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

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

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 37, normalized size = 1.03 \begin {gather*} \frac {b+2 c x^2}{\left (4 a c-b^2\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.31, size = 37, normalized size = 1.03 \begin {gather*} \frac {-b-2 c x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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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fricas [A]  time = 1.08, size = 67, normalized size = 1.86 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.20, size = 45, normalized size = 1.25 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 36, normalized size = 1.00 \begin {gather*} \frac {2 c \,x^{2}+b}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 4.36, size = 35, normalized size = 0.97 \begin {gather*} \frac {2\,c\,x^2+b}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2 + c*x^4)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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