3.1017 \(\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{8 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 x^{3/2}}+\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{5 x^{5/2}}+\frac{16 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 \sqrt{x}} \]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(5*x^(5/2)) + (8*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(15*x^(3/2)) +
(16*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(15*Sqrt[x])

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Rubi [A]  time = 0.0333635, 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} \[ \frac{8 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 x^{3/2}}+\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{5 x^{5/2}}+\frac{16 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2)),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(5*x^(5/2)) + (8*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(15*x^(3/2)) +
(16*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(15*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^{7/2}} \, dx &=\frac{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{5 x^{5/2}}+\frac{4}{5} \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}}}{5 x^{5/2}}+\frac{8 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{15 x^{3/2}}+\frac{8}{15} \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}}}{5 x^{5/2}}+\frac{8 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{15 x^{3/2}}+\frac{16 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{15 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.014021, size = 41, normalized size = 0.44 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left (8 x^2+4 x+3\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2)),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(3 + 4*x + 8*x^2))/(15*x^(5/2))

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Maple [A]  time = 0.01, size = 30, normalized size = 0.3 \begin{align*}{\frac{16\,{x}^{2}+8\,x+6}{15}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)

[Out]

2/15*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(8*x^2+4*x+3)/x^(5/2)

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Maxima [A]  time = 1.43081, size = 42, normalized size = 0.45 \begin{align*} \frac{16 \, \sqrt{x - 1}}{15 \, \sqrt{x}} + \frac{8 \, \sqrt{x - 1}}{15 \, x^{\frac{3}{2}}} + \frac{2 \, \sqrt{x - 1}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

16/15*sqrt(x - 1)/sqrt(x) + 8/15*sqrt(x - 1)/x^(3/2) + 2/5*sqrt(x - 1)/x^(5/2)

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Fricas [A]  time = 0.987849, size = 111, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (8 \, x^{3} +{\left (8 \, x^{2} + 4 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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

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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**(7/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.1361, size = 93, normalized size = 0.99 \begin{align*} \frac{4096 \,{\left (5 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 10 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 8\right )}}{15 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="giac")

[Out]

4096/15*(5*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 10*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 8)/((sqr
t(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^5