Optimal. Leaf size=42 \[ \frac {2 x}{5}+\frac {143 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{25 \sqrt {31}}-\frac {11}{50} \log \left (2+3 x+5 x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1671, 648, 632,
210, 642} \begin {gather*} \frac {143 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{25 \sqrt {31}}-\frac {11}{50} \log \left (5 x^2+3 x+2\right )+\frac {2 x}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
Rubi steps
\begin {align*} \int \frac {3-x+2 x^2}{2+3 x+5 x^2} \, dx &=\int \left (\frac {2}{5}+\frac {11 (1-x)}{5 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac {2 x}{5}+\frac {11}{5} \int \frac {1-x}{2+3 x+5 x^2} \, dx\\ &=\frac {2 x}{5}-\frac {11}{50} \int \frac {3+10 x}{2+3 x+5 x^2} \, dx+\frac {143}{50} \int \frac {1}{2+3 x+5 x^2} \, dx\\ &=\frac {2 x}{5}-\frac {11}{50} \log \left (2+3 x+5 x^2\right )-\frac {143}{25} \text {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac {2 x}{5}+\frac {143 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{25 \sqrt {31}}-\frac {11}{50} \log \left (2+3 x+5 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {2 x}{5}+\frac {143 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{25 \sqrt {31}}-\frac {11}{50} \log \left (2+3 x+5 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 34, normalized size = 0.81
method | result | size |
default | \(\frac {2 x}{5}-\frac {11 \ln \left (5 x^{2}+3 x +2\right )}{50}+\frac {143 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{775}\) | \(34\) |
risch | \(\frac {2 x}{5}-\frac {11 \ln \left (100 x^{2}+60 x +40\right )}{50}+\frac {143 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{775}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 33, normalized size = 0.79 \begin {gather*} \frac {143}{775} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {2}{5} \, x - \frac {11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.10, size = 33, normalized size = 0.79 \begin {gather*} \frac {143}{775} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {2}{5} \, x - \frac {11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 49, normalized size = 1.17 \begin {gather*} \frac {2 x}{5} - \frac {11 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{50} + \frac {143 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{775} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.98, size = 33, normalized size = 0.79 \begin {gather*} \frac {143}{775} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {2}{5} \, x - \frac {11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.39, size = 35, normalized size = 0.83 \begin {gather*} \frac {2\,x}{5}-\frac {11\,\ln \left (5\,x^2+3\,x+2\right )}{50}+\frac {143\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{775} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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