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

Optimal. Leaf size=63 \[ \frac {4 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{15 x^{3/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{5 x^{5/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {4 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{15 x^{3/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{5 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 {\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}}{5 x^{5/2}}+\frac {2}{5} \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}}{5 x^{5/2}}+\frac {4 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{15 x^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 36, normalized size = 0.57 \begin {gather*} \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2} (2 x+3)}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 10.09, size = 836, normalized size = 13.27 \begin {gather*} \frac {\left (\frac {\left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+1\right ) \left (\frac {45 \left (\sqrt {\sqrt {x}-1}-1\right )^{16}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{16}}+\frac {300 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{15}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{15}}+\frac {2960 \left (\sqrt {\sqrt {x}-1}-1\right )^{14}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{14}}+\frac {6620 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{13}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{13}}+\frac {34252 \left (\sqrt {\sqrt {x}-1}-1\right )^{12}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{12}}+\frac {46980 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{11}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{11}}+\frac {152688 \left (\sqrt {\sqrt {x}-1}-1\right )^{10}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{10}}+\frac {129140 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^9}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^9}+\frac {254222 \left (\sqrt {\sqrt {x}-1}-1\right )^8}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^8}+\frac {129140 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^7}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^7}+\frac {152688 \left (\sqrt {\sqrt {x}-1}-1\right )^6}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^6}+\frac {46980 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^5}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^5}+\frac {34252 \left (\sqrt {\sqrt {x}-1}-1\right )^4}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^4}+\frac {6620 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^3}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^3}+\frac {2960 \left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+\frac {300 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )}{\sqrt {3}-\sqrt {\sqrt {x}+1}}+45\right ) \left (\frac {1}{122880}-\frac {\sqrt {\sqrt {x}-1}}{122880}\right ) \left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{20}}{\left (\sqrt {\sqrt {x}+1}-\sqrt {3}\right ) \left (-2 x-2 \sqrt {\sqrt {x}-1} \sqrt {x}+\sqrt {3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+2 \sqrt {3} \sqrt {\sqrt {x}+1} \sqrt {x}-3 \sqrt {x}\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + (-1 + Sqrt[-1 + Sqrt[x]])^2/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^2)*(45 + (45*(-1 + Sqrt[-1 + Sqrt[x]])^16)/(Sq
rt[3] - Sqrt[1 + Sqrt[x]])^16 + (300*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^15)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^15 +
(2960*(-1 + Sqrt[-1 + Sqrt[x]])^14)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^14 + (6620*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])
^13)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^13 + (34252*(-1 + Sqrt[-1 + Sqrt[x]])^12)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^12
+ (46980*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^11)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^11 + (152688*(-1 + Sqrt[-1 + Sqrt
[x]])^10)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^10 + (129140*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^9)/(Sqrt[3] - Sqrt[1 +
Sqrt[x]])^9 + (254222*(-1 + Sqrt[-1 + Sqrt[x]])^8)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^8 + (129140*Sqrt[3]*(-1 + Sqr
t[-1 + Sqrt[x]])^7)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^7 + (152688*(-1 + Sqrt[-1 + Sqrt[x]])^6)/(Sqrt[3] - Sqrt[1 +
Sqrt[x]])^6 + (46980*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^5)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^5 + (34252*(-1 + Sqrt
[-1 + Sqrt[x]])^4)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^4 + (6620*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^3)/(Sqrt[3] - Sqr
t[1 + Sqrt[x]])^3 + (2960*(-1 + Sqrt[-1 + Sqrt[x]])^2)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^2 + (300*Sqrt[3]*(-1 + Sq
rt[-1 + Sqrt[x]]))/(Sqrt[3] - Sqrt[1 + Sqrt[x]]))*(1/122880 - Sqrt[-1 + Sqrt[x]]/122880)*(Sqrt[3] - Sqrt[1 + S
qrt[x]])^20)/((-Sqrt[3] + Sqrt[1 + Sqrt[x]])*(-3*Sqrt[x] - 2*Sqrt[-1 + Sqrt[x]]*Sqrt[x] + 2*Sqrt[3]*Sqrt[1 + S
qrt[x]]*Sqrt[x] + Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - 2*x)^5)

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fricas [A]  time = 0.41, size = 37, normalized size = 0.59 \begin {gather*} \frac {2 \, {\left (2 \, x^{3} + {\left (2 \, x^{2} + 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^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.28, size = 90, normalized size = 1.43 \begin {gather*} \frac {128 \, {\left (15 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{12} - 20 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 80 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 64\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^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x, algorithm="giac")

[Out]

128/15*(15*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 - 20*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 80*(sqr
t(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 64)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^5

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maple [A]  time = 0.05, size = 28, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right ) \left (2 x +3\right )}{15 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 21, normalized size = 0.33 \begin {gather*} \frac {4 \, {\left (x - 1\right )}^{\frac {3}{2}}}{15 \, x^{\frac {3}{2}}} + \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{5 \, x^{\frac {5}{2}}} \end {gather*}

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^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.07, size = 43, normalized size = 0.68 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{15}-\frac {2\,\sqrt {\sqrt {x}+1}}{5}+\frac {4\,x^2\,\sqrt {\sqrt {x}+1}}{15}\right )}{x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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**(7/2),x)

[Out]

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

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