Optimal. Leaf size=120 \[ \frac {2}{15} \sqrt {3 x^2-x+2} (2 x+1)^4+\frac {19}{60} \sqrt {3 x^2-x+2} (2 x+1)^3+\frac {44}{135} \sqrt {3 x^2-x+2} (2 x+1)^2-\frac {(6298 x+24897) \sqrt {3 x^2-x+2}}{3240}+\frac {9211 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{1296 \sqrt {3}} \]
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Rubi [A] time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1653, 832, 779, 619, 215} \[ \frac {2}{15} \sqrt {3 x^2-x+2} (2 x+1)^4+\frac {19}{60} \sqrt {3 x^2-x+2} (2 x+1)^3+\frac {44}{135} \sqrt {3 x^2-x+2} (2 x+1)^2-\frac {(6298 x+24897) \sqrt {3 x^2-x+2}}{3240}+\frac {9211 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{1296 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rule 779
Rule 832
Rule 1653
Rubi steps
\begin {align*} \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2-x+3 x^2}} \, dx &=\frac {2}{15} (1+2 x)^4 \sqrt {2-x+3 x^2}+\frac {1}{60} \int \frac {(1+2 x)^3 (-64+228 x)}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {19}{60} (1+2 x)^3 \sqrt {2-x+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2-x+3 x^2}+\frac {1}{720} \int \frac {(1+2 x)^2 (-3390+2112 x)}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {44}{135} (1+2 x)^2 \sqrt {2-x+3 x^2}+\frac {19}{60} (1+2 x)^3 \sqrt {2-x+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2-x+3 x^2}+\frac {\int \frac {(-46350-37788 x) (1+2 x)}{\sqrt {2-x+3 x^2}} \, dx}{6480}\\ &=\frac {44}{135} (1+2 x)^2 \sqrt {2-x+3 x^2}+\frac {19}{60} (1+2 x)^3 \sqrt {2-x+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2-x+3 x^2}-\frac {(24897+6298 x) \sqrt {2-x+3 x^2}}{3240}-\frac {9211 \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx}{1296}\\ &=\frac {44}{135} (1+2 x)^2 \sqrt {2-x+3 x^2}+\frac {19}{60} (1+2 x)^3 \sqrt {2-x+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2-x+3 x^2}-\frac {(24897+6298 x) \sqrt {2-x+3 x^2}}{3240}-\frac {9211 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{1296 \sqrt {69}}\\ &=\frac {44}{135} (1+2 x)^2 \sqrt {2-x+3 x^2}+\frac {19}{60} (1+2 x)^3 \sqrt {2-x+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2-x+3 x^2}-\frac {(24897+6298 x) \sqrt {2-x+3 x^2}}{3240}+\frac {9211 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{1296 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.50 \[ \frac {6 \sqrt {3 x^2-x+2} \left (6912 x^4+22032 x^3+26904 x^2+7538 x-22383\right )-46055 \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{19440} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 73, normalized size = 0.61 \[ \frac {1}{3240} \, {\left (6912 \, x^{4} + 22032 \, x^{3} + 26904 \, x^{2} + 7538 \, x - 22383\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {9211}{7776} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 68, normalized size = 0.57 \[ \frac {1}{3240} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (16 \, x + 51\right )} x + 1121\right )} x + 3769\right )} x - 22383\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {9211}{3888} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 96, normalized size = 0.80 \[ \frac {32 \sqrt {3 x^{2}-x +2}\, x^{4}}{15}+\frac {34 \sqrt {3 x^{2}-x +2}\, x^{3}}{5}+\frac {1121 \sqrt {3 x^{2}-x +2}\, x^{2}}{135}+\frac {3769 \sqrt {3 x^{2}-x +2}\, x}{1620}-\frac {9211 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{3888}-\frac {829 \sqrt {3 x^{2}-x +2}}{120} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 97, normalized size = 0.81 \[ \frac {32}{15} \, \sqrt {3 \, x^{2} - x + 2} x^{4} + \frac {34}{5} \, \sqrt {3 \, x^{2} - x + 2} x^{3} + \frac {1121}{135} \, \sqrt {3 \, x^{2} - x + 2} x^{2} + \frac {3769}{1620} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {9211}{3888} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {829}{120} \, \sqrt {3 \, x^{2} - x + 2} \]
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 \[ \int \frac {{\left (2\,x+1\right )}^3\,\left (4\,x^2+3\,x+1\right )}{\sqrt {3\,x^2-x+2}} \,d x \]
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 \[ \int \frac {\left (2 x + 1\right )^{3} \left (4 x^{2} + 3 x + 1\right )}{\sqrt {3 x^{2} - x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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