3.152 \(\int \frac{x^8}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^3}{3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{3}{2} \log \left (x^3+3\right ) \]

[Out]

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

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Rubi [A]  time = 0.0187006, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1357, 703, 632, 31} \[ \frac{x^3}{3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{3}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^8/(3 + 4*x^3 + x^6),x]

[Out]

x^3/3 + Log[1 + x^3]/6 - (3*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 703

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1))/(c*
(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(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] && GtQ[m, 1]

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^8}{3+4 x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac{x^3}{3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{-3-4 x}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac{x^3}{3}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^3\right )-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,x^3\right )\\ &=\frac{x^3}{3}+\frac{1}{6} \log \left (1+x^3\right )-\frac{3}{2} \log \left (3+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0050508, size = 28, normalized size = 1. \[ \frac{x^3}{3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{3}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(3 + 4*x^3 + x^6),x]

[Out]

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

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Maple [A]  time = 0.003, size = 23, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}-{\frac{3\,\ln \left ({x}^{3}+3 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^6+4*x^3+3),x)

[Out]

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

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Maxima [A]  time = 1.184, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{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^8/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47735, size = 63, normalized size = 2.25 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{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^8/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.127806, size = 22, normalized size = 0.79 \begin{align*} \frac{x^{3}}{3} + \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{3 \log{\left (x^{3} + 3 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**6+4*x**3+3),x)

[Out]

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

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Giac [A]  time = 1.11344, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{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^8/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

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