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

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

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Rubi [A]  time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 41, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \left (8 x^2+4 x+3\right )}{15 x^{5/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 1.37, size = 127, normalized size = 1.35 \begin {gather*} \frac {4 \left (\frac {15 \left (\sqrt {x}-1\right )^4}{\left (\sqrt {x}+1\right )^4}+\frac {20 \left (\sqrt {x}-1\right )^3}{\left (\sqrt {x}+1\right )^3}+\frac {58 \left (\sqrt {x}-1\right )^2}{\left (\sqrt {x}+1\right )^2}+\frac {20 \left (\sqrt {x}-1\right )}{\sqrt {x}+1}+15\right ) \sqrt {\sqrt {x}-1}}{15 \left (\frac {\sqrt {x}-1}{\sqrt {x}+1}+1\right )^5 \sqrt {\sqrt {x}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(15 + (15*(-1 + Sqrt[x])^4)/(1 + Sqrt[x])^4 + (20*(-1 + Sqrt[x])^3)/(1 + Sqrt[x])^3 + (58*(-1 + Sqrt[x])^2)
/(1 + Sqrt[x])^2 + (20*(-1 + Sqrt[x]))/(1 + Sqrt[x]))*Sqrt[-1 + Sqrt[x]])/(15*(1 + (-1 + Sqrt[x])/(1 + Sqrt[x]
))^5*Sqrt[1 + Sqrt[x]])

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fricas [A]  time = 0.41, size = 39, normalized size = 0.41 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.26, size = 69, normalized size = 0.73 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.06, size = 30, normalized size = 0.32 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (8 x^{2}+4 x +3\right )}{15 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 31, normalized size = 0.33 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 5.66, size = 43, normalized size = 0.46 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {8\,x}{15}+\frac {16\,x^2}{15}+\frac {2\,\sqrt {x}}{5}+\frac {8\,x^{3/2}}{15}+\frac {16\,x^{5/2}}{15}+\frac {2}{5}\right )}{x^{5/2}\,\sqrt {\sqrt {x}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {7}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \end {gather*}

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]

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

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