3.985 \(\int \frac{x}{(a+b x^2+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{b+2 c x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

[Out]

-((b + 2*c*x^2)/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]))

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Rubi [A]  time = 0.0231567, antiderivative size = 36, 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 = {1107, 613} \[ -\frac{b+2 c x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

-((b + 2*c*x^2)/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]))

Rule 1107

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

Rule 613

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

Rubi steps

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

Mathematica [A]  time = 0.0212924, size = 37, normalized size = 1.03 \[ \frac{b+2 c x^2}{\left (4 a c-b^2\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(b + 2*c*x^2)/((-b^2 + 4*a*c)*Sqrt[a + b*x^2 + c*x^4])

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Maple [A]  time = 0.046, size = 36, normalized size = 1. \begin{align*}{\frac{2\,c{x}^{2}+b}{4\,ac-{b}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

(2*c*x^2+b)/(c*x^4+b*x^2+a)^(1/2)/(4*a*c-b^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.70216, size = 140, normalized size = 3.89 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x/(a + b*x**2 + c*x**4)**(3/2), x)

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Giac [A]  time = 1.32656, size = 61, normalized size = 1.69 \begin{align*} -\frac{\frac{2 \, c x^{2}}{b^{2} - 4 \, a c} + \frac{b}{b^{2} - 4 \, a c}}{\sqrt{c x^{4} + b x^{2} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c*x^2/(b^2 - 4*a*c) + b/(b^2 - 4*a*c))/sqrt(c*x^4 + b*x^2 + a)