Optimal. Leaf size=60 \[ -\frac{5 x^6}{\left (x^4+x+3\right )^3}+\frac{x^4}{\left (x^4+x+3\right )^3}+\frac{5 x^2}{\left (x^4+x+3\right )^3}-\frac{3 x}{\left (x^4+x+3\right )^3}+\frac{2}{\left (x^4+x+3\right )^3} \]
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Rubi [A] time = 0.135984, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2102, 1588} \[ -\frac{5 x^6}{\left (x^4+x+3\right )^3}+\frac{x^4}{\left (x^4+x+3\right )^3}+\frac{5 x^2}{\left (x^4+x+3\right )^3}-\frac{3 x}{\left (x^4+x+3\right )^3}+\frac{2}{\left (x^4+x+3\right )^3} \]
Antiderivative was successfully verified.
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Rule 2102
Rule 1588
Rubi steps
\begin{align*} \int -\frac{15-36 x+5 x^2+12 x^3-34 x^4+140 x^5+15 x^6+8 x^7-30 x^9}{\left (3+x+x^4\right )^4} \, dx &=-\frac{5 x^6}{\left (3+x+x^4\right )^3}+\frac{1}{6} \int \frac{-90+216 x-30 x^2-72 x^3+204 x^4-300 x^5-48 x^7}{\left (3+x+x^4\right )^4} \, dx\\ &=\frac{x^4}{\left (3+x+x^4\right )^3}-\frac{5 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{48} \int \frac{720-1728 x+240 x^2+1152 x^3-1584 x^4+2400 x^5}{\left (3+x+x^4\right )^4} \, dx\\ &=\frac{5 x^2}{\left (3+x+x^4\right )^3}+\frac{x^4}{\left (3+x+x^4\right )^3}-\frac{5 x^6}{\left (3+x+x^4\right )^3}+\frac{1}{480} \int \frac{-7200+2880 x-11520 x^3+15840 x^4}{\left (3+x+x^4\right )^4} \, dx\\ &=-\frac{3 x}{\left (3+x+x^4\right )^3}+\frac{5 x^2}{\left (3+x+x^4\right )^3}+\frac{x^4}{\left (3+x+x^4\right )^3}-\frac{5 x^6}{\left (3+x+x^4\right )^3}-\frac{\int \frac{31680+126720 x^3}{\left (3+x+x^4\right )^4} \, dx}{5280}\\ &=\frac{2}{\left (3+x+x^4\right )^3}-\frac{3 x}{\left (3+x+x^4\right )^3}+\frac{5 x^2}{\left (3+x+x^4\right )^3}+\frac{x^4}{\left (3+x+x^4\right )^3}-\frac{5 x^6}{\left (3+x+x^4\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0149316, size = 27, normalized size = 0.45 \[ \frac{-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 28, normalized size = 0.5 \begin{align*}{\frac{-5\,{x}^{6}+{x}^{4}+5\,{x}^{2}-3\,x+2}{ \left ({x}^{4}+x+3 \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67129, size = 88, normalized size = 1.47 \begin{align*} -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.93845, size = 147, normalized size = 2.45 \begin{align*} -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.244175, size = 61, normalized size = 1.02 \begin{align*} - \frac{5 x^{6} - x^{4} - 5 x^{2} + 3 x - 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12653, size = 41, normalized size = 0.68 \begin{align*} -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{{\left (x^{4} + x + 3\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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