3.156 \(\int \frac{x^5}{-4+x^2+3 x^4} \, dx\)

Optimal. Leaf size=32 \[ \frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (3 x^2+4\right ) \]

[Out]

x^2/6 + Log[1 - x^2]/14 - (8*Log[4 + 3*x^2])/63

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Rubi [A]  time = 0.0245061, antiderivative size = 32, 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 = {1114, 703, 632, 31} \[ \frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (3 x^2+4\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(-4 + x^2 + 3*x^4),x]

[Out]

x^2/6 + Log[1 - x^2]/14 - (8*Log[4 + 3*x^2])/63

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

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^5}{-4+x^2+3 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{-4+x+3 x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{6}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{4-x}{-4+x+3 x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{6}+\frac{3}{14} \operatorname{Subst}\left (\int \frac{1}{-3+3 x} \, dx,x,x^2\right )-\frac{8}{21} \operatorname{Subst}\left (\int \frac{1}{4+3 x} \, dx,x,x^2\right )\\ &=\frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (4+3 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0053258, size = 32, normalized size = 1. \[ \frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (3 x^2+4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(-4 + x^2 + 3*x^4),x]

[Out]

x^2/6 + Log[1 - x^2]/14 - (8*Log[4 + 3*x^2])/63

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Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{6}}-{\frac{8\,\ln \left ( 3\,{x}^{2}+4 \right ) }{63}}+{\frac{\ln \left ({x}^{2}-1 \right ) }{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(3*x^4+x^2-4),x)

[Out]

1/6*x^2-8/63*ln(3*x^2+4)+1/14*ln(x^2-1)

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Maxima [A]  time = 0.925519, size = 32, normalized size = 1. \begin{align*} \frac{1}{6} \, x^{2} - \frac{8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac{1}{14} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3*x^4+x^2-4),x, algorithm="maxima")

[Out]

1/6*x^2 - 8/63*log(3*x^2 + 4) + 1/14*log(x^2 - 1)

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Fricas [A]  time = 2.01062, size = 69, normalized size = 2.16 \begin{align*} \frac{1}{6} \, x^{2} - \frac{8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac{1}{14} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3*x^4+x^2-4),x, algorithm="fricas")

[Out]

1/6*x^2 - 8/63*log(3*x^2 + 4) + 1/14*log(x^2 - 1)

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Sympy [A]  time = 0.107478, size = 24, normalized size = 0.75 \begin{align*} \frac{x^{2}}{6} + \frac{\log{\left (x^{2} - 1 \right )}}{14} - \frac{8 \log{\left (x^{2} + \frac{4}{3} \right )}}{63} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(3*x**4+x**2-4),x)

[Out]

x**2/6 + log(x**2 - 1)/14 - 8*log(x**2 + 4/3)/63

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Giac [A]  time = 1.05862, size = 34, normalized size = 1.06 \begin{align*} \frac{1}{6} \, x^{2} - \frac{8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac{1}{14} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3*x^4+x^2-4),x, algorithm="giac")

[Out]

1/6*x^2 - 8/63*log(3*x^2 + 4) + 1/14*log(abs(x^2 - 1))