3.1015 \(\int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {265} \[ \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 265

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*
x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*c*(m + 1)), x] /; FreeQ[{a1, b1, a2, b2, c, m,
n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx &=\frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 25, normalized size = 0.86 \[ \frac {2 \, {\left (\sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + x\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + x)/x

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giac [A]  time = 0.22, size = 25, normalized size = 0.86 \[ \frac {16}{{\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)

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maple [A]  time = 0.06, size = 20, normalized size = 0.69 \[ \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(x^(1/2)-1)^(1/2)/(x^(1/2)+1)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.21, size = 10, normalized size = 0.34 \[ \frac {2 \, \sqrt {x - 1}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x - 1)/sqrt(x)

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mupad [B]  time = 5.56, size = 19, normalized size = 0.66 \[ \frac {2\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {3}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

Integral(1/(x**(3/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)), x)

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