Optimal. Leaf size=105 \[ \frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}+\frac {107}{240} \left (2 x^2-x+3\right )^{5/2}-\frac {179 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{1536}-\frac {4117 (1-4 x) \sqrt {2 x^2-x+3}}{8192}-\frac {94691 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16384 \sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1661, 640, 612, 619, 215} \[ \frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}+\frac {107}{240} \left (2 x^2-x+3\right )^{5/2}-\frac {179 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{1536}-\frac {4117 (1-4 x) \sqrt {2 x^2-x+3}}{8192}-\frac {94691 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16384 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx &=\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac {1}{12} \int \left (9+\frac {107 x}{2}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx\\ &=\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac {179}{96} \int \left (3-x+2 x^2\right )^{3/2} \, dx\\ &=-\frac {179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac {4117 \int \sqrt {3-x+2 x^2} \, dx}{1024}\\ &=-\frac {4117 (1-4 x) \sqrt {3-x+2 x^2}}{8192}-\frac {179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac {94691 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{16384}\\ &=-\frac {4117 (1-4 x) \sqrt {3-x+2 x^2}}{8192}-\frac {179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac {\left (4117 \sqrt {\frac {23}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{16384}\\ &=-\frac {4117 (1-4 x) \sqrt {3-x+2 x^2}}{8192}-\frac {179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}-\frac {94691 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16384 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 65, normalized size = 0.62 \[ \frac {4 \sqrt {2 x^2-x+3} \left (204800 x^5+14336 x^4+561024 x^3+319072 x^2+565276 x+388341\right )-1420365 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{491520} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 78, normalized size = 0.74 \[ \frac {1}{122880} \, {\left (204800 \, x^{5} + 14336 \, x^{4} + 561024 \, x^{3} + 319072 \, x^{2} + 565276 \, x + 388341\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {94691}{65536} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 73, normalized size = 0.70 \[ \frac {1}{122880} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x + 7\right )} x + 4383\right )} x + 9971\right )} x + 141319\right )} x + 388341\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {94691}{32768} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.79 \[ \frac {5 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x}{12}+\frac {94691 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{32768}+\frac {107 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{240}+\frac {179 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1536}+\frac {4117 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{8192} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 104, normalized size = 0.99 \[ \frac {5}{12} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {107}{240} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {179}{384} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {179}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {4117}{2048} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {94691}{32768} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {4117}{8192} \, \sqrt {2 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^2+3\,x+2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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