Optimal. Leaf size=172 \[ \frac{5}{96} (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}-\frac{839}{960} \left (2 x^2-x+3\right )^{5/2}-\frac{(909513-226052 x) \left (2 x^2-x+3\right )^{3/2}}{18432}-\frac{(85448933-14243732 x) \sqrt{2 x^2-x+3}}{32768}+\frac{959625 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{64 \sqrt{2}}-\frac{982669459 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{65536 \sqrt{2}} \]
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Rubi [A] time = 0.281693, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1650, 1653, 814, 843, 619, 215, 724, 206} \[ \frac{5}{96} (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}-\frac{839}{960} \left (2 x^2-x+3\right )^{5/2}-\frac{(909513-226052 x) \left (2 x^2-x+3\right )^{3/2}}{18432}-\frac{(85448933-14243732 x) \sqrt{2 x^2-x+3}}{32768}+\frac{959625 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{64 \sqrt{2}}-\frac{982669459 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{65536 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 1653
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)^2} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}-\frac{1}{72} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{26675}{16}-4990 x+486 x^2-180 x^3\right )}{5+2 x} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (148350-406320 x+120816 x^2\right )}{5+2 x} \, dx}{6912}\\ &=-\frac{839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac{\int \frac{(8954400-27126240 x) \left (3-x+2 x^2\right )^{3/2}}{5+2 x} \, dx}{276480}\\ &=-\frac{(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac{839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac{\int \frac{(-11522887200+30766461120 x) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{17694720}\\ &=-\frac{(85448933-14243732 x) \sqrt{3-x+2 x^2}}{32768}-\frac{(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac{839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac{\int \frac{8489566411200-16980528251520 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{566231040}\\ &=-\frac{(85448933-14243732 x) \sqrt{3-x+2 x^2}}{32768}-\frac{(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac{839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac{982669459 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{65536}-\frac{2878875}{32} \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{(85448933-14243732 x) \sqrt{3-x+2 x^2}}{32768}-\frac{(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac{839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac{2878875}{16} \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )+\frac{982669459 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{65536 \sqrt{46}}\\ &=-\frac{(85448933-14243732 x) \sqrt{3-x+2 x^2}}{32768}-\frac{(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac{839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac{5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac{982669459 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{65536 \sqrt{2}}+\frac{959625 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{64 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.224764, size = 108, normalized size = 0.63 \[ \frac{\frac{4 \sqrt{2 x^2-x+3} \left (409600 x^6-1798144 x^5+8283904 x^4-35369408 x^3+182033816 x^2-1404323114 x-6814208295\right )}{2 x+5}+14739840000 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-14740041885 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1966080} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 208, normalized size = 1.2 \begin{align*}{\frac{5\,x}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{589}{960} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-9059+36236\,x}{6144} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-208357+833428\,x}{32768}\sqrt{2\,{x}^{2}-x+3}}+{\frac{982669459\,\sqrt{2}}{131072}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{106625}{2304} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{-1637+6548\,x}{16}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{319875}{128}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{959625\,\sqrt{2}}{128}{\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{3667}{1152} \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{-3667+14668\,x}{2304} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58311, size = 217, normalized size = 1.26 \begin{align*} \frac{5}{48} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{589}{960} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{9059}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{185827}{6144} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3560933}{8192} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{982669459}{131072} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{959625}{128} \, \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{85448933}{32768} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46846, size = 504, normalized size = 2.93 \begin{align*} \frac{14740041885 \, \sqrt{2}{\left (2 \, x + 5\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 14739840000 \, \sqrt{2}{\left (2 \, x + 5\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 ) + 8 \,{\left (409600 \, x^{6} - 1798144 \, x^{5} + 8283904 \, x^{4} - 35369408 \, x^{3} + 182033816 \, x^{2} - 1404323114 \, x - 6814208295\right )} \sqrt{2 \, x^{2} - x + 3}}{3932160 \,{\left (2 \, x + 5\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 )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36325, size = 954, normalized size = 5.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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