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]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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]

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

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fricas [B]  time = 1.02, size = 39, normalized size = 2.17 \[ -\frac {{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{5 \, b x^{4}} \]

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)

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giac [B]  time = 0.22, size = 92, normalized size = 5.11 \[ \frac {2 \, {\left (5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{8} a^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 10 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {5}{2}} b^{2} \mathrm {sgn}\relax (x) + a^{\frac {5}{2}} b^{4} \mathrm {sgn}\relax (x)\right )}}{5 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{5}} \]

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

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maple [A]  time = 0.00, size = 29, normalized size = 1.61 \[ -\frac {\left (a \,x^{2}+b \right ) \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}{5 b \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.86, size = 14, normalized size = 0.78 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, b} \]

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

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mupad [B]  time = 1.46, size = 26, normalized size = 1.44 \[ -\frac {\sqrt {a+\frac {b}{x^2}}\,{\left (a\,x^2+b\right )}^2}{5\,b\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)^(3/2)/x^3,x)

[Out]

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

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sympy [B]  time = 1.18, size = 68, normalized size = 3.78 \[ - \frac {a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x^{2}}}}{5 b} - \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{2}}}}{5 x^{2}} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{2}}}}{5 x^{4}} \]

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)

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