Optimal. Leaf size=47 \[ -\frac {2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (3889-4290 x)}{14283 \sqrt {2-x+3 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1674, 650}
\begin {gather*} -\frac {4 (3889-4290 x)}{14283 \sqrt {3 x^2-x+2}}-\frac {2 (367 x+73)}{621 \left (3 x^2-x+2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 1674
Rubi steps
\begin {align*} \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {577}{9}+92 x}{\left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (3889-4290 x)}{14283 \sqrt {2-x+3 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 33, normalized size = 0.70 \begin {gather*} \frac {2 \left (-1915+1833 x-3546 x^2+2860 x^3\right )}{1587 \left (2-x+3 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs.
\(2(39)=78\).
time = 0.18, size = 86, normalized size = 1.83
method | result | size |
gosper | \(\frac {\frac {5720}{1587} x^{3}-\frac {2364}{529} x^{2}+\frac {1222}{529} x -\frac {3830}{1587}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}\) | \(30\) |
trager | \(\frac {\frac {5720}{1587} x^{3}-\frac {2364}{529} x^{2}+\frac {1222}{529} x -\frac {3830}{1587}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}\) | \(30\) |
risch | \(\frac {\frac {5720}{1587} x^{3}-\frac {2364}{529} x^{2}+\frac {1222}{529} x -\frac {3830}{1587}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}\) | \(30\) |
default | \(-\frac {8 x^{2}}{3 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {13 x}{9 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {295}{162 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {\frac {715 x}{621}-\frac {715}{3726}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {\frac {5720 x}{4761}-\frac {2860}{14283}}{\sqrt {3 x^{2}-x +2}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 76, normalized size = 1.62 \begin {gather*} \frac {5720 \, x}{4761 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {8 \, x^{2}}{3 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {2860}{14283 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {182 \, x}{621 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {1250}{621 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 51, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (2860 \, x^{3} - 3546 \, x^{2} + 1833 \, x - 1915\right )} \sqrt {3 \, x^{2} - x + 2}}{1587 \, {\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\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 ) \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.92, size = 28, normalized size = 0.60 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (1430 \, x - 1773\right )} x + 1833\right )} x - 1915\right )}}{1587 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.20, size = 49, normalized size = 1.04 \begin {gather*} -\frac {442\,x-5720\,x\,\left (3\,x^2-x+2\right )+15556\,x^2+11490}{\sqrt {3\,x^2-x+2}\,\left (14283\,x^2-4761\,x+9522\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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