3.5.92 \(\int -\frac {15-36 x+5 x^2+12 x^3-34 x^4+140 x^5+15 x^6+8 x^7-30 x^9}{(3+x+x^4)^4} \, dx\) [492]

Optimal. Leaf size=60 \[ \frac {2}{\left (3+x+x^4\right )^3}-\frac {3 x}{\left (3+x+x^4\right )^3}+\frac {5 x^2}{\left (3+x+x^4\right )^3}+\frac {x^4}{\left (3+x+x^4\right )^3}-\frac {5 x^6}{\left (3+x+x^4\right )^3} \]

[Out]

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

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Rubi [A]
time = 0.09, 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 = {2127, 1602} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-((15 - 36*x + 5*x^2 + 12*x^3 - 34*x^4 + 140*x^5 + 15*x^6 + 8*x^7 - 30*x^9)/(3 + x + x^4)^4),x]

[Out]

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

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 2127

Int[(Pm_)*(Qn_)^(p_.), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*x^(m - n +
1)*(Qn^(p + 1)/((m + n*p + 1)*Coeff[Qn, x, n])), x] + Dist[1/((m + n*p + 1)*Coeff[Qn, x, n]), Int[ExpandToSum
[(m + n*p + 1)*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*x^(m - n)*((m - n + 1)*Qn + (p + 1)*x*D[Qn, x]), x]*Qn^p,
x], x] /; LtQ[1, n, m + 1] && m + n*p + 1 < 0] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && LtQ[p, -1]

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*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.45 \begin {gather*} \frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-((15 - 36*x + 5*x^2 + 12*x^3 - 34*x^4 + 140*x^5 + 15*x^6 + 8*x^7 - 30*x^9)/(3 + x + x^4)^4),x]

[Out]

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

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Maple [A]
time = 0.02, size = 28, normalized size = 0.47

method	result	size
default	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
norman	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
risch	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
gosper	\(-\frac {5 x^{6}-x^{4}-5 x^{2}+3 x -2}{\left (x^{4}+x +3\right )^{3}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*x^9-8*x^7-15*x^6-140*x^5+34*x^4-12*x^3-5*x^2+36*x-15)/(x^4+x+3)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 65, normalized size = 1.08 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x^9-8*x^7-15*x^6-140*x^5+34*x^4-12*x^3-5*x^2+36*x-15)/(x^4+x+3)^4,x, algorithm="maxima")

[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)

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Fricas [A]
time = 0.38, size = 65, normalized size = 1.08 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x^9-8*x^7-15*x^6-140*x^5+34*x^4-12*x^3-5*x^2+36*x-15)/(x^4+x+3)^4,x, algorithm="fricas")

[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)

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Sympy [A]
time = 0.10, size = 60, normalized size = 1.00 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x**9-8*x**7-15*x**6-140*x**5+34*x**4-12*x**3-5*x**2+36*x-15)/(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)

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Giac [A]
time = 4.14, size = 30, normalized size = 0.50 \begin {gather*} -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{{\left (x^{4} + x + 3\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((30*x^9-8*x^7-15*x^6-140*x^5+34*x^4-12*x^3-5*x^2+36*x-15)/(x^4+x+3)^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.34, size = 27, normalized size = 0.45 \begin {gather*} \frac {-5\,x^6+x^4+5\,x^2-3\,x+2}{{\left (x^4+x+3\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x^2 - 36*x + 12*x^3 - 34*x^4 + 140*x^5 + 15*x^6 + 8*x^7 - 30*x^9 + 15)/(x + x^4 + 3)^4,x)

[Out]

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

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