3.1009 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{9/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{3/2}}+\frac{8 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{35 x^{5/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{7 x^{7/2}} \]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(7*x^(7/2)) + (8*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(35*x^
(5/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(105*x^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0312135, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{3/2}}+\frac{8 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{35 x^{5/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(7*x^(7/2)) + (8*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(35*x^
(5/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(105*x^(3/2))

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{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{9/2}} \, dx &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{7 x^{7/2}}+\frac{4}{7} \int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{7/2}} \, dx\\ &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{7 x^{7/2}}+\frac{8 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{35 x^{5/2}}+\frac{8}{35} \int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{5/2}} \, dx\\ &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{7 x^{7/2}}+\frac{8 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{35 x^{5/2}}+\frac{16 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{105 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0162812, size = 41, normalized size = 0.44 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2} \left (8 x^2+12 x+15\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(15 + 12*x + 8*x^2))/(105*x^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 33, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 8\,{x}^{2}+12\,x+15 \right ) }{105}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{7}{2}}}} \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^(9/2),x)

[Out]

2/105*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(-1+x)*(8*x^2+12*x+15)/x^(7/2)

________________________________________________________________________________________

Maxima [A]  time = 1.40464, size = 42, normalized size = 0.45 \begin{align*} \frac{16 \,{\left (x - 1\right )}^{\frac{3}{2}}}{105 \, x^{\frac{3}{2}}} + \frac{8 \,{\left (x - 1\right )}^{\frac{3}{2}}}{35 \, x^{\frac{5}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{7 \, x^{\frac{7}{2}}} \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^(9/2),x, algorithm="maxima")

[Out]

16/105*(x - 1)^(3/2)/x^(3/2) + 8/35*(x - 1)^(3/2)/x^(5/2) + 2/7*(x - 1)^(3/2)/x^(7/2)

________________________________________________________________________________________

Fricas [A]  time = 0.971025, size = 124, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (8 \, x^{4} +{\left (8 \, x^{3} + 4 \, x^{2} + 3 \, x - 15\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}}{105 \, x^{4}} \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^(9/2),x, algorithm="fricas")

[Out]

2/105*(8*x^4 + (8*x^3 + 4*x^2 + 3*x - 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1))/x^4

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.12485, size = 150, normalized size = 1.6 \begin{align*} \frac{4096 \,{\left (35 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{16} - 70 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} + 168 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 224 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 128\right )}}{105 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{7}} \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^(9/2),x, algorithm="giac")

[Out]

4096/105*(35*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^16 - 70*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 + 168*
(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 224*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 128)/((sqrt(sqrt(x
) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^7