Optimal. Leaf size=60 \[ \frac {x^4}{\left (x^4+x+3\right )^3}-\frac {3 x}{\left (x^4+x+3\right )^3}+\frac {2}{\left (x^4+x+3\right )^3}-\frac {5 x^6}{\left (x^4+x+3\right )^3}+\frac {5 x^2}{\left (x^4+x+3\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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.
[In]
[Out]
Rule 1588
Rule 2102
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 = 0.02, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 65, normalized size = 1.08 \[ -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 30, normalized size = 0.50 \[ -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{{\left (x^{4} + x + 3\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 28, normalized size = 0.47 \[ \frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.25, size = 65, normalized size = 1.08 \[ -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.34, size = 27, normalized size = 0.45 \[ \frac {-5\,x^6+x^4+5\,x^2-3\,x+2}{{\left (x^4+x+3\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 60, normalized size = 1.00 \[ \frac {- 5 x^{6} + x^{4} + 5 x^{2} - 3 x + 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________