Optimal. Leaf size=70 \[ \frac {8 x^5}{25}-\frac {21 x^4}{25}+\frac {1222 x^3}{375}-\frac {7451 x^2}{1250}-\frac {158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac {49508 x}{3125}+\frac {328757 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{15625 \sqrt {31}} \]
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Rubi [A] time = 0.05, antiderivative size = 70, 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} \begin {gather*} \frac {8 x^5}{25}-\frac {21 x^4}{25}+\frac {1222 x^3}{375}-\frac {7451 x^2}{1250}-\frac {158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac {49508 x}{3125}+\frac {328757 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{15625 \sqrt {31}} \end {gather*}
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 )^3}{2+3 x+5 x^2} \, dx &=\int \left (\frac {49508}{3125}-\frac {7451 x}{625}+\frac {1222 x^2}{125}-\frac {84 x^3}{25}+\frac {8 x^4}{5}-\frac {1331 (11+119 x)}{3125 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac {49508 x}{3125}-\frac {7451 x^2}{1250}+\frac {1222 x^3}{375}-\frac {21 x^4}{25}+\frac {8 x^5}{25}-\frac {1331 \int \frac {11+119 x}{2+3 x+5 x^2} \, dx}{3125}\\ &=\frac {49508 x}{3125}-\frac {7451 x^2}{1250}+\frac {1222 x^3}{375}-\frac {21 x^4}{25}+\frac {8 x^5}{25}-\frac {158389 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{31250}+\frac {328757 \int \frac {1}{2+3 x+5 x^2} \, dx}{31250}\\ &=\frac {49508 x}{3125}-\frac {7451 x^2}{1250}+\frac {1222 x^3}{375}-\frac {21 x^4}{25}+\frac {8 x^5}{25}-\frac {158389 \log \left (2+3 x+5 x^2\right )}{31250}-\frac {328757 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{15625}\\ &=\frac {49508 x}{3125}-\frac {7451 x^2}{1250}+\frac {1222 x^3}{375}-\frac {21 x^4}{25}+\frac {8 x^5}{25}+\frac {328757 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{15625 \sqrt {31}}-\frac {158389 \log \left (2+3 x+5 x^2\right )}{31250}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.90 \begin {gather*} \frac {31 \left (5 x \left (6000 x^4-15750 x^3+61100 x^2-111765 x+297048\right )-475167 \log \left (5 x^2+3 x+2\right )\right )+1972542 \sqrt {31} \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{2906250} \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 {\left (3-x+2 x^2\right )^3}{2+3 x+5 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 53, normalized size = 0.76 \begin {gather*} \frac {8}{25} \, x^{5} - \frac {21}{25} \, x^{4} + \frac {1222}{375} \, x^{3} - \frac {7451}{1250} \, x^{2} + \frac {328757}{484375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {49508}{3125} \, x - \frac {158389}{31250} \, \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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giac [A] time = 0.21, size = 53, normalized size = 0.76 \begin {gather*} \frac {8}{25} \, x^{5} - \frac {21}{25} \, x^{4} + \frac {1222}{375} \, x^{3} - \frac {7451}{1250} \, x^{2} + \frac {328757}{484375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {49508}{3125} \, x - \frac {158389}{31250} \, \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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maple [A] time = 0.00, size = 54, normalized size = 0.77 \begin {gather*} \frac {8 x^{5}}{25}-\frac {21 x^{4}}{25}+\frac {1222 x^{3}}{375}-\frac {7451 x^{2}}{1250}+\frac {49508 x}{3125}+\frac {328757 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{484375}-\frac {158389 \ln \left (5 x^{2}+3 x +2\right )}{31250} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 53, normalized size = 0.76 \begin {gather*} \frac {8}{25} \, x^{5} - \frac {21}{25} \, x^{4} + \frac {1222}{375} \, x^{3} - \frac {7451}{1250} \, x^{2} + \frac {328757}{484375} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {49508}{3125} \, x - \frac {158389}{31250} \, \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 = 0.04, size = 55, normalized size = 0.79 \begin {gather*} \frac {49508\,x}{3125}-\frac {158389\,\ln \left (5\,x^2+3\,x+2\right )}{31250}+\frac {328757\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{484375}-\frac {7451\,x^2}{1250}+\frac {1222\,x^3}{375}-\frac {21\,x^4}{25}+\frac {8\,x^5}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 76, normalized size = 1.09 \begin {gather*} \frac {8 x^{5}}{25} - \frac {21 x^{4}}{25} + \frac {1222 x^{3}}{375} - \frac {7451 x^{2}}{1250} + \frac {49508 x}{3125} - \frac {158389 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{31250} + \frac {328757 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{484375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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