Optimal. Leaf size=43 \[ -\frac {11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac {82 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{23 \sqrt {23}} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1660, 12, 618, 204} \begin {gather*} -\frac {11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac {82 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{23 \sqrt {23}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^2} \, dx &=-\frac {11 (5+3 x)}{46 \left (3-x+2 x^2\right )}+\frac {1}{23} \int \frac {41}{3-x+2 x^2} \, dx\\ &=-\frac {11 (5+3 x)}{46 \left (3-x+2 x^2\right )}+\frac {41}{23} \int \frac {1}{3-x+2 x^2} \, dx\\ &=-\frac {11 (5+3 x)}{46 \left (3-x+2 x^2\right )}-\frac {82}{23} \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=-\frac {11 (5+3 x)}{46 \left (3-x+2 x^2\right )}-\frac {82 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{23 \sqrt {23}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} \frac {82 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{23 \sqrt {23}}-\frac {11 (3 x+5)}{46 \left (2 x^2-x+3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x+5 x^2}{\left (3-x+2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 45, normalized size = 1.05 \begin {gather*} \frac {164 \, \sqrt {23} {\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 759 \, x - 1265}{1058 \, {\left (2 \, x^{2} - x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 36, normalized size = 0.84 \begin {gather*} \frac {82}{529} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11 \, {\left (3 \, x + 5\right )}}{46 \, {\left (2 \, x^{2} - x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.79 \begin {gather*} \frac {82 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{529}+\frac {-\frac {33 x}{92}-\frac {55}{92}}{x^{2}-\frac {1}{2} x +\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 36, normalized size = 0.84 \begin {gather*} \frac {82}{529} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11 \, {\left (3 \, x + 5\right )}}{46 \, {\left (2 \, x^{2} - x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.84 \begin {gather*} \frac {82\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{529}-\frac {\frac {33\,x}{92}+\frac {55}{92}}{x^2-\frac {x}{2}+\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 42, normalized size = 0.98 \begin {gather*} \frac {- 33 x - 55}{92 x^{2} - 46 x + 138} + \frac {82 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{529} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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