Optimal. Leaf size=82 \[ \frac {5}{8} x \left (2 x^2-x+3\right )^{3/2}+\frac {73}{96} \left (2 x^2-x+3\right )^{3/2}-\frac {81}{512} (1-4 x) \sqrt {2 x^2-x+3}-\frac {1863 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1024 \sqrt {2}} \]
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Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {5}{8} x \left (2 x^2-x+3\right )^{3/2}+\frac {73}{96} \left (2 x^2-x+3\right )^{3/2}-\frac {81}{512} (1-4 x) \sqrt {2 x^2-x+3}-\frac {1863 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1024 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right ) \, dx &=\frac {5}{8} x \left (3-x+2 x^2\right )^{3/2}+\frac {1}{8} \int \left (1+\frac {73 x}{2}\right ) \sqrt {3-x+2 x^2} \, dx\\ &=\frac {73}{96} \left (3-x+2 x^2\right )^{3/2}+\frac {5}{8} x \left (3-x+2 x^2\right )^{3/2}+\frac {81}{64} \int \sqrt {3-x+2 x^2} \, dx\\ &=-\frac {81}{512} (1-4 x) \sqrt {3-x+2 x^2}+\frac {73}{96} \left (3-x+2 x^2\right )^{3/2}+\frac {5}{8} x \left (3-x+2 x^2\right )^{3/2}+\frac {1863 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{1024}\\ &=-\frac {81}{512} (1-4 x) \sqrt {3-x+2 x^2}+\frac {73}{96} \left (3-x+2 x^2\right )^{3/2}+\frac {5}{8} x \left (3-x+2 x^2\right )^{3/2}+\frac {\left (81 \sqrt {\frac {23}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{1024}\\ &=-\frac {81}{512} (1-4 x) \sqrt {3-x+2 x^2}+\frac {73}{96} \left (3-x+2 x^2\right )^{3/2}+\frac {5}{8} x \left (3-x+2 x^2\right )^{3/2}-\frac {1863 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1024 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.67 \[ \frac {4 \sqrt {2 x^2-x+3} \left (1920 x^3+1376 x^2+2684 x+3261\right )-5589 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{6144} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 68, normalized size = 0.83 \[ \frac {1}{1536} \, {\left (1920 \, x^{3} + 1376 \, x^{2} + 2684 \, x + 3261\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1863}{4096} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 63, normalized size = 0.77 \[ \frac {1}{1536} \, {\left (4 \, {\left (8 \, {\left (60 \, x + 43\right )} x + 671\right )} x + 3261\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {1863}{2048} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 0.78 \[ \frac {5 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x}{8}+\frac {1863 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2048}+\frac {73 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{96}+\frac {81 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{512} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 75, normalized size = 0.91 \[ \frac {5}{8} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {73}{96} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {81}{128} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {1863}{2048} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {81}{512} \, \sqrt {2 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.84, size = 119, normalized size = 1.45 \[ \frac {23\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{256}+\frac {\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{8}+\frac {73\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{1536}+\frac {5\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{8}+\frac {1679\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{2048} \]
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 \[ \int \sqrt {2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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