3.2091 \(\int \frac {x}{(a+\frac {b}{x^4})^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac {x^2 \sqrt {a+\frac {b}{x^4}}}{a^2}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {273, 264} \[ \frac {x^2 \sqrt {a+\frac {b}{x^4}}}{a^2}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^4)^(3/2),x]

[Out]

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

Rule 264

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

Rule 273

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 30, normalized size = 0.75 \[ \frac {a x^4+2 b}{2 a^2 x^2 \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^4)^(3/2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 42, normalized size = 1.05 \[ \frac {{\left (a x^{6} + 2 \, b x^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{2 \, {\left (a^{3} x^{4} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^6 + 2*b*x^2)*sqrt((a*x^4 + b)/x^4)/(a^3*x^4 + a^2*b)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 42, normalized size = 1.05 \[ \frac {\frac {\sqrt {a x^{4} + b}}{a} + \frac {b}{\sqrt {a x^{4} + b} a}}{2 \, a} - \frac {\sqrt {b}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 38, normalized size = 0.95 \[ \frac {\left (a \,x^{4}+b \right ) \left (a \,x^{4}+2 b \right )}{2 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^4+b)*(a*x^4+2*b)/a^2/x^6/((a*x^4+b)/x^4)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 0.82, size = 36, normalized size = 0.90 \[ \frac {\sqrt {a + \frac {b}{x^{4}}} x^{2}}{2 \, a^{2}} + \frac {b}{2 \, \sqrt {a + \frac {b}{x^{4}}} a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.39, size = 24, normalized size = 0.60 \[ \frac {\frac {a\,x^4}{2}+b}{a^2\,x^2\,\sqrt {a+\frac {b}{x^4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x^4)^(3/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.21, size = 42, normalized size = 1.05 \[ \frac {x^{4}}{2 a \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} + \frac {\sqrt {b}}{a^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/(2*a*sqrt(b)*sqrt(a*x**4/b + 1)) + sqrt(b)/(a**2*sqrt(a*x**4/b + 1))

________________________________________________________________________________________