3.288 \(\int \frac{x^3}{\sqrt{3 x^2+4 x^4}} \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{8} \sqrt{4 x^4+3 x^2}-\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4+3 x^2}}\right ) \]

[Out]

Sqrt[3*x^2 + 4*x^4]/8 - (3*ArcTanh[(2*x^2)/Sqrt[3*x^2 + 4*x^4]])/16

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Rubi [A]  time = 0.0561752, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2018, 640, 620, 206} \[ \frac{1}{8} \sqrt{4 x^4+3 x^2}-\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4+3 x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[3*x^2 + 4*x^4],x]

[Out]

Sqrt[3*x^2 + 4*x^4]/8 - (3*ArcTanh[(2*x^2)/Sqrt[3*x^2 + 4*x^4]])/16

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 640

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

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{\sqrt{3 x^2+4 x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{3 x+4 x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{8} \sqrt{3 x^2+4 x^4}-\frac{3}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3 x+4 x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{8} \sqrt{3 x^2+4 x^4}-\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1-4 x^2} \, dx,x,\frac{x^2}{\sqrt{3 x^2+4 x^4}}\right )\\ &=\frac{1}{8} \sqrt{3 x^2+4 x^4}-\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{3 x^2+4 x^4}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0097691, size = 51, normalized size = 1.13 \[ \frac{x \left (8 x^3-3 \sqrt{4 x^2+3} \sinh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )+6 x\right )}{16 \sqrt{x^2 \left (4 x^2+3\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[3*x^2 + 4*x^4],x]

[Out]

(x*(6*x + 8*x^3 - 3*Sqrt[3 + 4*x^2]*ArcSinh[(2*x)/Sqrt[3]]))/(16*Sqrt[x^2*(3 + 4*x^2)])

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Maple [A]  time = 0.046, size = 48, normalized size = 1.1 \begin{align*} -{\frac{x}{16}\sqrt{4\,{x}^{2}+3} \left ( -2\,x\sqrt{4\,{x}^{2}+3}+3\,{\it Arcsinh} \left ( 2/3\,x\sqrt{3} \right ) \right ){\frac{1}{\sqrt{4\,{x}^{4}+3\,{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(4*x^4+3*x^2)^(1/2),x)

[Out]

-1/16*x*(4*x^2+3)^(1/2)*(-2*x*(4*x^2+3)^(1/2)+3*arcsinh(2/3*x*3^(1/2)))/(4*x^4+3*x^2)^(1/2)

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Maxima [A]  time = 1.45947, size = 55, normalized size = 1.22 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} - \frac{3}{32} \, \log \left (8 \, x^{2} + 4 \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(4*x^4+3*x^2)^(1/2),x, algorithm="maxima")

[Out]

1/8*sqrt(4*x^4 + 3*x^2) - 3/32*log(8*x^2 + 4*sqrt(4*x^4 + 3*x^2) + 3)

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Fricas [A]  time = 1.23217, size = 95, normalized size = 2.11 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + \frac{3}{16} \, \log \left (-\frac{2 \, x^{2} - \sqrt{4 \, x^{4} + 3 \, x^{2}}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(4*x^4+3*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(4*x^4 + 3*x^2) + 3/16*log(-(2*x^2 - sqrt(4*x^4 + 3*x^2))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{x^{2} \left (4 x^{2} + 3\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(4*x**4+3*x**2)**(1/2),x)

[Out]

Integral(x**3/sqrt(x**2*(4*x**2 + 3)), x)

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Giac [A]  time = 1.1693, size = 55, normalized size = 1.22 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + \frac{3}{32} \, \log \left (8 \, x^{2} - 4 \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(4*x^4+3*x^2)^(1/2),x, algorithm="giac")

[Out]

1/8*sqrt(4*x^4 + 3*x^2) + 3/32*log(8*x^2 - 4*sqrt(4*x^4 + 3*x^2) + 3)