Optimal. Leaf size=169 \[ \frac{87677717 \left (2 x^2-x+3\right )^{3/2}}{8599633920 (2 x+5)^3}-\frac{5703277 \left (2 x^2-x+3\right )^{3/2}}{39813120 (2 x+5)^4}+\frac{92239 \left (2 x^2-x+3\right )^{3/2}}{138240 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{3456 (2 x+5)^6}-\frac{1172725 (17-22 x) \sqrt{2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac{26972675 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{3962711310336 \sqrt{2}} \]
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Rubi [A] time = 0.218244, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1650, 806, 720, 724, 206} \[ \frac{87677717 \left (2 x^2-x+3\right )^{3/2}}{8599633920 (2 x+5)^3}-\frac{5703277 \left (2 x^2-x+3\right )^{3/2}}{39813120 (2 x+5)^4}+\frac{92239 \left (2 x^2-x+3\right )^{3/2}}{138240 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{3456 (2 x+5)^6}-\frac{1172725 (17-22 x) \sqrt{2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac{26972675 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{3962711310336 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
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)^7} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}-\frac{1}{432} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{61041}{16}-\frac{20751 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{8057313}{16}-\frac{1191609 x}{2}+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}-\frac{\int \frac{\left (\frac{182650383}{16}-\frac{60644907 x}{4}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac{1172725 \int \frac{\sqrt{3-x+2 x^2}}{(5+2 x)^3} \, dx}{1146617856}\\ &=-\frac{1172725 (17-22 x) \sqrt{3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac{26972675 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{660451885056}\\ &=-\frac{1172725 (17-22 x) \sqrt{3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac{26972675 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{330225942528}\\ &=-\frac{1172725 (17-22 x) \sqrt{3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac{92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac{5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac{87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac{26972675 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{3962711310336 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.172825, size = 91, normalized size = 0.54 \[ \frac{24 \sqrt{2 x^2-x+3} \left (271409942624 x^5+12256250416 x^4+397498825328 x^3+158340720344 x^2+27245373694 x-219337079305\right )-134863375 \sqrt{2} (2 x+5)^6 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{39627113103360 (2 x+5)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 195, normalized size = 1.2 \begin{align*} -{\frac{3667}{221184} \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{12899975}{11888133931008} \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{-12899975+51599900\,x}{23776267862016}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{1172725}{330225942528} \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{26972675\,\sqrt{2}}{7925422620672}{\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{26972675}{23776267862016}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{5703277}{637009920} \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{87677717}{68797071360} \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{92239}{4423680} \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 = 1.60522, size = 338, normalized size = 2. \begin{align*} \frac{26972675}{7925422620672} \, \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{1172725}{165112971264} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3456 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac{92239 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{138240 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac{5703277 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{39813120 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{87677717 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8599633920 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{1172725 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{82556485632 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{12899975 \, \sqrt{2 \, x^{2} - x + 3}}{330225942528 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35957, size = 555, normalized size = 3.28 \begin{align*} \frac{134863375 \, \sqrt{2}{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\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 (271409942624 \, x^{5} + 12256250416 \, x^{4} + 397498825328 \, x^{3} + 158340720344 \, x^{2} + 27245373694 \, x - 219337079305\right )} \sqrt{2 \, x^{2} - x + 3}}{79254226206720 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\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 )^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30763, size = 547, normalized size = 3.24 \begin{align*} -\frac{26972675}{7925422620672} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{26972675}{7925422620672} \, \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 (16506981498400 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} + 389429252643040 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} + 2263923918689840 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 11663651054548560 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 902212326134736 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} - 84192729519861840 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 4317200555009448 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 351543414066518760 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 376787166452923830 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 356306707647610982 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 82348353128195465 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 15499394004553969\right )}}{3302259425280 \,{\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 )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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