Optimal. Leaf size=166 \[ \frac{5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4-\frac{1121 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^3}{2304}+\frac{69415 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^2}{32256}-\frac{3 (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}}{143360}-\frac{92727 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{131072}-\frac{6398163 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{147157749 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}} \]
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Rubi [A] time = 0.194405, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {1653, 779, 612, 619, 215} \[ \frac{5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4-\frac{1121 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^3}{2304}+\frac{69415 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^2}{32256}-\frac{3 (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}}{143360}-\frac{92727 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{131072}-\frac{6398163 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{147157749 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 779
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{288} \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (-2299-11262 x-15996 x^2-8968 x^3\right ) \, dx\\ &=-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}+\frac{\int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (198968+2253280 x+2221280 x^2\right ) \, dx}{36864}\\ &=\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}+\frac{\int (13363488-55961280 x) (5+2 x) \left (3-x+2 x^2\right )^{3/2} \, dx}{2064384}\\ &=\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{92727 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{8192}\\ &=-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{6398163 \int \sqrt{3-x+2 x^2} \, dx}{262144}\\ &=-\frac{6398163 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{147157749 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4194304}\\ &=-\frac{6398163 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{\left (6398163 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4194304}\\ &=-\frac{6398163 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}-\frac{147157749 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.190871, size = 80, normalized size = 0.48 \[ \frac{4 \sqrt{2 x^2-x+3} \left (1468006400 x^8+2926837760 x^7+1033175040 x^6+12117893120 x^5+379086848 x^4+12669290112 x^3+4870637856 x^2+12357760788 x+1592737263\right )-46354690935 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2642411520} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 134, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{4}}{9} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{479\,{x}^{3}}{288} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{2005\,{x}^{2}}{8064} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{5645\,x}{21504} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-6398163+25592652\,x}{2097152}\sqrt{2\,{x}^{2}-x+3}}+{\frac{147157749\,\sqrt{2}}{8388608}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-92727+370908\,x}{131072} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{120809}{143360} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58366, size = 209, normalized size = 1.26 \begin{align*} \frac{5}{9} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{4} + \frac{479}{288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{2005}{8064} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{5645}{21504} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{120809}{143360} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{92727}{32768} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{92727}{131072} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{6398163}{524288} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{147157749}{8388608} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{6398163}{2097152} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32289, size = 375, normalized size = 2.26 \begin{align*} \frac{1}{660602880} \,{\left (1468006400 \, x^{8} + 2926837760 \, x^{7} + 1033175040 \, x^{6} + 12117893120 \, x^{5} + 379086848 \, x^{4} + 12669290112 \, x^{3} + 4870637856 \, x^{2} + 12357760788 \, x + 1592737263\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{147157749}{16777216} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\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 \left (2 x + 5\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22369, size = 119, normalized size = 0.72 \begin{align*} \frac{1}{660602880} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (8 \,{\left (28 \,{\left (160 \, x + 319\right )} x + 3153\right )} x + 295847\right )} x + 185101\right )} x + 98978829\right )} x + 152207433\right )} x + 3089440197\right )} x + 1592737263\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{147157749}{8388608} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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