3.11.17 \(\int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}} \, dx\) [1017]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {278, 271} \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 271

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 278

Int[(x_)^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(a1
+ 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 [B] Leaf count is larger than twice the leaf count of optimal. \(693\) vs. \(2(94)=188\).
time = 6.93, size = 693, normalized size = 7.37 \begin {gather*} \frac {\left (-1+\sqrt {-1+\sqrt {x}}\right ) \left (\sqrt {3}-\sqrt {1+\sqrt {x}}\right ) \left (-2+\sqrt {-1+\sqrt {x}}+\sqrt {3} \sqrt {1+\sqrt {x}}-\sqrt {x}\right ) \left (384 \left (97-168 \sqrt {-1+\sqrt {x}}-56 \sqrt {3} \sqrt {1+\sqrt {x}}+97 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right )+192 \left (-499-1112 \sqrt {-1+\sqrt {x}}+344 \sqrt {3} \sqrt {1+\sqrt {x}}+545 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) \sqrt {x}+32 \left (-7985-5016 \sqrt {-1+\sqrt {x}}+3496 \sqrt {3} \sqrt {1+\sqrt {x}}+1405 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x+16 \left (-12223-2144 \sqrt {-1+\sqrt {x}}+4160 \sqrt {3} \sqrt {1+\sqrt {x}}+515 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{3/2}+8 \left (-14415-3248 \sqrt {-1+\sqrt {x}}+5264 \sqrt {3} \sqrt {1+\sqrt {x}}+1405 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^2+8 \left (-6511-2908 \sqrt {-1+\sqrt {x}}+1740 \sqrt {3} \sqrt {1+\sqrt {x}}+1141 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{5/2}+4 \left (-5007-3692 \sqrt {-1+\sqrt {x}}+1852 \sqrt {3} \sqrt {1+\sqrt {x}}+1155 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^3+2 \left (-4080-1468 \sqrt {-1+\sqrt {x}}+852 \sqrt {3} \sqrt {1+\sqrt {x}}+145 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{7/2}-503 x^4\right )}{240 \left (-3-2 \sqrt {-1+\sqrt {x}}+2 \sqrt {3} \sqrt {1+\sqrt {x}}+\sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}-2 \sqrt {x}\right )^5 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]

((-1 + Sqrt[-1 + Sqrt[x]])*(Sqrt[3] - Sqrt[1 + Sqrt[x]])*(-2 + Sqrt[-1 + Sqrt[x]] + Sqrt[3]*Sqrt[1 + Sqrt[x]]
- Sqrt[x])*(384*(97 - 168*Sqrt[-1 + Sqrt[x]] - 56*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 97*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sq
rt[1 + Sqrt[x]]) + 192*(-499 - 1112*Sqrt[-1 + Sqrt[x]] + 344*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 545*Sqrt[3]*Sqrt[-1 +
Sqrt[x]]*Sqrt[1 + Sqrt[x]])*Sqrt[x] + 32*(-7985 - 5016*Sqrt[-1 + Sqrt[x]] + 3496*Sqrt[3]*Sqrt[1 + Sqrt[x]] +
1405*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x + 16*(-12223 - 2144*Sqrt[-1 + Sqrt[x]] + 4160*Sqrt[3]*Sqr
t[1 + Sqrt[x]] + 515*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^(3/2) + 8*(-14415 - 3248*Sqrt[-1 + Sqrt[x
]] + 5264*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 1405*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^2 + 8*(-6511 - 2908
*Sqrt[-1 + Sqrt[x]] + 1740*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 1141*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^(5
/2) + 4*(-5007 - 3692*Sqrt[-1 + Sqrt[x]] + 1852*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 1155*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sq
rt[1 + Sqrt[x]])*x^3 + 2*(-4080 - 1468*Sqrt[-1 + Sqrt[x]] + 852*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 145*Sqrt[3]*Sqrt[-
1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^(7/2) - 503*x^4))/(240*(-3 - 2*Sqrt[-1 + Sqrt[x]] + 2*Sqrt[3]*Sqrt[1 + Sqrt[
x]] + Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]] - 2*Sqrt[x])^5*x^(5/2))

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Maple [A]
time = 0.40, size = 30, normalized size = 0.32

method	result	size
derivativedivides	\(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (8 x^{2}+4 x +3\right )}{15 x^{\frac {5}{2}}}\)	\(30\)
default	\(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (8 x^{2}+4 x +3\right )}{15 x^{\frac {5}{2}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.54, 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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Fricas [A]
time = 2.47, 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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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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Giac [A]
time = 1.06, 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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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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