Optimal. Leaf size=43 \[ \frac {11 (13 x+7)}{155 \left (5 x^2+3 x+2\right )}+\frac {82 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{31 \sqrt {31}} \]
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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} \[ \frac {11 (13 x+7)}{155 \left (5 x^2+3 x+2\right )}+\frac {82 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{31 \sqrt {31}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {41}{2+3 x+5 x^2} \, dx\\ &=\frac {11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}+\frac {41}{31} \int \frac {1}{2+3 x+5 x^2} \, dx\\ &=\frac {11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}-\frac {82}{31} \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac {11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}+\frac {82 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{31 \sqrt {31}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \[ \frac {11 (13 x+7)}{155 \left (5 x^2+3 x+2\right )}+\frac {82 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{31 \sqrt {31}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 45, normalized size = 1.05 \[ \frac {410 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 4433 \, x + 2387}{4805 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 36, normalized size = 0.84 \[ \frac {82}{961} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {11 \, {\left (13 \, x + 7\right )}}{155 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
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 \[ \frac {82 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{961}+\frac {\frac {143 x}{775}+\frac {77}{775}}{x^{2}+\frac {3}{5} x +\frac {2}{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 36, normalized size = 0.84 \[ \frac {82}{961} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {11 \, {\left (13 \, x + 7\right )}}{155 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 0.81 \[ \frac {\frac {143\,x}{775}+\frac {77}{775}}{x^2+\frac {3\,x}{5}+\frac {2}{5}}+\frac {82\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{961} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 42, normalized size = 0.98 \[ \frac {143 x + 77}{775 x^{2} + 465 x + 310} + \frac {82 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{961} \]
Verification of antiderivative is not currently implemented for this CAS.
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