3.11.16 \(\int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}} \, dx\) [1016]

Optimal. Leaf size=63 \[ \frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 x^{3/2}}+\frac {4 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.01, 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 = {278, 271} \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 x^{3/2}}+\frac {4 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2)),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(3*x^(3/2)) + (4*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(3*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^{5/2}} \, dx &=\frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 x^{3/2}}+\frac {2}{3} \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}}}{3 x^{3/2}}+\frac {4 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 \sqrt {x}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(407\) vs. \(2(63)=126\).
time = 1.82, size = 407, normalized size = 6.46 \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 (8 \left (-7-12 \sqrt {-1+\sqrt {x}}+4 \sqrt {3} \sqrt {1+\sqrt {x}}+7 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right )-4 \left (49+8 \sqrt {-1+\sqrt {x}}-24 \sqrt {3} \sqrt {1+\sqrt {x}}+3 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) \sqrt {x}+2 \left (-61+16 \sqrt {3} \sqrt {1+\sqrt {x}}+7 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x+\left (-56-28 \sqrt {-1+\sqrt {x}}+20 \sqrt {3} \sqrt {1+\sqrt {x}}+6 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{3/2}-11 x^2\right )}{12 \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 )^3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2)),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])*(8*(-7 - 12*Sqrt[-1 + Sqrt[x]] + 4*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 7*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1
+ Sqrt[x]]) - 4*(49 + 8*Sqrt[-1 + Sqrt[x]] - 24*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 3*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[
1 + Sqrt[x]])*Sqrt[x] + 2*(-61 + 16*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 7*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]
)*x + (-56 - 28*Sqrt[-1 + Sqrt[x]] + 20*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 6*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt
[x]])*x^(3/2) - 11*x^2))/(12*(-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])^3*x^(3/2))

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Maple [A]
time = 0.33, size = 25, normalized size = 0.40

method	result	size
derivativedivides	\(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (2 x +1\right )}{3 x^{\frac {3}{2}}}\)	\(25\)
default	\(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (2 x +1\right )}{3 x^{\frac {3}{2}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(x^(1/2)+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 21, normalized size = 0.33 \begin {gather*} \frac {4 \, \sqrt {x - 1}}{3 \, \sqrt {x}} + \frac {2 \, \sqrt {x - 1}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.81, size = 34, normalized size = 0.54 \begin {gather*} \frac {2 \, {\left ({\left (2 \, x + 1\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + 2 \, x^{2}\right )}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {5}{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**(5/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

Integral(1/(x**(5/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)), x)

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Giac [A]
time = 1.67, size = 48, normalized size = 0.76 \begin {gather*} \frac {128 \, {\left (3 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}}{3 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="giac")

[Out]

128/3*(3*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^3

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Mupad [B]
time = 5.50, size = 33, normalized size = 0.52 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {4\,x}{3}+\frac {2\,\sqrt {x}}{3}+\frac {4\,x^{3/2}}{3}+\frac {2}{3}\right )}{x^{3/2}\,\sqrt {\sqrt {x}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2)),x)

[Out]

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

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