3.151 \(\int \frac{x^{11}}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{9}{2} \log \left (x^3+3\right ) \]

[Out]

(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2

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Rubi [A]  time = 0.024081, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1357, 701, 632, 31} \[ \frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{9}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^11/(3 + 4*x^3 + x^6),x]

[Out]

(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2

Rule 1357

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

Rule 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2,
0] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^{11}}{3+4 x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-4+x+\frac{12+13 x}{3+4 x+x^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{4 x^3}{3}+\frac{x^6}{6}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{12+13 x}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=-\frac{4 x^3}{3}+\frac{x^6}{6}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^3\right )+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,x^3\right )\\ &=-\frac{4 x^3}{3}+\frac{x^6}{6}-\frac{1}{6} \log \left (1+x^3\right )+\frac{9}{2} \log \left (3+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0061133, size = 35, normalized size = 1. \[ \frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{9}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(3 + 4*x^3 + x^6),x]

[Out]

(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2

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Maple [A]  time = 0.005, size = 28, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{3}}{3}}+{\frac{{x}^{6}}{6}}-{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}+{\frac{9\,\ln \left ({x}^{3}+3 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^6+4*x^3+3),x)

[Out]

-4/3*x^3+1/6*x^6-1/6*ln(x^3+1)+9/2*ln(x^3+3)

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Maxima [A]  time = 1.2259, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{6} \, x^{6} - \frac{4}{3} \, x^{3} + \frac{9}{2} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/6*x^6 - 4/3*x^3 + 9/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Fricas [A]  time = 1.41851, size = 77, normalized size = 2.2 \begin{align*} \frac{1}{6} \, x^{6} - \frac{4}{3} \, x^{3} + \frac{9}{2} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/6*x^6 - 4/3*x^3 + 9/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Sympy [A]  time = 0.120889, size = 29, normalized size = 0.83 \begin{align*} \frac{x^{6}}{6} - \frac{4 x^{3}}{3} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{9 \log{\left (x^{3} + 3 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(x**6+4*x**3+3),x)

[Out]

x**6/6 - 4*x**3/3 - log(x**3 + 1)/6 + 9*log(x**3 + 3)/2

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Giac [A]  time = 1.09476, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{6} \, x^{6} - \frac{4}{3} \, x^{3} + \frac{9}{2} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

1/6*x^6 - 4/3*x^3 + 9/2*log(abs(x^3 + 3)) - 1/6*log(abs(x^3 + 1))