3.3068 \(\int \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx\)

Optimal. Leaf size=26 \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]

[Out]

(4*(Sqrt[x^(-1)] + x^(-1))^(3/2))/(3*(x^(-1))^(3/2))

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Rubi [A]  time = 0.0259805, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1966, 2014} \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sqrt[x^(-1)] + x^(-1)],x]

[Out]

(4*(Sqrt[x^(-1)] + x^(-1))^(3/2))/(3*(x^(-1))^(3/2))

Rule 1966

Int[((a_.) + (c_.)*((d_.)/(x_))^(n2_.) + (b_.)*((d_.)/(x_))^(n_))^(p_.), x_Symbol] :> -Dist[d, Subst[Int[(a +
b*x^n + c*x^(2*n))^p/x^2, x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, 2*n]

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{\sqrt{x}+x}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0175556, size = 26, normalized size = 1. \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sqrt[x^(-1)] + x^(-1)],x]

[Out]

(4*(Sqrt[x^(-1)] + x^(-1))^(3/2))/(3*(x^(-1))^(3/2))

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Maple [A]  time = 0.104, size = 32, normalized size = 1.2 \begin{align*}{\frac{4}{3}\sqrt{{\frac{1}{x} \left ( \sqrt{{x}^{-1}}x+1 \right ) }} \left ( \sqrt{{x}^{-1}}x+1 \right ){\frac{1}{\sqrt{{x}^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(((1/x)^(1/2)*x+1)/x)^(1/2)*((1/x)^(1/2)*x+1)/(1/x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{1}{\sqrt{x}} + \frac{1}{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(1/sqrt(x) + 1/x), x)

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Fricas [A]  time = 2.44551, size = 55, normalized size = 2.12 \begin{align*} \frac{4}{3} \,{\left (x + \sqrt{x}\right )} \sqrt{\frac{\sqrt{x} + 1}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(x + sqrt(x))*sqrt((sqrt(x) + 1)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{\frac{1}{x}} + \frac{1}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sqrt(1/x) + 1/x), x)

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Giac [A]  time = 1.09235, size = 15, normalized size = 0.58 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - \frac{4}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2) - 4/3