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

Optimal. Leaf size=28 \[ -\frac{b+2 c x^2}{b^2 \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0464225, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2013, 613} \[ -\frac{b+2 c x^2}{b^2 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

Rule 2013

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[(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]] && EqQ[Simplify[m - n + 1], 0]

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

Mathematica [A]  time = 0.0093953, size = 29, normalized size = 1.04 \[ \frac{-b-2 c x^2}{b^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.044, size = 37, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2} \left ( c{x}^{2}+b \right ) \left ( 2\,c{x}^{2}+b \right ) }{{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0129, size = 55, normalized size = 1.96 \begin{align*} -\frac{2 \, c x^{2}}{\sqrt{c x^{4} + b x^{2}} b^{2}} - \frac{1}{\sqrt{c x^{4} + b x^{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.21007, size = 78, normalized size = 2.79 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}{\left (2 \, c x^{2} + b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.208, size = 38, normalized size = 1.36 \begin{align*} -\frac{\frac{2 \, c x^{2}}{b^{2}} + \frac{1}{b}}{\sqrt{c x^{4} + b x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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