3.61.94 \(\int \frac {-36-54 x-36 x^4-36 x^5+2 x^9}{9-6 x^4+x^8} \, dx\)

Optimal. Leaf size=27 \[ x^2-4 \left (\frac {3 \left (x+x^2\right )}{3-x^4}-\log (4)\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 0.70, number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {28, 1858, 1586, 12, 30} \begin {gather*} x^2-\frac {12 x (x+1)}{3-x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36 - 54*x - 36*x^4 - 36*x^5 + 2*x^9)/(9 - 6*x^4 + x^8),x]

[Out]

x^2 - (12*x*(1 + x))/(3 - x^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-36-54 x-36 x^4-36 x^5+2 x^9}{\left (-3+x^4\right )^2} \, dx\\ &=-\frac {12 x (1+x)}{3-x^4}+\frac {1}{12} \int \frac {-72 x+24 x^5}{-3+x^4} \, dx\\ &=-\frac {12 x (1+x)}{3-x^4}+\frac {1}{12} \int 24 x \, dx\\ &=-\frac {12 x (1+x)}{3-x^4}+2 \int x \, dx\\ &=x^2-\frac {12 x (1+x)}{3-x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} 2 \left (\frac {x^2}{2}+\frac {6 \left (x+x^2\right )}{-3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 - 54*x - 36*x^4 - 36*x^5 + 2*x^9)/(9 - 6*x^4 + x^8),x]

[Out]

2*(x^2/2 + (6*(x + x^2))/(-3 + x^4))

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fricas [A]  time = 0.62, size = 20, normalized size = 0.74 \begin {gather*} \frac {x^{6} + 9 \, x^{2} + 12 \, x}{x^{4} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^9-36*x^5-36*x^4-54*x-36)/(x^8-6*x^4+9),x, algorithm="fricas")

[Out]

(x^6 + 9*x^2 + 12*x)/(x^4 - 3)

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giac [A]  time = 0.16, size = 18, normalized size = 0.67 \begin {gather*} x^{2} + \frac {12 \, {\left (x^{2} + x\right )}}{x^{4} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^9-36*x^5-36*x^4-54*x-36)/(x^8-6*x^4+9),x, algorithm="giac")

[Out]

x^2 + 12*(x^2 + x)/(x^4 - 3)

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maple [A]  time = 0.04, size = 18, normalized size = 0.67

method	result	size
gosper	\(\frac {x \left (x^{5}+9 x +12\right )}{x^{4}-3}\)	\(18\)
norman	\(\frac {x^{6}+9 x^{2}+12 x}{x^{4}-3}\)	\(21\)
risch	\(x^{2}+\frac {12 x^{2}+12 x}{x^{4}-3}\)	\(22\)
default	\(x^{2}-\frac {12 \left (-x^{2}-x \right )}{x^{4}-3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^9-36*x^5-36*x^4-54*x-36)/(x^8-6*x^4+9),x,method=_RETURNVERBOSE)

[Out]

x*(x^5+9*x+12)/(x^4-3)

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maxima [A]  time = 0.34, size = 18, normalized size = 0.67 \begin {gather*} x^{2} + \frac {12 \, {\left (x^{2} + x\right )}}{x^{4} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^9-36*x^5-36*x^4-54*x-36)/(x^8-6*x^4+9),x, algorithm="maxima")

[Out]

x^2 + 12*(x^2 + x)/(x^4 - 3)

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mupad [B]  time = 0.05, size = 17, normalized size = 0.63 \begin {gather*} x^2+\frac {12\,x\,\left (x+1\right )}{x^4-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(54*x + 36*x^4 + 36*x^5 - 2*x^9 + 36)/(x^8 - 6*x^4 + 9),x)

[Out]

x^2 + (12*x*(x + 1))/(x^4 - 3)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.56 \begin {gather*} x^{2} + \frac {12 x^{2} + 12 x}{x^{4} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**9-36*x**5-36*x**4-54*x-36)/(x**8-6*x**4+9),x)

[Out]

x**2 + (12*x**2 + 12*x)/(x**4 - 3)

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