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

Optimal. Leaf size=38 \[ -\frac {1}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1107, 607} \begin {gather*} -\frac {1}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a+b x^2}{4 b \left (\left (a+b x^2\right )^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a + b*x^2)/(b*((a + b*x^2)^2)^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.52, size = 137, normalized size = 3.61 \begin {gather*} \frac {\sqrt {b^2} \left (a-b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}+a^2 b+b^3 x^4}{2 b \sqrt {b^2} x^4 \left (2 a^2 b^2+4 a b^3 x^2+2 b^4 x^4\right )+2 b x^4 \left (-2 a b^3-2 b^4 x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*b + b^3*x^4 + Sqrt[b^2]*(a - b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*b*x^4*(-2*a*b^3 - 2*b^4*x^2)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4] + 2*b*Sqrt[b^2]*x^4*(2*a^2*b^2 + 4*a*b^3*x^2 + 2*b^4*x^4))

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 26, normalized size = 0.68 \begin {gather*} -\frac {1}{4 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(b^3*x^4 + 2*a*b^2*x^2 + a^2*b)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 24, normalized size = 0.63 \begin {gather*} -\frac {1}{4 \, {\left (b x^{2} + a\right )}^{2} b \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/((b*x^2 + a)^2*b*sgn(b*x^2 + a))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 24, normalized size = 0.63 \begin {gather*} -\frac {b \,x^{2}+a}{4 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b*x^2+a)/b/((b*x^2+a)^2)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 26, normalized size = 0.68 \begin {gather*} -\frac {1}{4 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(b^3*x^4 + 2*a*b^2*x^2 + a^2*b)

________________________________________________________________________________________

mupad [B]  time = 4.34, size = 34, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,b\,{\left (b\,x^2+a\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2)/(4*b*(a + b*x^2)^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________