3.1016 \(\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0201345, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2)),x]

[Out]

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

Rule 272

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

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^{5/2}} \, dx &=\frac{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{3 x^{3/2}}+\frac{2}{3} \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}}}{3 x^{3/2}}+\frac{4 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{3 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.011744, size = 36, normalized size = 0.57 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} (2 x+1)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 25, normalized size = 0.4 \begin{align*}{\frac{4\,x+2}{3}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.40682, size = 28, normalized size = 0.44 \begin{align*} \frac{4 \, \sqrt{x - 1}}{3 \, \sqrt{x}} + \frac{2 \, \sqrt{x - 1}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.00233, size = 99, normalized size = 1.57 \begin{align*} \frac{2 \,{\left ({\left (2 \, x + 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 2 \, x^{2}\right )}}{3 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

Integral(1/(x**(5/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)), x)

________________________________________________________________________________________

Giac [A]  time = 1.06921, size = 65, normalized size = 1.03 \begin{align*} \frac{128 \,{\left (3 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}}{3 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="giac")

[Out]

128/3*(3*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^3