3.72.33 \(\int \frac {24 x^2+8 x^4-2 x^6}{48+96 x^2+24 x^3+72 x^4+24 x^5+27 x^6+6 x^7+3 x^8} \, dx\)

Optimal. Leaf size=29 \[ e^2+\frac {2}{3} \left (2+\frac {x}{x+\left (\frac {2}{x}+x\right )^2}\right )-\log (25) \]

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Rubi [A]  time = 0.28, antiderivative size = 22, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1594, 6688, 12, 1588} \begin {gather*} \frac {2 x^3}{3 \left (x^4+x^3+4 x^2+4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24*x^2 + 8*x^4 - 2*x^6)/(48 + 96*x^2 + 24*x^3 + 72*x^4 + 24*x^5 + 27*x^6 + 6*x^7 + 3*x^8),x]

[Out]

(2*x^3)/(3*(4 + 4*x^2 + 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 1588

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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (24+8 x^2-2 x^4\right )}{48+96 x^2+24 x^3+72 x^4+24 x^5+27 x^6+6 x^7+3 x^8} \, dx\\ &=\int \frac {2 x^2 \left (12+4 x^2-x^4\right )}{3 \left (4+4 x^2+x^3+x^4\right )^2} \, dx\\ &=\frac {2}{3} \int \frac {x^2 \left (12+4 x^2-x^4\right )}{\left (4+4 x^2+x^3+x^4\right )^2} \, dx\\ &=\frac {2 x^3}{3 \left (4+4 x^2+x^3+x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.76 \begin {gather*} \frac {2 x^3}{3 \left (4+4 x^2+x^3+x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x^2 + 8*x^4 - 2*x^6)/(48 + 96*x^2 + 24*x^3 + 72*x^4 + 24*x^5 + 27*x^6 + 6*x^7 + 3*x^8),x]

[Out]

(2*x^3)/(3*(4 + 4*x^2 + x^3 + x^4))

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fricas [A]  time = 0.67, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 \, x^{3}}{3 \, {\left (x^{4} + x^{3} + 4 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+8*x^4+24*x^2)/(3*x^8+6*x^7+27*x^6+24*x^5+72*x^4+24*x^3+96*x^2+48),x, algorithm="fricas")

[Out]

2/3*x^3/(x^4 + x^3 + 4*x^2 + 4)

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giac [A]  time = 0.22, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 \, x^{3}}{3 \, {\left (x^{4} + x^{3} + 4 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+8*x^4+24*x^2)/(3*x^8+6*x^7+27*x^6+24*x^5+72*x^4+24*x^3+96*x^2+48),x, algorithm="giac")

[Out]

2/3*x^3/(x^4 + x^3 + 4*x^2 + 4)

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maple [A]  time = 0.03, size = 21, normalized size = 0.72

method	result	size
gosper	\(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\)	\(21\)
default	\(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\)	\(21\)
norman	\(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\)	\(21\)
risch	\(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^6+8*x^4+24*x^2)/(3*x^8+6*x^7+27*x^6+24*x^5+72*x^4+24*x^3+96*x^2+48),x,method=_RETURNVERBOSE)

[Out]

2/3*x^3/(x^4+x^3+4*x^2+4)

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maxima [A]  time = 0.34, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 \, x^{3}}{3 \, {\left (x^{4} + x^{3} + 4 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+8*x^4+24*x^2)/(3*x^8+6*x^7+27*x^6+24*x^5+72*x^4+24*x^3+96*x^2+48),x, algorithm="maxima")

[Out]

2/3*x^3/(x^4 + x^3 + 4*x^2 + 4)

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mupad [B]  time = 0.05, size = 24, normalized size = 0.83 \begin {gather*} \frac {2\,x^3}{3\,\left (x^4+x^3+4\,x^2+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^2 + 8*x^4 - 2*x^6)/(96*x^2 + 24*x^3 + 72*x^4 + 24*x^5 + 27*x^6 + 6*x^7 + 3*x^8 + 48),x)

[Out]

(2*x^3)/(3*(4*x^2 + x^3 + x^4 + 4))

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sympy [A]  time = 0.12, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 x^{3}}{3 x^{4} + 3 x^{3} + 12 x^{2} + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**6+8*x**4+24*x**2)/(3*x**8+6*x**7+27*x**6+24*x**5+72*x**4+24*x**3+96*x**2+48),x)

[Out]

2*x**3/(3*x**4 + 3*x**3 + 12*x**2 + 12)

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