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

Optimal. Leaf size=25 \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.0191952, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1247, 636} \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{x \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{2+3 x}{\left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{8+11 x^2}{13 \sqrt{3+5 x^2+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0954649, size = 25, normalized size = 1. \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

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Maple [A]  time = 0.005, size = 22, normalized size = 0.9 \begin{align*}{\frac{11\,{x}^{2}+8}{13}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*(11*x^2+8)/(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.952234, size = 43, normalized size = 1.72 \begin{align*} \frac{11 \, x^{2}}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} + \frac{8}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/13*x^2/sqrt(x^4 + 5*x^2 + 3) + 8/13/sqrt(x^4 + 5*x^2 + 3)

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Fricas [B]  time = 1.33069, size = 113, normalized size = 4.52 \begin{align*} \frac{11 \, x^{4} + 55 \, x^{2} + \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (11 \, x^{2} + 8\right )} + 33}{13 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*(11*x^4 + 55*x^2 + sqrt(x^4 + 5*x^2 + 3)*(11*x^2 + 8) + 33)/(x^4 + 5*x^2 + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(3*x**2 + 2)/(x**4 + 5*x**2 + 3)**(3/2), x)

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Giac [A]  time = 1.16028, size = 28, normalized size = 1.12 \begin{align*} \frac{11 \, x^{2} + 8}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*(11*x^2 + 8)/sqrt(x^4 + 5*x^2 + 3)