3.1004 \(\int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x} \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0324258, antiderivative size = 73, 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 = {280, 323, 330, 52} \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x],x]

[Out]

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

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 323

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(2
*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a
1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 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] && GtQ[m, 2*n - 1] && 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 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x} \, dx &=\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{4} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{8} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=-\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{4} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}-\frac{1}{4} \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0298912, size = 87, normalized size = 1.19 \[ \frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left (2 x^{3/2}-2 x-\sqrt{x}+1\right )+2 \sqrt{1-\sqrt{x}} \sin ^{-1}\left (\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right )}{4 \sqrt{\sqrt{x}-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.007, size = 52, normalized size = 0.7 \begin{align*} -{\frac{1}{4}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( -2\,{x}^{3/2}\sqrt{-1+x}+\sqrt{x}\sqrt{-1+x}+\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.920499, size = 50, normalized size = 0.68 \begin{align*} \frac{1}{2} \,{\left (x - 1\right )}^{\frac{3}{2}} \sqrt{x} + \frac{1}{4} \, \sqrt{x - 1} \sqrt{x} - \frac{1}{4} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x - 1)^(3/2)*sqrt(x) + 1/4*sqrt(x - 1)*sqrt(x) - 1/4*log(2*sqrt(x - 1) + 2*sqrt(x))

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Fricas [A]  time = 0.982566, size = 170, normalized size = 2.33 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \frac{1}{8} \, \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 1/8*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) -
1) - 2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError