Optimal. Leaf size=55 \[ \frac {1}{8} (2 x+1) \left (x^2+x+1\right )^{3/2}+\frac {9}{64} (2 x+1) \sqrt {x^2+x+1}+\frac {27}{128} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {612, 619, 215} \[ \frac {1}{8} (2 x+1) \left (x^2+x+1\right )^{3/2}+\frac {9}{64} (2 x+1) \sqrt {x^2+x+1}+\frac {27}{128} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rubi steps
\begin {align*} \int \left (1+x+x^2\right )^{3/2} \, dx &=\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {9}{16} \int \sqrt {1+x+x^2} \, dx\\ &=\frac {9}{64} (1+2 x) \sqrt {1+x+x^2}+\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {27}{128} \int \frac {1}{\sqrt {1+x+x^2}} \, dx\\ &=\frac {9}{64} (1+2 x) \sqrt {1+x+x^2}+\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {1}{128} \left (9 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )\\ &=\frac {9}{64} (1+2 x) \sqrt {1+x+x^2}+\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {27}{128} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.84 \[ \frac {1}{128} \left (2 \sqrt {x^2+x+1} \left (16 x^3+24 x^2+42 x+17\right )+27 \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 52, normalized size = 0.95 \[ \frac {1}{64} \sqrt {x^2+x+1} \left (16 x^3+24 x^2+42 x+17\right )-\frac {27}{128} \log \left (2 \sqrt {x^2+x+1}-2 x-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 44, normalized size = 0.80 \[ \frac {1}{64} \, {\left (16 \, x^{3} + 24 \, x^{2} + 42 \, x + 17\right )} \sqrt {x^{2} + x + 1} - \frac {27}{128} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 44, normalized size = 0.80 \[ \frac {1}{64} \, {\left (2 \, {\left (4 \, {\left (2 \, x + 3\right )} x + 21\right )} x + 17\right )} \sqrt {x^{2} + x + 1} - \frac {27}{128} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 38, normalized size = 0.69
method | result | size |
risch | \(\frac {\left (16 x^{3}+24 x^{2}+42 x +17\right ) \sqrt {x^{2}+x +1}}{64}+\frac {27 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{128}\) | \(38\) |
default | \(\frac {\left (1+2 x \right ) \left (x^{2}+x +1\right )^{\frac {3}{2}}}{8}+\frac {9 \left (1+2 x \right ) \sqrt {x^{2}+x +1}}{64}+\frac {27 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{128}\) | \(43\) |
trager | \(\left (\frac {1}{4} x^{3}+\frac {3}{8} x^{2}+\frac {21}{32} x +\frac {17}{64}\right ) \sqrt {x^{2}+x +1}+\frac {27 \ln \left (1+2 x +2 \sqrt {x^{2}+x +1}\right )}{128}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 56, normalized size = 1.02 \[ \frac {1}{4} \, {\left (x^{2} + x + 1\right )}^{\frac {3}{2}} x + \frac {1}{8} \, {\left (x^{2} + x + 1\right )}^{\frac {3}{2}} + \frac {9}{32} \, \sqrt {x^{2} + x + 1} x + \frac {9}{64} \, \sqrt {x^{2} + x + 1} + \frac {27}{128} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 43, normalized size = 0.78 \[ \frac {27\,\ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right )}{128}+\frac {\left (x+\frac {1}{2}\right )\,{\left (x^2+x+1\right )}^{3/2}}{4}+\frac {9\,\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {x^2+x+1}}{16} \]
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 \[ \int \left (x^{2} + x + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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