3.1781 \(\int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx\)

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

[Out]

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

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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} \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}}}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*Sqrt[x]),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

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}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx &=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x}}{a}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*Sqrt[x]),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

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fricas [A]  time = 0.58, size = 19, normalized size = 0.90 \[ \frac {2 \, \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*sqrt((a*x + b)/x)/a

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giac [A]  time = 0.18, size = 21, normalized size = 1.00 \[ \frac {2 \, \sqrt {a x + b}}{a} - \frac {2 \, \sqrt {b}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*x + b)/a - 2*sqrt(b)/a

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maple [A]  time = 0.00, size = 25, normalized size = 1.19 \[ \frac {2 a x +2 b}{a \sqrt {\frac {a x +b}{x}}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 17, normalized size = 0.81 \[ \frac {2 \, \sqrt {a + \frac {b}{x}} \sqrt {x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a + b/x)*sqrt(x)/a

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mupad [B]  time = 1.41, size = 17, normalized size = 0.81 \[ \frac {2\,\sqrt {x}\,\sqrt {a+\frac {b}{x}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b/x)^(1/2)),x)

[Out]

(2*x^(1/2)*(a + b/x)^(1/2))/a

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sympy [A]  time = 1.72, size = 17, normalized size = 0.81 \[ \frac {2 \sqrt {b} \sqrt {\frac {a x}{b} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b)*sqrt(a*x/b + 1)/a

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