Optimal. Leaf size=80 \[ \frac {35}{72} (5+6 x) \sqrt {2+5 x+3 x^2}-\frac {1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac {35 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{144 \sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 635,
212} \begin {gather*} -\frac {1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac {35}{72} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {35 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{144 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rubi steps
\begin {align*} \int (5-x) \sqrt {2+5 x+3 x^2} \, dx &=-\frac {1}{9} \left (2+5 x+3 x^2\right )^{3/2}+\frac {35}{6} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {35}{72} (5+6 x) \sqrt {2+5 x+3 x^2}-\frac {1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac {35}{144} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {35}{72} (5+6 x) \sqrt {2+5 x+3 x^2}-\frac {1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac {35}{72} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {35}{72} (5+6 x) \sqrt {2+5 x+3 x^2}-\frac {1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac {35 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{144 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 61, normalized size = 0.76 \begin {gather*} \frac {1}{216} \left (-3 \sqrt {2+5 x+3 x^2} \left (-159-170 x+24 x^2\right )-35 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 64, normalized size = 0.80
method | result | size |
risch | \(-\frac {\left (24 x^{2}-170 x -159\right ) \sqrt {3 x^{2}+5 x +2}}{72}-\frac {35 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{432}\) | \(55\) |
default | \(\frac {35 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{72}-\frac {35 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{432}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{9}\) | \(64\) |
trager | \(\left (-\frac {1}{3} x^{2}+\frac {85}{36} x +\frac {53}{24}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {35 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{432}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 72, normalized size = 0.90 \begin {gather*} -\frac {1}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {35}{12} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {35}{432} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {175}{72} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.22, size = 63, normalized size = 0.79 \begin {gather*} -\frac {1}{72} \, {\left (24 \, x^{2} - 170 \, x - 159\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {35}{864} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end {gather*}
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 \begin {gather*} - \int x \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 5 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 59, normalized size = 0.74 \begin {gather*} -\frac {1}{72} \, {\left (2 \, {\left (12 \, x - 85\right )} x - 159\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {35}{432} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.66, size = 104, normalized size = 1.30 \begin {gather*} 5\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}-\frac {5\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{72}-\frac {\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{216}-\frac {5\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{432} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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