Optimal. Leaf size=27 \[ \frac{-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]
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Rubi [F] time = 0.433148, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (\frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac{3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \left (\frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac{3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx &=-\left (3 \int \frac{\left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4} \, dx\right )+\int \frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3} \, dx\\ &=-\frac{10 x^6}{\left (3+x+x^4\right )^3}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-\frac{1}{6} \int \frac{18+120 x-24 x^3}{\left (3+x+x^4\right )^3} \, dx+\frac{1}{2} \int \frac{-12+18 x-30 x^2-48 x^3+66 x^4+240 x^5+90 x^6-24 x^7}{\left (3+x+x^4\right )^4} \, dx\\ &=\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-\frac{1}{24} \int \frac{96+480 x}{\left (3+x+x^4\right )^3} \, dx-\frac{1}{16} \int \frac{96-144 x+240 x^2+672 x^3-504 x^4-1920 x^5-720 x^6}{\left (3+x+x^4\right )^4} \, dx\\ &=-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{1}{144} \int \frac{-864+1296 x+4320 x^2-6048 x^3+4536 x^4+17280 x^5}{\left (3+x+x^4\right )^4} \, dx-\frac{1}{24} \int \left (\frac{96}{\left (3+x+x^4\right )^3}+\frac{480 x}{\left (3+x+x^4\right )^3}\right ) \, dx\\ &=-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-\frac{\int \frac{8640-116640 x-25920 x^2+60480 x^3-45360 x^4}{\left (3+x+x^4\right )^4} \, dx}{1440}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{\int \frac{41040+1192320 x+285120 x^2-665280 x^3}{\left (3+x+x^4\right )^4} \, dx}{15840}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=\frac{7}{2 \left (3+x+x^4\right )^3}-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{\int \frac{829440+4769280 x+1140480 x^2}{\left (3+x+x^4\right )^4} \, dx}{63360}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=\frac{7}{2 \left (3+x+x^4\right )^3}-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{\int \left (\frac{829440}{\left (3+x+x^4\right )^4}+\frac{4769280 x}{\left (3+x+x^4\right )^4}+\frac{1140480 x^2}{\left (3+x+x^4\right )^4}\right ) \, dx}{63360}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=\frac{7}{2 \left (3+x+x^4\right )^3}-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx+\frac{144}{11} \int \frac{1}{\left (3+x+x^4\right )^4} \, dx+18 \int \frac{x^2}{\left (3+x+x^4\right )^4} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx+\frac{828}{11} \int \frac{x}{\left (3+x+x^4\right )^4} \, dx\\ \end{align*}
Mathematica [A] time = 0.0105844, size = 27, normalized size = 1. \[ \frac{-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 112, normalized size = 4.2 \begin{align*} -{\frac{1}{ \left ({x}^{4}+x+3 \right ) ^{2}} \left ( -{\frac{34568\,{x}^{7}}{195075}}+{\frac{73672\,{x}^{6}}{195075}}+{\frac{15392\,{x}^{5}}{195075}}-{\frac{60494\,{x}^{4}}{195075}}-{\frac{68792\,{x}^{3}}{195075}}-{\frac{583927\,{x}^{2}}{195075}}+{\frac{3356\,x}{13005}}-{\frac{2069}{43350}} \right ) }+3\,{\frac{1}{ \left ({x}^{4}+x+3 \right ) ^{3}} \left ( -{\frac{34568\,{x}^{11}}{585225}}+{\frac{73672\,{x}^{10}}{585225}}+{\frac{15392\,{x}^{9}}{585225}}-{\frac{95062\,{x}^{8}}{585225}}-{\frac{98824\,{x}^{7}}{585225}}-{\frac{1322894\,{x}^{6}}{585225}}+{\frac{36022\,{x}^{5}}{585225}}-{\frac{129019\,{x}^{4}}{1170450}}-{\frac{790303\,{x}^{3}}{585225}}-{\frac{80674\,{x}^{2}}{65025}}-{\frac{10951\,x}{14450}}+{\frac{26831}{43350}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.31228, size = 88, normalized size = 3.26 \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 [B] time = 0.940147, size = 147, normalized size = 5.44 \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 [B] time = 0.285799, size = 61, normalized size = 2.26 \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 [B] time = 1.1459, size = 150, normalized size = 5.56 \begin{align*} \frac{69136 \, x^{7} - 147344 \, x^{6} - 30784 \, x^{5} + 120988 \, x^{4} + 137584 \, x^{3} + 1167854 \, x^{2} - 100680 \, x + 18621}{390150 \,{\left (x^{4} + x + 3\right )}^{2}} - \frac{69136 \, x^{11} - 147344 \, x^{10} - 30784 \, x^{9} + 190124 \, x^{8} + 197648 \, x^{7} + 2645788 \, x^{6} - 72044 \, x^{5} + 129019 \, x^{4} + 1580606 \, x^{3} + 1452132 \, x^{2} + 887031 \, x - 724437}{390150 \,{\left (x^{4} + x + 3\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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