3.1947 \(\int \frac{1}{(a+\frac{b}{x^2})^{5/2} x^3} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

1/(3*b*(a + b/x^2)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0061118, antiderivative size = 18, 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 = {261} \[ \frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(5/2)*x^3),x]

[Out]

1/(3*b*(a + b/x^2)^(3/2))

Rule 261

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^3} \, dx &=\frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0091765, size = 28, normalized size = 1.56 \[ \frac{a x^2+b}{3 b x^2 \left (a+\frac{b}{x^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(5/2)*x^3),x]

[Out]

(b + a*x^2)/(3*b*(a + b/x^2)^(5/2)*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 29, normalized size = 1.6 \begin{align*}{\frac{a{x}^{2}+b}{3\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^(5/2)/x^3,x)

[Out]

1/3*(a*x^2+b)/x^2/b/((a*x^2+b)/x^2)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.02597, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^3,x, algorithm="maxima")

[Out]

1/3/((a + b/x^2)^(3/2)*b)

________________________________________________________________________________________

Fricas [B]  time = 1.52651, size = 85, normalized size = 4.72 \begin{align*} \frac{x^{4} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 3.93812, size = 48, normalized size = 2.67 \begin{align*} \begin{cases} \frac{1}{3 a b \sqrt{a + \frac{b}{x^{2}}} + \frac{3 b^{2} \sqrt{a + \frac{b}{x^{2}}}}{x^{2}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(5/2)/x**3,x)

[Out]

Piecewise((1/(3*a*b*sqrt(a + b/x**2) + 3*b**2*sqrt(a + b/x**2)/x**2), Ne(b, 0)), (-1/(2*a**(5/2)*x**2), True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^3,x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(5/2)*x^3), x)