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/(a+b/x)^(1/2)/x^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {2}{a \sqrt {x} \sqrt {a+\frac {b}{x}}} \]

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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fricas [A]  time = 0.78, size = 27, normalized size = 1.29 \[ -\frac {2 \, \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a^{2} x + a b} \]

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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giac [A]  time = 0.17, size = 21, normalized size = 1.00 \[ -\frac {2}{\sqrt {a x + b} a} + \frac {2}{a \sqrt {b}} \]

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))

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maple [A]  time = 0.00, size = 25, normalized size = 1.19 \[ -\frac {2 \left (a x +b \right )}{\left (\frac {a x +b}{x}\right )^{\frac {3}{2}} a \,x^{\frac {3}{2}}} \]

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 = 1.10, size = 17, normalized size = 0.81 \[ -\frac {2}{\sqrt {a + \frac {b}{x}} a \sqrt {x}} \]

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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mupad [B]  time = 1.43, size = 25, normalized size = 1.19 \[ -\frac {2\,\sqrt {x}\,\sqrt {a+\frac {b}{x}}}{x\,a^2+b\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a + b/x)^(3/2)),x)

[Out]

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

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sympy [A]  time = 5.40, size = 19, normalized size = 0.90 \[ - \frac {2}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} \]

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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