3.1905 \(\int \frac{(a+\frac{b}{x^2})^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b} \]

[Out]

-(a + b/x^2)^(5/2)/(5*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0060208, 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} \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(3/2)/x^3,x]

[Out]

-(a + b/x^2)^(5/2)/(5*b)

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

Mathematica [A]  time = 0.009633, size = 28, normalized size = 1.56 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2} \left (a x^2+b\right )}{5 b x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(3/2)/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{a{x}^{2}+b}{5\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.989785, size = 19, normalized size = 1.06 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(a + b/x^2)^(5/2)/b

________________________________________________________________________________________

Fricas [B]  time = 1.50641, size = 86, normalized size = 4.78 \begin{align*} -\frac{{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \, b x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(a^2*x^4 + 2*a*b*x^2 + b^2)*sqrt((a*x^2 + b)/x^2)/(b*x^4)

________________________________________________________________________________________

Sympy [B]  time = 1.13738, size = 68, normalized size = 3.78 \begin{align*} - \frac{a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{5 b} - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{2}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**(5/2)*sqrt(1 + b/(a*x**2))/(5*b) - 2*a**(3/2)*sqrt(1 + b/(a*x**2))/(5*x**2) - sqrt(a)*b*sqrt(1 + b/(a*x**2
))/(5*x**4)

________________________________________________________________________________________

Giac [B]  time = 1.21479, size = 124, normalized size = 6.89 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{5}{2}} b^{2} \mathrm{sgn}\left (x\right ) + a^{\frac{5}{2}} b^{4} \mathrm{sgn}\left (x\right )\right )}}{5 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(5*(sqrt(a)*x - sqrt(a*x^2 + b))^8*a^(5/2)*sgn(x) + 10*(sqrt(a)*x - sqrt(a*x^2 + b))^4*a^(5/2)*b^2*sgn(x)
+ a^(5/2)*b^4*sgn(x))/((sqrt(a)*x - sqrt(a*x^2 + b))^2 - b)^5