3.7.47 \(\int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{9/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac {16 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{105 x^{3/2}}+\frac {8 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{35 x^{5/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{7 x^{7/2}} \]

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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 {16 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{105 x^{3/2}}+\frac {8 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{35 x^{5/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{7 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(7*x^(7/2)) + (8*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(35*x^
(5/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(105*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^{9/2}} \, dx &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{7 x^{7/2}}+\frac {4}{7} \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}}{7 x^{7/2}}+\frac {8 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{35 x^{5/2}}+\frac {8}{35} \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}}{7 x^{7/2}}+\frac {8 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{35 x^{5/2}}+\frac {16 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{105 x^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 41, normalized size = 0.44 \begin {gather*} \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2} \left (8 x^2+12 x+15\right )}{105 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(15 + 12*x + 8*x^2))/(105*x^(7/2))

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IntegrateAlgebraic [B]  time = 36.64, size = 1160, normalized size = 12.34 \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 {315 \left (\sqrt {\sqrt {x}-1}-1\right )^{24}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{24}}+\frac {2730 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{23}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{23}}+\frac {37520 \left (\sqrt {\sqrt {x}-1}-1\right )^{22}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{22}}+\frac {123550 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{21}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{21}}+\frac {965902 \left (\sqrt {\sqrt {x}-1}-1\right )^{20}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{20}}+\frac {2042838 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{19}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{19}}+\frac {10643536 \left (\sqrt {\sqrt {x}-1}-1\right )^{18}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{18}}+\frac {15378258 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{17}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{17}}+\frac {56109493 \left (\sqrt {\sqrt {x}-1}-1\right )^{16}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{16}}+\frac {57833188 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{15}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{15}}+\frac {151811360 \left (\sqrt {\sqrt {x}-1}-1\right )^{14}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{14}}+\frac {112621740 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{13}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{13}}+\frac {212123652 \left (\sqrt {\sqrt {x}-1}-1\right )^{12}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{12}}+\frac {112621740 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{11}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{11}}+\frac {151811360 \left (\sqrt {\sqrt {x}-1}-1\right )^{10}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{10}}+\frac {57833188 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^9}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^9}+\frac {56109493 \left (\sqrt {\sqrt {x}-1}-1\right )^8}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^8}+\frac {15378258 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^7}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^7}+\frac {10643536 \left (\sqrt {\sqrt {x}-1}-1\right )^6}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^6}+\frac {2042838 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^5}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^5}+\frac {965902 \left (\sqrt {\sqrt {x}-1}-1\right )^4}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^4}+\frac {123550 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^3}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^3}+\frac {37520 \left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+\frac {2730 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )}{\sqrt {3}-\sqrt {\sqrt {x}+1}}+315\right ) \left (\frac {1}{55050240}-\frac {\sqrt {\sqrt {x}-1}}{55050240}\right ) \left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{28}}{\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 )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

((1 + (-1 + Sqrt[-1 + Sqrt[x]])^2/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^2)*(315 + (315*(-1 + Sqrt[-1 + Sqrt[x]])^24)/(
Sqrt[3] - Sqrt[1 + Sqrt[x]])^24 + (2730*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^23)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^23
+ (37520*(-1 + Sqrt[-1 + Sqrt[x]])^22)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^22 + (123550*Sqrt[3]*(-1 + Sqrt[-1 + Sqr
t[x]])^21)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^21 + (965902*(-1 + Sqrt[-1 + Sqrt[x]])^20)/(Sqrt[3] - Sqrt[1 + Sqrt[x
]])^20 + (2042838*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^19)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^19 + (10643536*(-1 + Sqr
t[-1 + Sqrt[x]])^18)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^18 + (15378258*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^17)/(Sqrt[
3] - Sqrt[1 + Sqrt[x]])^17 + (56109493*(-1 + Sqrt[-1 + Sqrt[x]])^16)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^16 + (57833
188*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^15)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^15 + (151811360*(-1 + Sqrt[-1 + Sqrt[x
]])^14)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^14 + (112621740*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^13)/(Sqrt[3] - Sqrt[1
+ Sqrt[x]])^13 + (212123652*(-1 + Sqrt[-1 + Sqrt[x]])^12)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^12 + (112621740*Sqrt[3
]*(-1 + Sqrt[-1 + Sqrt[x]])^11)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^11 + (151811360*(-1 + Sqrt[-1 + Sqrt[x]])^10)/(S
qrt[3] - Sqrt[1 + Sqrt[x]])^10 + (57833188*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^9)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^
9 + (56109493*(-1 + Sqrt[-1 + Sqrt[x]])^8)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^8 + (15378258*Sqrt[3]*(-1 + Sqrt[-1 +
Sqrt[x]])^7)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^7 + (10643536*(-1 + Sqrt[-1 + Sqrt[x]])^6)/(Sqrt[3] - Sqrt[1 + Sqr
t[x]])^6 + (2042838*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^5)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^5 + (965902*(-1 + Sqrt[
-1 + Sqrt[x]])^4)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^4 + (123550*Sqrt[3]*(-1 + Sqrt[-1 + Sqrt[x]])^3)/(Sqrt[3] - Sq
rt[1 + Sqrt[x]])^3 + (37520*(-1 + Sqrt[-1 + Sqrt[x]])^2)/(Sqrt[3] - Sqrt[1 + Sqrt[x]])^2 + (2730*Sqrt[3]*(-1 +
Sqrt[-1 + Sqrt[x]]))/(Sqrt[3] - Sqrt[1 + Sqrt[x]]))*(1/55050240 - Sqrt[-1 + Sqrt[x]]/55050240)*(Sqrt[3] - Sqr
t[1 + Sqrt[x]])^28)/((-Sqrt[3] + Sqrt[1 + Sqrt[x]])*(-3*Sqrt[x] - 2*Sqrt[-1 + Sqrt[x]]*Sqrt[x] + 2*Sqrt[3]*Sqr
t[1 + Sqrt[x]]*Sqrt[x] + Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - 2*x)^7)

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fricas [A]  time = 0.40, size = 44, normalized size = 0.47 \begin {gather*} \frac {2 \, {\left (8 \, x^{4} + {\left (8 \, x^{3} + 4 \, x^{2} + 3 \, x - 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1}\right )}}{105 \, x^{4}} \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^(9/2),x, algorithm="fricas")

[Out]

2/105*(8*x^4 + (8*x^3 + 4*x^2 + 3*x - 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1))/x^4

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giac [A]  time = 0.29, size = 111, normalized size = 1.18 \begin {gather*} \frac {4096 \, {\left (35 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{16} - 70 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{12} + 168 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 224 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 128\right )}}{105 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{7}} \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^(9/2),x, algorithm="giac")

[Out]

4096/105*(35*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^16 - 70*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 + 168*
(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 224*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 128)/((sqrt(sqrt(x
) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^7

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maple [A]  time = 0.05, size = 33, normalized size = 0.35 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right ) \left (8 x^{2}+12 x +15\right )}{105 x^{\frac {7}{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^(9/2),x)

[Out]

2/105*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)*(x-1)*(8*x^2+12*x+15)/x^(7/2)

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maxima [A]  time = 1.55, size = 31, normalized size = 0.33 \begin {gather*} \frac {16 \, {\left (x - 1\right )}^{\frac {3}{2}}}{105 \, x^{\frac {3}{2}}} + \frac {8 \, {\left (x - 1\right )}^{\frac {3}{2}}}{35 \, x^{\frac {5}{2}}} + \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{7 \, x^{\frac {7}{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^(9/2),x, algorithm="maxima")

[Out]

16/105*(x - 1)^(3/2)/x^(3/2) + 8/35*(x - 1)^(3/2)/x^(5/2) + 2/7*(x - 1)^(3/2)/x^(7/2)

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mupad [B]  time = 5.04, size = 55, normalized size = 0.59 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{35}-\frac {2\,\sqrt {\sqrt {x}+1}}{7}+\frac {8\,x^2\,\sqrt {\sqrt {x}+1}}{105}+\frac {16\,x^3\,\sqrt {\sqrt {x}+1}}{105}\right )}{x^{7/2}} \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^(9/2),x)

[Out]

((x^(1/2) - 1)^(1/2)*((2*x*(x^(1/2) + 1)^(1/2))/35 - (2*(x^(1/2) + 1)^(1/2))/7 + (8*x^2*(x^(1/2) + 1)^(1/2))/1
05 + (16*x^3*(x^(1/2) + 1)^(1/2))/105))/x^(7/2)

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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**(9/2),x)

[Out]

Timed out

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