\(\int (\frac {-3+10 x+4 x^3-30 x^5}{(3+x+x^4)^3}-\frac {3 (1+4 x^3) (2-3 x+5 x^2+x^4-5 x^6)}{(3+x+x^4)^4}) \, dx\) [494]

Optimal result

Integrand size = 60, antiderivative size = 27 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=\frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \]

[Out]

(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^3

Rubi [F]

\[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=\int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx \]

[In]

Int[(-3 + 10*x + 4*x^3 - 30*x^5)/(3 + x + x^4)^3 - (3*(1 + 4*x^3)*(2 - 3*x + 5*x^2 + x^4 - 5*x^6))/(3 + x + x^
4)^4,x]

[Out]

7/(2*(3 + x + x^4)^3) - (63*x)/(22*(3 + x + x^4)^3) - (12*x^2)/(3 + x + x^4)^3 - (5*x^3)/(3 + x + x^4)^3 + (3*
x^4)/(2*(3 + x + x^4)^3) - (10*x^6)/(3 + x + x^4)^3 - 1/(2*(3 + x + x^4)^2) + (5*x^2)/(3 + x + x^4)^2 + (144*D
efer[Int][(3 + x + x^4)^(-4), x])/11 + (828*Defer[Int][x/(3 + x + x^4)^4, x])/11 + 18*Defer[Int][x^2/(3 + x +
x^4)^4, x] - 4*Defer[Int][(3 + x + x^4)^(-3), x] - 20*Defer[Int][x/(3 + x + x^4)^3, x]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=\frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \]

[In]

Integrate[(-3 + 10*x + 4*x^3 - 30*x^5)/(3 + x + x^4)^3 - (3*(1 + 4*x^3)*(2 - 3*x + 5*x^2 + x^4 - 5*x^6))/(3 +
x + x^4)^4,x]

[Out]

(2 - 3*x + 5*x^2 + x^4 - 5*x^6)/(3 + x + x^4)^3

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04

method	result	size
norman	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
parallelrisch	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
gosper	\(-\frac {5 x^{6}-x^{4}-5 x^{2}+3 x -2}{\left (x^{4}+x +3\right )^{3}}\)	\(31\)
risch	\(-\frac {5 x^{6}-x^{4}-5 x^{2}+3 x -2}{\left (x^{4}+x +3\right )^{3}}\)	\(31\)
default	\(-\frac {-\frac {34568}{195075} x^{7}+\frac {73672}{195075} x^{6}+\frac {15392}{195075} x^{5}-\frac {60494}{195075} x^{4}-\frac {68792}{195075} x^{3}-\frac {583927}{195075} x^{2}+\frac {3356}{13005} x -\frac {2069}{43350}}{\left (x^{4}+x +3\right )^{2}}+\frac {-\frac {34568}{195075} x^{11}+\frac {73672}{195075} x^{10}+\frac {15392}{195075} x^{9}-\frac {95062}{195075} x^{8}-\frac {98824}{195075} x^{7}-\frac {1322894}{195075} x^{6}+\frac {36022}{195075} x^{5}-\frac {129019}{390150} x^{4}-\frac {790303}{195075} x^{3}-\frac {80674}{21675} x^{2}-\frac {32853}{14450} x +\frac {26831}{14450}}{\left (x^{4}+x +3\right )^{3}}\)	\(112\)

[In]

int((-30*x^5+4*x^3+10*x-3)/(x^4+x+3)^3-3*(4*x^3+1)*(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^4,x,method=_RETURNVERBOS
E)

[Out]

(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=-\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

[In]

integrate((-30*x^5+4*x^3+10*x-3)/(x^4+x+3)^3-3*(4*x^3+1)*(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^4,x, algorithm="fr
icas")

[Out]

-(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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=\frac {- 5 x^{6} + x^{4} + 5 x^{2} - 3 x + 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \]

[In]

integrate((-30*x**5+4*x**3+10*x-3)/(x**4+x+3)**3-3*(4*x**3+1)*(-5*x**6+x**4+5*x**2-3*x+2)/(x**4+x+3)**4,x)

[Out]

(-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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).

Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=-\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

[In]

integrate((-30*x^5+4*x^3+10*x-3)/(x^4+x+3)^3-3*(4*x^3+1)*(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^4,x, algorithm="ma
xima")

[Out]

-(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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (30) = 60\).

Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.11 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=\frac {69136 \, x^{7} - 147344 \, x^{6} - 30784 \, x^{5} + 120988 \, x^{4} + 137584 \, x^{3} + 1167854 \, x^{2} - 100680 \, x + 18621}{390150 \, {\left (x^{4} + x + 3\right )}^{2}} - \frac {69136 \, x^{11} - 147344 \, x^{10} - 30784 \, x^{9} + 190124 \, x^{8} + 197648 \, x^{7} + 2645788 \, x^{6} - 72044 \, x^{5} + 129019 \, x^{4} + 1580606 \, x^{3} + 1452132 \, x^{2} + 887031 \, x - 724437}{390150 \, {\left (x^{4} + x + 3\right )}^{3}} \]

[In]

integrate((-30*x^5+4*x^3+10*x-3)/(x^4+x+3)^3-3*(4*x^3+1)*(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^4,x, algorithm="gi
ac")

[Out]

1/390150*(69136*x^7 - 147344*x^6 - 30784*x^5 + 120988*x^4 + 137584*x^3 + 1167854*x^2 - 100680*x + 18621)/(x^4
+ x + 3)^2 - 1/390150*(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)/(x^4 + x + 3)^3

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (\frac {-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac {3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx=\frac {-5\,x^6+x^4+5\,x^2-3\,x+2}{{\left (x^4+x+3\right )}^3} \]

[In]

int((10*x + 4*x^3 - 30*x^5 - 3)/(x + x^4 + 3)^3 - (3*(4*x^3 + 1)*(5*x^2 - 3*x + x^4 - 5*x^6 + 2))/(x + x^4 + 3
)^4,x)

[Out]

(5*x^2 - 3*x + x^4 - 5*x^6 + 2)/(x + x^4 + 3)^3