Optimal. Leaf size=64 \[ -\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac {131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}-\frac {262 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}} \]
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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1674, 12, 628,
632, 210} \begin {gather*} -\frac {262 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}}-\frac {131 (1-4 x)}{2116 \left (2 x^2-x+3\right )}-\frac {11 (3 x+5)}{92 \left (2 x^2-x+3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 628
Rule 632
Rule 1674
Rubi steps
\begin {align*} \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^3} \, dx &=-\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}+\frac {1}{46} \int \frac {131}{2 \left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}+\frac {131}{92} \int \frac {1}{\left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac {131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}+\frac {131}{529} \int \frac {1}{3-x+2 x^2} \, dx\\ &=-\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac {131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}-\frac {262}{529} \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=-\frac {11 (5+3 x)}{92 \left (3-x+2 x^2\right )^2}-\frac {131 (1-4 x)}{2116 \left (3-x+2 x^2\right )}-\frac {262 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 0.80 \begin {gather*} \frac {\frac {46 \left (-829+472 x-393 x^2+524 x^3\right )}{\left (-3+x-2 x^2\right )^2}+1048 \sqrt {23} \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )}{48668} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 47, normalized size = 0.73
method | result | size |
default | \(\frac {\frac {262}{529} x^{3}-\frac {393}{1058} x^{2}+\frac {236}{529} x -\frac {829}{1058}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {262 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}\) | \(47\) |
risch | \(\frac {\frac {262}{529} x^{3}-\frac {393}{1058} x^{2}+\frac {236}{529} x -\frac {829}{1058}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {262 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 56, normalized size = 0.88 \begin {gather*} \frac {262}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {524 \, x^{3} - 393 \, x^{2} + 472 \, x - 829}{1058 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.57, size = 75, normalized size = 1.17 \begin {gather*} \frac {12052 \, x^{3} + 524 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 9039 \, x^{2} + 10856 \, x - 19067}{24334 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 61, normalized size = 0.95 \begin {gather*} \frac {524 x^{3} - 393 x^{2} + 472 x - 829}{4232 x^{4} - 4232 x^{3} + 13754 x^{2} - 6348 x + 9522} + \frac {262 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{12167} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 46, normalized size = 0.72 \begin {gather*} \frac {262}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {524 \, x^{3} - 393 \, x^{2} + 472 \, x - 829}{1058 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 55, normalized size = 0.86 \begin {gather*} \frac {262\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{12167}+\frac {\frac {131\,x^3}{1058}-\frac {393\,x^2}{4232}+\frac {59\,x}{529}-\frac {829}{4232}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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