Optimal. Leaf size=93 \[ \frac {2}{15} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2+\frac {(738 x+745) \left (3 x^2-x+2\right )^{3/2}}{1620}+\frac {19 (1-6 x) \sqrt {3 x^2-x+2}}{2592}+\frac {437 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{5184 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1653, 779, 612, 619, 215} \[ \frac {2}{15} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2+\frac {(738 x+745) \left (3 x^2-x+2\right )^{3/2}}{1620}+\frac {19 (1-6 x) \sqrt {3 x^2-x+2}}{2592}+\frac {437 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{5184 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 779
Rule 1653
Rubi steps
\begin {align*} \int (1+2 x) \sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx &=\frac {2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac {1}{60} \int (1+2 x) (8+164 x) \sqrt {2-x+3 x^2} \, dx\\ &=\frac {2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac {(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}-\frac {19}{216} \int \sqrt {2-x+3 x^2} \, dx\\ &=\frac {19 (1-6 x) \sqrt {2-x+3 x^2}}{2592}+\frac {2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac {(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}-\frac {437 \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx}{5184}\\ &=\frac {19 (1-6 x) \sqrt {2-x+3 x^2}}{2592}+\frac {2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac {(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}-\frac {\left (19 \sqrt {\frac {23}{3}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{5184}\\ &=\frac {19 (1-6 x) \sqrt {2-x+3 x^2}}{2592}+\frac {2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac {(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}+\frac {437 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{5184 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.65 \[ \frac {6 \sqrt {3 x^2-x+2} \left (20736 x^4+31536 x^3+24072 x^2+17374 x+15471\right )-2185 \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{77760} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 73, normalized size = 0.78 \[ \frac {1}{12960} \, {\left (20736 \, x^{4} + 31536 \, x^{3} + 24072 \, x^{2} + 17374 \, x + 15471\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {437}{31104} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 68, normalized size = 0.73 \[ \frac {1}{12960} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (48 \, x + 73\right )} x + 1003\right )} x + 8687\right )} x + 15471\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {437}{15552} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 0.87 \[ \frac {8 \left (3 x^{2}-x +2\right )^{\frac {3}{2}} x^{2}}{15}+\frac {89 \left (3 x^{2}-x +2\right )^{\frac {3}{2}} x}{90}-\frac {437 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{15552}+\frac {961 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{1620}-\frac {19 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{2592} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 92, normalized size = 0.99 \[ \frac {8}{15} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {89}{90} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + \frac {961}{1620} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {19}{432} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {437}{15552} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) + \frac {19}{2592} \, \sqrt {3 \, x^{2} - x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.88, size = 136, normalized size = 1.46 \[ \frac {8\,x^2\,{\left (3\,x^2-x+2\right )}^{3/2}}{15}-\frac {253\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2-x+2}+\frac {\sqrt {3}\,\left (3\,x-\frac {1}{2}\right )}{3}\right )}{810}-\frac {44\,\left (\frac {x}{2}-\frac {1}{12}\right )\,\sqrt {3\,x^2-x+2}}{45}+\frac {961\,\sqrt {3\,x^2-x+2}\,\left (72\,x^2-6\,x+45\right )}{38880}+\frac {89\,x\,{\left (3\,x^2-x+2\right )}^{3/2}}{90}+\frac {22103\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2-x+2}+\frac {\sqrt {3}\,\left (6\,x-1\right )}{3}\right )}{77760} \]
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 \[ \int \left (2 x + 1\right ) \sqrt {3 x^{2} - x + 2} \left (4 x^{2} + 3 x + 1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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