Optimal. Leaf size=56 \[ \frac {4 x^3}{15}-\frac {16 x^2}{25}-\frac {1573 \log \left (5 x^2+3 x+2\right )}{1250}+\frac {381 x}{125}+\frac {8349 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{625 \sqrt {31}} \]
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Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1657, 634, 618, 204, 628} \[ \frac {4 x^3}{15}-\frac {16 x^2}{25}-\frac {1573 \log \left (5 x^2+3 x+2\right )}{1250}+\frac {381 x}{125}+\frac {8349 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{625 \sqrt {31}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^2}{2+3 x+5 x^2} \, dx &=\int \left (\frac {381}{125}-\frac {32 x}{25}+\frac {4 x^2}{5}+\frac {121 (3-13 x)}{125 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac {381 x}{125}-\frac {16 x^2}{25}+\frac {4 x^3}{15}+\frac {121}{125} \int \frac {3-13 x}{2+3 x+5 x^2} \, dx\\ &=\frac {381 x}{125}-\frac {16 x^2}{25}+\frac {4 x^3}{15}-\frac {1573 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{1250}+\frac {8349 \int \frac {1}{2+3 x+5 x^2} \, dx}{1250}\\ &=\frac {381 x}{125}-\frac {16 x^2}{25}+\frac {4 x^3}{15}-\frac {1573 \log \left (2+3 x+5 x^2\right )}{1250}-\frac {8349}{625} \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac {381 x}{125}-\frac {16 x^2}{25}+\frac {4 x^3}{15}+\frac {8349 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{625 \sqrt {31}}-\frac {1573 \log \left (2+3 x+5 x^2\right )}{1250}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.95 \[ \frac {10 x \left (100 x^2-240 x+1143\right )-4719 \log \left (5 x^2+3 x+2\right )}{3750}+\frac {8349 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{625 \sqrt {31}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 43, normalized size = 0.77 \[ \frac {4}{15} \, x^{3} - \frac {16}{25} \, x^{2} + \frac {8349}{19375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {381}{125} \, x - \frac {1573}{1250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 43, normalized size = 0.77 \[ \frac {4}{15} \, x^{3} - \frac {16}{25} \, x^{2} + \frac {8349}{19375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {381}{125} \, x - \frac {1573}{1250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.79 \[ \frac {4 x^{3}}{15}-\frac {16 x^{2}}{25}+\frac {381 x}{125}+\frac {8349 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{19375}-\frac {1573 \ln \left (5 x^{2}+3 x +2\right )}{1250} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 43, normalized size = 0.77 \[ \frac {4}{15} \, x^{3} - \frac {16}{25} \, x^{2} + \frac {8349}{19375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {381}{125} \, x - \frac {1573}{1250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.45, size = 45, normalized size = 0.80 \[ \frac {381\,x}{125}-\frac {1573\,\ln \left (5\,x^2+3\,x+2\right )}{1250}+\frac {8349\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{19375}-\frac {16\,x^2}{25}+\frac {4\,x^3}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 63, normalized size = 1.12 \[ \frac {4 x^{3}}{15} - \frac {16 x^{2}}{25} + \frac {381 x}{125} - \frac {1573 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{1250} + \frac {8349 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{19375} \]
Verification of antiderivative is not currently implemented for this CAS.
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