3.1006 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{\sqrt{x}}-2 \sqrt{\sqrt{x}-1} \sqrt{x} \sqrt{\sqrt{x}+1}+2 \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0307674, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {327, 280, 330, 52} \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{\sqrt{x}}-2 \sqrt{\sqrt{x}-1} \sqrt{x} \sqrt{\sqrt{x}+1}+2 \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 327

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] - Dist[(b1*b2*(m + 2*n*(p + 1) +
1))/(a1*a2*c^(2*n)*(m + 1)), Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2,
b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[m, -1] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m,
p, x]

Rule 280

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*(a2 + b2*x^n)^p)/(c*(m + 2*n*p + 1)), x] + Dist[(2*a1*a2*n*p)/(m + 2*n*p + 1), Int[
(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 +
a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x
]

Rule 330

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x]
, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.0385511, size = 74, normalized size = 1.1 \[ \frac{2 \left (-\frac{\sqrt{\sqrt{x}+1} \left (\sqrt{x}-1\right )}{\sqrt{x}}-2 \sqrt{1-\sqrt{x}} \sin ^{-1}\left (\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right )\right )}{\sqrt{\sqrt{x}-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 47, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( \ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \sqrt{x}-\sqrt{-1+x} \right ) }{\sqrt{x}\sqrt{-1+x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39772, size = 36, normalized size = 0.54 \begin{align*} -\frac{2 \, \sqrt{x - 1}}{\sqrt{x}} + 2 \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.01254, size = 166, normalized size = 2.48 \begin{align*} -\frac{x \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) + 2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 2 \, x}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{x^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 3.15214, size = 65, normalized size = 0.97 \begin{align*} -\frac{16}{{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4} - \log \left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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