Optimal. Leaf size=181 \[ -\frac{32865365 \left (2 x^2-x+3\right )^{5/2}}{17915904 (2 x+5)}+\frac{556255 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^2}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}-\frac{(138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}}{17915904}-\frac{(135068604-22512089 x) \sqrt{2 x^2-x+3}}{331776}+\frac{517762327 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{221184 \sqrt{2}}-\frac{19176431 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}} \]
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Rubi [A] time = 0.267145, antiderivative size = 181, 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, 814, 843, 619, 215, 724, 206} \[ -\frac{32865365 \left (2 x^2-x+3\right )^{5/2}}{17915904 (2 x+5)}+\frac{556255 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^2}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}-\frac{(138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}}{17915904}-\frac{(135068604-22512089 x) \sqrt{2 x^2-x+3}}{331776}+\frac{517762327 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{221184 \sqrt{2}}-\frac{19176431 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 814
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)^4} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}-\frac{1}{216} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{43355}{16}-\frac{11605 x}{2}+1458 x^2-540 x^3\right )}{(5+2 x)^3} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{4202675}{16}-\frac{2477469 x}{4}+38880 x^2\right )}{(5+2 x)^2} \, dx}{31104}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac{\int \frac{\left (\frac{182685181}{16}-34265045 x\right ) \left (3-x+2 x^2\right )^{3/2}}{5+2 x} \, dx}{2239488}\\ &=-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}+\frac{\int \frac{(-14584438152+38900889792 x) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{143327232}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac{\int \frac{10736183791872-21472693553664 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{4586471424}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}+\frac{19176431 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{8192}-\frac{517762327 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{36864}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}+\frac{517762327 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{18432}+\frac{19176431 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{8192 \sqrt{46}}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac{19176431 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}}+\frac{517762327 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{221184 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.220023, size = 108, normalized size = 0.6 \[ \frac{\frac{12 \sqrt{2 x^2-x+3} \left (46080 x^6-315648 x^5+2626848 x^4-33595416 x^3-594798908 x^2-2006873194 x-1994650739\right )}{(2 x+5)^3}+517762327 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-517763637 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{442368} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 221, normalized size = 1.2 \begin{align*}{\frac{-22400309+89601236\,x}{1327104}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{556255}{995328} \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{517762327\,\sqrt{2}}{442368}{\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{-345+1380\,x}{4096}\sqrt{2\,{x}^{2}-x+3}}+{\frac{19176431\,\sqrt{2}}{16384}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-5+20\,x}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{517762327}{71663616} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{517762327}{1327104}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3667}{13824} \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{-32865365+131461460\,x}{71663616} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{32865365}{35831808} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5567, size = 255, normalized size = 1.41 \begin{align*} \frac{5}{64} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{1094743}{497664} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1728 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{556255 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{248832 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{22512089}{331776} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{19176431}{16384} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{517762327}{442368} \, \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{11255717}{27648} \, \sqrt{2 \, x^{2} - x + 3} - \frac{32865365 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{995328 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48414, size = 581, normalized size = 3.21 \begin{align*} \frac{517763637 \, \sqrt{2}{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 517762327 \, \sqrt{2}{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\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 ) + 24 \,{\left (46080 \, x^{6} - 315648 \, x^{5} + 2626848 \, x^{4} - 33595416 \, x^{3} - 594798908 \, x^{2} - 2006873194 \, x - 1994650739\right )} \sqrt{2 \, x^{2} - x + 3}}{884736 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\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 )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24558, size = 424, normalized size = 2.34 \begin{align*} \frac{1}{4096} \,{\left (4 \,{\left (8 \,{\left (20 \, x - 287\right )} x + 23341\right )} x - 1004633\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{19176431}{16384} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{517762327}{442368} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{517762327}{442368} \, \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 (1092794276 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 18284336132 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 20314214356 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 151449344092 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 102529692109 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 41882448755\right )}}{36864 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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