Optimal. Leaf size=103 \[ \frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {353 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{81 \sqrt {3}} \]
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Rubi [A]
time = 0.08, antiderivative size = 103, 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 = {1674, 1675,
654, 633, 221} \begin {gather*} \frac {32}{27} \sqrt {3 x^2-x+2} x^2+\frac {412}{81} \sqrt {3 x^2-x+2} x+\frac {746}{81} \sqrt {3 x^2-x+2}+\frac {2 (12839-3871 x)}{1863 \sqrt {3 x^2-x+2}}+\frac {353 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{81 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
Rule 1674
Rule 1675
Rubi steps
\begin {align*} \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {2}{23} \int \frac {\frac {1127}{81}+\frac {7682 x}{27}+\frac {2852 x^2}{9}+\frac {368 x^3}{3}}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {2}{207} \int \frac {\frac {1127}{9}+2070 x+\frac {9476 x^2}{3}}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {1}{621} \int \frac {-5566+17158 x}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}-\frac {353}{81} \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}-\frac {353 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{81 \sqrt {69}}\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {353 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{81 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 70, normalized size = 0.68 \begin {gather*} \frac {\frac {6 \left (29997-2974 x+23207 x^2+13110 x^3+3312 x^4\right )}{\sqrt {2-x+3 x^2}}+8119 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{5589} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 115, normalized size = 1.12
method | result | size |
risch | \(\frac {\frac {32}{9} x^{4}+\frac {380}{27} x^{3}+\frac {2018}{81} x^{2}-\frac {5948}{1863} x +\frac {2222}{69}}{\sqrt {3 x^{2}-x +2}}-\frac {353 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{243}\) | \(50\) |
trager | \(\frac {\frac {32}{9} x^{4}+\frac {380}{27} x^{3}+\frac {2018}{81} x^{2}-\frac {5948}{1863} x +\frac {2222}{69}}{\sqrt {3 x^{2}-x +2}}-\frac {353 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}-\RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{243}\) | \(77\) |
default | \(\frac {557}{18 \sqrt {3 x^{2}-x +2}}-\frac {353 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{243}+\frac {2018 x^{2}}{81 \sqrt {3 x^{2}-x +2}}+\frac {353 x}{81 \sqrt {3 x^{2}-x +2}}+\frac {32 x^{4}}{9 \sqrt {3 x^{2}-x +2}}+\frac {380 x^{3}}{27 \sqrt {3 x^{2}-x +2}}-\frac {521 \left (6 x -1\right )}{414 \sqrt {3 x^{2}-x +2}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 97, normalized size = 0.94 \begin {gather*} \frac {32 \, x^{4}}{9 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {380 \, x^{3}}{27 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {2018 \, x^{2}}{81 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {353}{243} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {5948 \, x}{1863 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {2222}{69 \, \sqrt {3 \, x^{2} - x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 97, normalized size = 0.94 \begin {gather*} \frac {8119 \, \sqrt {3} {\left (3 \, x^{2} - x + 2\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 12 \, {\left (3312 \, x^{4} + 13110 \, x^{3} + 23207 \, x^{2} - 2974 \, x + 29997\right )} \sqrt {3 \, x^{2} - x + 2}}{11178 \, {\left (3 \, x^{2} - x + 2\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 (2 x + 1\right )^{3} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.12, size = 67, normalized size = 0.65 \begin {gather*} \frac {353}{243} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac {2 \, {\left ({\left (23 \, {\left (6 \, {\left (24 \, x + 95\right )} x + 1009\right )} x - 2974\right )} x + 29997\right )}}{1863 \, \sqrt {3 \, x^{2} - x + 2}} \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 (2\,x+1\right )}^3\,\left (4\,x^2+3\,x+1\right )}{{\left (3\,x^2-x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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