Optimal. Leaf size=147 \[ \frac{5}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac{23}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac{125 \left (2 x^2-x+3\right )^{5/2} x}{3584}+\frac{1167 \left (2 x^2-x+3\right )^{5/2}}{14336}-\frac{8597 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{65536}-\frac{593193 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}-\frac{13643439 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}} \]
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Rubi [A] time = 0.121456, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1661, 640, 612, 619, 215} \[ \frac{5}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac{23}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac{125 \left (2 x^2-x+3\right )^{5/2} x}{3584}+\frac{1167 \left (2 x^2-x+3\right )^{5/2}}{14336}-\frac{8597 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{65536}-\frac{593193 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}-\frac{13643439 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{16} \int \left (3-x+2 x^2\right )^{3/2} \left (32+16 x+3 x^2+\frac{23 x^3}{2}\right ) \, dx\\ &=\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{224} \int \left (3-x+2 x^2\right )^{3/2} \left (448+155 x+\frac{375 x^2}{4}\right ) \, dx\\ &=\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (\frac{20379}{4}+\frac{17505 x}{8}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{2688}\\ &=\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{8597 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{4096}\\ &=-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{593193 \int \sqrt{3-x+2 x^2} \, dx}{131072}\\ &=-\frac{593193 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{13643439 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{2097152}\\ &=-\frac{593193 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{\left (593193 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{2097152}\\ &=-\frac{593193 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac{13643439 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.129866, size = 75, normalized size = 0.51 \[ \frac{4 \sqrt{2 x^2-x+3} \left (9175040 x^7-7667712 x^6+29335552 x^5-7497728 x^4+27023744 x^3+3845856 x^2+27845612 x-1663407\right )-95504073 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{29360128} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 117, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{23\,{x}^{2}}{448} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{125\,x}{3584} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-593193+2372772\,x}{1048576}\sqrt{2\,{x}^{2}-x+3}}+{\frac{13643439\,\sqrt{2}}{4194304}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-8597+34388\,x}{65536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1167}{14336} \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 = 1.47795, size = 186, normalized size = 1.27 \begin{align*} \frac{5}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{23}{448} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{125}{3584} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{1167}{14336} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{8597}{16384} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{8597}{65536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{593193}{262144} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{13643439}{4194304} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{593193}{1048576} \, \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 = 1.2855, size = 316, normalized size = 2.15 \begin{align*} \frac{1}{7340032} \,{\left (9175040 \, x^{7} - 7667712 \, x^{6} + 29335552 \, x^{5} - 7497728 \, x^{4} + 27023744 \, x^{3} + 3845856 \, x^{2} + 27845612 \, x - 1663407\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{13643439}{8388608} \, \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^{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 = 1.38808, size = 112, normalized size = 0.76 \begin{align*} \frac{1}{7340032} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (140 \, x - 117\right )} x + 3581\right )} x - 3661\right )} x + 211123\right )} x + 120183\right )} x + 6961403\right )} x - 1663407\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{13643439}{4194304} \, \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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