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

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 ) \end {gather*}

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 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]

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 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 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]

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 = 0.08, size = 74, normalized size = 1.10 \begin {gather*} \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}} \end {gather*}

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]]

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 1.13, size = 184, normalized size = 2.75 \begin {gather*} \frac {\left (\sqrt {\sqrt {x}-1}-1\right ) \left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right ) \left (\sqrt {\sqrt {x}-1}+\sqrt {3} \sqrt {\sqrt {x}+1}-\sqrt {x}-2\right )}{\left (\sqrt {3} \sqrt {\sqrt {x}+1} \sqrt {\sqrt {x}-1}-2 \sqrt {\sqrt {x}-1}+2 \sqrt {3} \sqrt {\sqrt {x}+1}-2 \sqrt {x}-3\right ) \sqrt {x}}-8 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}-1}{\sqrt {3}-\sqrt {\sqrt {x}+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + Sqrt[-1 + Sqrt[x]])*(Sqrt[3] - Sqrt[1 + Sqrt[x]])*(-2 + Sqrt[-1 + Sqrt[x]] + Sqrt[3]*Sqrt[1 + Sqrt[x]]
- Sqrt[x]))/((-3 - 2*Sqrt[-1 + Sqrt[x]] + 2*Sqrt[3]*Sqrt[1 + Sqrt[x]] + Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sq
rt[x]] - 2*Sqrt[x])*Sqrt[x]) - 8*ArcTanh[(-1 + Sqrt[-1 + Sqrt[x]])/(Sqrt[3] - Sqrt[1 + Sqrt[x]])]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 55, normalized size = 0.82 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.05, size = 47, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (\sqrt {x}\, \ln \left (\sqrt {x}+\sqrt {x -1}\right )-\sqrt {x -1}\right )}{\sqrt {x -1}\, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.28, size = 27, normalized size = 0.40 \begin {gather*} -\frac {2 \, \sqrt {x - 1}}{\sqrt {x}} + 2 \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}

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))

________________________________________________________________________________________

mupad [B]  time = 6.25, size = 129, normalized size = 1.93 \begin {gather*} 8\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )-\frac {\frac {5\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}+\frac {1}{2}}{\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}}-\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{2\,\left (\sqrt {\sqrt {x}+1}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1)) - ((5*((x^(1/2) - 1)^(1/2) - 1i)^2)/(2*((x^(1/2)
+ 1)^(1/2) - 1)^2) + 1/2)/(((x^(1/2) - 1)^(1/2) - 1i)^3/((x^(1/2) + 1)^(1/2) - 1)^3 + ((x^(1/2) - 1)^(1/2) -
1i)/((x^(1/2) + 1)^(1/2) - 1)) - ((x^(1/2) - 1)^(1/2) - 1i)/(2*((x^(1/2) + 1)^(1/2) - 1))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________