Optimal. Leaf size=138 \[ \frac {1}{624} (1858-771 x) \sqrt {2-x+3 x^2}+\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}+\frac {1519 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{192 \sqrt {3}}-\frac {1153 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{64 \sqrt {13}} \]
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Rubi [A]
time = 0.08, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1664, 826, 828,
857, 633, 221, 738, 212} \begin {gather*} -\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{312 (2 x+1)}+\frac {1}{624} (1858-771 x) \sqrt {3 x^2-x+2}-\frac {1153 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{64 \sqrt {13}}+\frac {1519 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{192 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 633
Rule 738
Rule 826
Rule 828
Rule 857
Rule 1664
Rubi steps
\begin {align*} \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx &=-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}-\frac {1}{26} \int \frac {\left (-\frac {31}{2}-61 x\right ) \left (2-x+3 x^2\right )^{3/2}}{(1+2 x)^2} \, dx\\ &=\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}+\frac {1}{208} \int \frac {(639-1028 x) \sqrt {2-x+3 x^2}}{1+2 x} \, dx\\ &=\frac {1}{624} (1858-771 x) \sqrt {2-x+3 x^2}+\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}-\frac {\int \frac {-100880+157976 x}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx}{9984}\\ &=\frac {1}{624} (1858-771 x) \sqrt {2-x+3 x^2}+\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}-\frac {1519}{192} \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx+\frac {1153}{64} \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx\\ &=\frac {1}{624} (1858-771 x) \sqrt {2-x+3 x^2}+\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}-\frac {1153}{32} \text {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )-\frac {1519 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{192 \sqrt {69}}\\ &=\frac {1}{624} (1858-771 x) \sqrt {2-x+3 x^2}+\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}+\frac {1519 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{192 \sqrt {3}}-\frac {1153 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{64 \sqrt {13}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 121, normalized size = 0.88 \begin {gather*} \frac {\frac {156 \sqrt {2-x+3 x^2} \left (182+627 x+390 x^2-68 x^3+96 x^4\right )}{(1+2 x)^2}+20754 \sqrt {13} \tanh ^{-1}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )+19747 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{7488} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 162, normalized size = 1.17
method | result | size |
risch | \(\frac {288 x^{6}-300 x^{5}+1430 x^{4}+1355 x^{3}+699 x^{2}+1072 x +364}{48 \left (2 x +1\right )^{2} \sqrt {3 x^{2}-x +2}}-\frac {1519 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{576}-\frac {1153 \sqrt {13}\, \arctanh \left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{832}\) | \(97\) |
trager | \(\frac {\left (96 x^{4}-68 x^{3}+390 x^{2}+627 x +182\right ) \sqrt {3 x^{2}-x +2}}{48 \left (2 x +1\right )^{2}}+\frac {1153 \RootOf \left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \RootOf \left (\textit {\_Z}^{2}-13\right )}{2 x +1}\right )}{832}-\frac {1519 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}-\RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{576}\) | \(133\) |
default | \(\frac {1153 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{4056}-\frac {257 \left (6 x -1\right ) \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}{1248}-\frac {1519 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{576}+\frac {1153 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}{832}-\frac {1153 \sqrt {13}\, \arctanh \left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{832}+\frac {15 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {5}{2}}}{338 \left (x +\frac {1}{2}\right )}-\frac {15 \left (6 x -1\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{676}-\frac {\left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {5}{2}}}{104 \left (x +\frac {1}{2}\right )^{2}}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 143, normalized size = 1.04 \begin {gather*} \frac {61}{312} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {{\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}}}{26 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} - \frac {257}{208} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {1519}{576} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1153}{832} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {929}{312} \, \sqrt {3 \, x^{2} - x + 2} + \frac {15 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}}{52 \, {\left (2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 159, normalized size = 1.15 \begin {gather*} \frac {19747 \, \sqrt {3} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 10377 \, \sqrt {13} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 312 \, {\left (96 \, x^{4} - 68 \, x^{3} + 390 \, x^{2} + 627 \, x + 182\right )} \sqrt {3 \, x^{2} - x + 2}}{14976 \, {\left (4 \, x^{2} + 4 \, x + 1\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 \frac {\left (3 x^{2} - x + 2\right )^{\frac {3}{2}} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (2 x + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (3\,x^2-x+2\right )}^{3/2}\,\left (4\,x^2+3\,x+1\right )}{{\left (2\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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