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

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

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Rubi [A]  time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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^{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*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 0.37 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 111.25, size = 1484, normalized size = 11.87

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + (-1 + Sqrt[-1 + Sqrt[x]])^2/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^2)*(945 + (945*(-1 + Sqrt[-1 + Sqrt[x]])^32)/(
Sqrt[3] - Sqrt[1 + Sqrt[x]])^32 + (10080*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^31)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^3
1 + (178080*(-1 + Sqrt[-1 + Sqrt[x]])^30)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^30 + (771120*Sqrt[3]*(-1 + Sqrt[-1 + S
qrt[x]])^29)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^29 + (7981176*(-1 + Sqrt[-1 + Sqrt[x]])^28)/(Sqrt[3] - Sqrt[1 + Sqr
t[x]])^28 + (22539888*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^27)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^27 + (159375840*(-1
+ Sqrt[-1 + Sqrt[x]])^26)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^26 + (319049280*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^25)/
(Sqrt[3] - Sqrt[1 + Sqrt[x]])^25 + (1649481052*(-1 + Sqrt[-1 + Sqrt[x]])^24)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^24
+ (2469292416*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^23)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^23 + (9691565216*(-1 + Sqrt[
-1 + Sqrt[x]])^22)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^22 + (11128795920*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^21)/(Sqrt
[3] - Sqrt[1 + Sqrt[x]])^21 + (33755226952*(-1 + Sqrt[-1 + Sqrt[x]])^20)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^20 + (3
0109229136*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^19)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^19 + (71150262752*(-1 + Sqrt[-1
+ Sqrt[x]])^18)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^18 + (49500527328*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^17)/(Sqrt[3
] - Sqrt[1 + Sqrt[x]])^17 + (91239315046*(-1 + Sqrt[-1 + Sqrt[x]])^16)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^16 + (495
00527328*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^15)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^15 + (71150262752*(-1 + Sqrt[-1 +
Sqrt[x]])^14)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^14 + (30109229136*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^13)/(Sqrt[3]
- Sqrt[1 + Sqrt[x]])^13 + (33755226952*(-1 + Sqrt[-1 + Sqrt[x]])^12)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^12 + (11128
795920*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^11)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^11 + (9691565216*(-1 + Sqrt[-1 + Sq
rt[x]])^10)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^10 + (2469292416*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^9)/(Sqrt[3] - Sqr
t[1 + Sqrt[x]])^9 + (1649481052*(-1 + Sqrt[-1 + Sqrt[x]])^8)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^8 + (319049280*Sqrt
[3]*(-1 + Sqrt[-1 + Sqrt[x]])^7)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^7 + (159375840*(-1 + Sqrt[-1 + Sqrt[x]])^6)/(Sq
rt[3] - Sqrt[1 + Sqrt[x]])^6 + (22539888*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^5)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^5
+ (7981176*(-1 + Sqrt[-1 + Sqrt[x]])^4)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^4 + (771120*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt
[x]])^3)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^3 + (178080*(-1 + Sqrt[-1 + Sqrt[x]])^2)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^
2 + (10080*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]]))/(Sqrt[3] - Sqrt[1 + Sqrt[x]]))*(1/10569646080 - Sqrt[-1 + Sqrt[x
]]/10569646080)*(Sqrt[3] - Sqrt[1 + Sqrt[x]])^36)/((-Sqrt[3] + Sqrt[1 + Sqrt[x]])*(-3*Sqrt[x] - 2*Sqrt[-1 + Sq
rt[x]]*Sqrt[x] + 2*Sqrt[3]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] -
2*x)^9)

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fricas [A]  time = 0.40, size = 49, normalized size = 0.39 \begin {gather*} \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 {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^(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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giac [A]  time = 0.43, size = 132, normalized size = 1.06 \begin {gather*} \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 {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^(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

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maple [A]  time = 0.06, size = 38, normalized size = 0.30 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right ) \left (16 x^{3}+24 x^{2}+30 x +35\right )}{315 x^{\frac {9}{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^(11/2),x)

[Out]

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

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maxima [A]  time = 1.22, size = 41, normalized size = 0.33 \begin {gather*} \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 {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^(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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mupad [B]  time = 5.02, size = 67, normalized size = 0.54 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{63}-\frac {2\,\sqrt {\sqrt {x}+1}}{9}+\frac {4\,x^2\,\sqrt {\sqrt {x}+1}}{105}+\frac {16\,x^3\,\sqrt {\sqrt {x}+1}}{315}+\frac {32\,x^4\,\sqrt {\sqrt {x}+1}}{315}\right )}{x^{9/2}} \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^(11/2),x)

[Out]

((x^(1/2) - 1)^(1/2)*((2*x*(x^(1/2) + 1)^(1/2))/63 - (2*(x^(1/2) + 1)^(1/2))/9 + (4*x^2*(x^(1/2) + 1)^(1/2))/1
05 + (16*x^3*(x^(1/2) + 1)^(1/2))/315 + (32*x^4*(x^(1/2) + 1)^(1/2))/315))/x^(9/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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**(11/2),x)

[Out]

Timed out

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