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

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

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Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {4 \left (\sqrt {\frac {1}{x}}+\frac {1}{x}\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \begin {gather*} \frac {4 \left (\sqrt {\frac {1}{x}}+\frac {1}{x}\right )^{3/2}}{3 \left (\frac {1}{x}\right )^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.07, size = 29, normalized size = 1.12 \begin {gather*} \frac {4}{3} \left (\sqrt {\frac {1}{x}}+1\right ) \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 18, normalized size = 0.69 \begin {gather*} \frac {4}{3} \, {\left (x + \sqrt {x}\right )} \sqrt {\frac {\sqrt {x} + 1}{x}} \end {gather*}

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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giac [A]  time = 0.17, size = 11, normalized size = 0.42 \begin {gather*} \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \end {gather*}

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

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maple [A]  time = 0.11, size = 32, normalized size = 1.23 \begin {gather*} \frac {4 \sqrt {\frac {\sqrt {\frac {1}{x}}\, x +1}{x}}\, \left (\sqrt {\frac {1}{x}}\, x +1\right )}{3 \sqrt {\frac {1}{x}}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {1}{\sqrt {x}} + \frac {1}{x}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1/x)^(1/2) + 1/x)^(1/2),x)

[Out]

int(((1/x)^(1/2) + 1/x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {\frac {1}{x}} + \frac {1}{x}}\, dx \end {gather*}

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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