3.1790 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^{3/2}} \, dx\)

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

[Out]

-2/(a*Sqrt[a + b/x]*Sqrt[x])

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Rubi [A]  time = 0.0056761, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^(3/2)),x]

[Out]

-2/(a*Sqrt[a + b/x]*Sqrt[x])

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

Mathematica [A]  time = 0.0128592, size = 21, normalized size = 1. \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a*Sqrt[a + b/x]*Sqrt[x])

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Maple [A]  time = 0.002, size = 25, normalized size = 1.2 \begin{align*} -2\,{\frac{ax+b}{a{x}^{3/2}} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964417, size = 23, normalized size = 1.1 \begin{align*} -\frac{2}{\sqrt{a + \frac{b}{x}} a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(a + b/x)*a*sqrt(x))

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Fricas [A]  time = 1.44867, size = 59, normalized size = 2.81 \begin{align*} -\frac{2 \, \sqrt{x} \sqrt{\frac{a x + b}{x}}}{a^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 15.5344, size = 19, normalized size = 0.9 \begin{align*} - \frac{2}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*sqrt(b)*sqrt(a*x/b + 1))

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Giac [A]  time = 1.2211, size = 28, normalized size = 1.33 \begin{align*} -\frac{2}{\sqrt{a x + b} a} + \frac{2}{a \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(a*x + b)*a) + 2/(a*sqrt(b))