac{99869175}{16}-\frac{50485029 x}{4}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^2} \, dx}{8957952}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}+\frac{\int \frac{\frac{2321210451}{8}-580027392 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{71663616}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}-\frac{259}{64} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{4640586097 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{191102976}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}-\frac{4640586097 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{95551488}-\frac{259 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{64 \sqrt{46}}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}+\frac{259 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}-\frac{4640586097 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1146617856 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.178584, size = 98, normalized size = 0.59 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (238878720 x^4+6105343976 x^3+31323229164 x^2+62847867486 x+44676885233\right )}{(2 x+5)^4}-4640586097 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+4640219136 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2293235712} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 167, normalized size = 1. \begin{align*}{\frac{201573155}{3439853568} \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{-201573155+806292620\,x}{6879707136}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{9363383}{95551488} \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{4640586097\,\sqrt{2}}{2293235712}{\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{259\,\sqrt{2}}{128}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{4640586097}{6879707136}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3667}{36864} \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{593771}{3981312} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59831, size = 244, normalized size = 1.48 \begin{align*} -\frac{259}{128} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{4640586097}{2293235712} \, \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{16828343}{47775744} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2304 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{593771 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{497664 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{9363383 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{23887872 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{201573155 \, \sqrt{2 \, x^{2} - x + 3}}{95551488 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4576, size = 618, normalized size = 3.75 \begin{align*} \frac{4640219136 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 4640586097 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 (238878720 \, x^{4} + 6105343976 \, x^{3} + 31323229164 \, x^{2} + 62847867486 \, x + 44676885233\right )} \sqrt{2 \, x^{2} - x + 3}}{4586471424 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 )^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33849, size = 441, normalized size = 2.67 \begin{align*} -\frac{1}{2293235712} \, \sqrt{2}{\left (4640586097 \, \log \left (12 \, \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{72}{2 \, x + 5} - 11\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) + 4640219136 \, \log \left ({\left | \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5} + 1 \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) - 4640219136 \, \log \left ({\left | \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5} - 1 \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) + 12 \,{\left (\frac{24 \,{\left (\frac{144 \,{\left (\frac{792072 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )}{2 \, x + 5} - 835793 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )}}{2 \, x + 5} + 57384361 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )}}{2 \, x + 5} - 464569597 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )} \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{179159040 \,{\left (11 \,{\left (\sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5}\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) - 12 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )}}{{\left (\sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5}\right )}^{2} - 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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