3.1010 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{11/2}} \, dx\)

Optimal. Leaf size=125 \[ \frac{32 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{315 x^{3/2}}+\frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{5/2}}+\frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{21 x^{7/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{9 x^{9/2}} \]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(9*x^(9/2)) + (4*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(21*x^
(7/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(105*x^(5/2)) + (32*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^
(3/2))/(315*x^(3/2))

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Rubi [A]  time = 0.041845, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac{32 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{315 x^{3/2}}+\frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{5/2}}+\frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{21 x^{7/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{9 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(9*x^(9/2)) + (4*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(21*x^
(7/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(105*x^(5/2)) + (32*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^
(3/2))/(315*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^{11/2}} \, dx &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{9 x^{9/2}}+\frac{2}{3} \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}}{9 x^{9/2}}+\frac{4 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{21 x^{7/2}}+\frac{8}{21} \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}}{9 x^{9/2}}+\frac{4 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{21 x^{7/2}}+\frac{16 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{105 x^{5/2}}+\frac{16}{105} \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}}{9 x^{9/2}}+\frac{4 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{21 x^{7/2}}+\frac{16 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{105 x^{5/2}}+\frac{32 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{315 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.018402, size = 46, normalized size = 0.37 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2} \left (16 x^3+24 x^2+30 x+35\right )}{315 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(35 + 30*x + 24*x^2 + 16*x^3))/(315*x^(9/2))

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Maple [A]  time = 0.013, size = 38, normalized size = 0.3 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 16\,{x}^{3}+24\,{x}^{2}+30\,x+35 \right ) }{315}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{9}{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^(11/2),x)

[Out]

2/315*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(-1+x)*(16*x^3+24*x^2+30*x+35)/x^(9/2)

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Maxima [A]  time = 1.4097, size = 55, normalized size = 0.44 \begin{align*} \frac{32 \,{\left (x - 1\right )}^{\frac{3}{2}}}{315 \, x^{\frac{3}{2}}} + \frac{16 \,{\left (x - 1\right )}^{\frac{3}{2}}}{105 \, x^{\frac{5}{2}}} + \frac{4 \,{\left (x - 1\right )}^{\frac{3}{2}}}{21 \, x^{\frac{7}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{9 \, x^{\frac{9}{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^(11/2),x, algorithm="maxima")

[Out]

32/315*(x - 1)^(3/2)/x^(3/2) + 16/105*(x - 1)^(3/2)/x^(5/2) + 4/21*(x - 1)^(3/2)/x^(7/2) + 2/9*(x - 1)^(3/2)/x
^(9/2)

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Fricas [A]  time = 0.918929, size = 138, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (16 \, x^{5} +{\left (16 \, x^{4} + 8 \, x^{3} + 6 \, x^{2} + 5 \, x - 35\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}}{315 \, x^{5}} \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^(11/2),x, algorithm="fricas")

[Out]

2/315*(16*x^5 + (16*x^4 + 8*x^3 + 6*x^2 + 5*x - 35)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1))/x^5

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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**(11/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15701, size = 178, normalized size = 1.42 \begin{align*} \frac{16384 \,{\left (315 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{20} - 756 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{16} + 1344 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} + 2304 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 2304 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 1024\right )}}{315 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{9}} \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^(11/2),x, algorithm="giac")

[Out]

16384/315*(315*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^20 - 756*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^16 + 1
344*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 + 2304*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 2304*(sqrt(s
qrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 1024)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^9