3.153 \(\int \frac{x^5}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]

[Out]

-Log[1 + x^3]/6 + Log[3 + x^3]/2

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Rubi [A]  time = 0.0141699, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1357, 632, 31} \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(3 + 4*x^3 + x^6),x]

[Out]

-Log[1 + x^3]/6 + 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 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^5}{3+4 x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^3\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,x^3\right )\\ &=-\frac{1}{6} \log \left (1+x^3\right )+\frac{1}{2} \log \left (3+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0041518, size = 21, normalized size = 1. \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(3 + 4*x^3 + x^6),x]

[Out]

-Log[1 + x^3]/6 + Log[3 + x^3]/2

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Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+3 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^6+4*x^3+3),x)

[Out]

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

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Maxima [A]  time = 1.30684, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{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^5/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Fricas [A]  time = 1.48159, size = 50, normalized size = 2.38 \begin{align*} \frac{1}{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^5/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Sympy [A]  time = 0.116274, size = 15, normalized size = 0.71 \begin{align*} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**6+4*x**3+3),x)

[Out]

-log(x**3 + 1)/6 + log(x**3 + 3)/2

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Giac [A]  time = 1.09246, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{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^5/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

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