3.2093 \(\int \frac{1}{(a+\frac{b}{x^4})^{3/2} x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0066597, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^4)^(3/2)*x^3),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^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]

Rubi steps

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

Mathematica [A]  time = 0.0065657, size = 21, normalized size = 1. \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^4)^(3/2)*x^3),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 29, normalized size = 1.4 \begin{align*} -{\frac{a{x}^{4}+b}{2\,{x}^{6}a} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^4)^(3/2)/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.960122, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{2 \, \sqrt{a + \frac{b}{x^{4}}} a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2)/x^3,x, algorithm="maxima")

[Out]

-1/2/(sqrt(a + b/x^4)*a*x^2)

________________________________________________________________________________________

Fricas [A]  time = 1.46497, size = 65, normalized size = 3.1 \begin{align*} -\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \,{\left (a^{2} x^{4} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2)/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.11494, size = 22, normalized size = 1.05 \begin{align*} - \frac{1}{2 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**4)**(3/2)/x**3,x)

[Out]

-1/(2*a*sqrt(b)*sqrt(a*x**4/b + 1))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2)/x^3,x, algorithm="giac")

[Out]

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