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

Optimal. Leaf size=20 \[ -\frac{15-2 x^2}{10 \sqrt{x^4+5}} \]

[Out]

-(15 - 2*x^2)/(10*Sqrt[5 + x^4])

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Rubi [A]  time = 0.0156476, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1248, 637} \[ -\frac{15-2 x^2}{10 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(15 - 2*x^2)/(10*Sqrt[5 + x^4])

Rule 1248

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

Rule 637

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

Rubi steps

\begin{align*} \int \frac{x \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{2+3 x}{\left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{15-2 x^2}{10 \sqrt{5+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0099904, size = 20, normalized size = 1. \[ \frac{2 x^2-15}{10 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15 + 2*x^2)/(10*Sqrt[5 + x^4])

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Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \begin{align*}{\frac{2\,{x}^{2}-15}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*x^2-15)/(x^4+5)^(1/2)

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Maxima [A]  time = 1.41095, size = 30, normalized size = 1.5 \begin{align*} \frac{x^{2}}{5 \, \sqrt{x^{4} + 5}} - \frac{3}{2 \, \sqrt{x^{4} + 5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^2/sqrt(x^4 + 5) - 3/2/sqrt(x^4 + 5)

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Fricas [A]  time = 1.50904, size = 78, normalized size = 3.9 \begin{align*} \frac{2 \, x^{4} + \sqrt{x^{4} + 5}{\left (2 \, x^{2} - 15\right )} + 10}{10 \,{\left (x^{4} + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*x^4 + sqrt(x^4 + 5)*(2*x^2 - 15) + 10)/(x^4 + 5)

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Sympy [B]  time = 4.29222, size = 31, normalized size = 1.55 \begin{align*} \frac{\sqrt{5} x^{2}}{5 \sqrt{5 x^{4} + 25}} - \frac{3}{2 \sqrt{x^{4} + 5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(5)*x**2/(5*sqrt(5*x**4 + 25)) - 3/(2*sqrt(x**4 + 5))

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Giac [A]  time = 1.13046, size = 22, normalized size = 1.1 \begin{align*} \frac{2 \, x^{2} - 15}{10 \, \sqrt{x^{4} + 5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*x^2 - 15)/sqrt(x^4 + 5)