3.3.41 \(\int \sqrt {1-\sqrt {x}} \, dx\) [241]

Optimal. Leaf size=35 \[ -\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {4}{5} \left (1-\sqrt {x}\right )^{5/2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {196, 45} \begin {gather*} \frac {4}{5} \left (1-\sqrt {x}\right )^{5/2}-\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sqrt[x]],x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=2 \text {Subst}\left (\int \sqrt {1-x} x \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\sqrt {1-x}-(1-x)^{3/2}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {4}{5} \left (1-\sqrt {x}\right )^{5/2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.74 \begin {gather*} -\frac {4}{15} \left (1-\sqrt {x}\right )^{3/2} \left (2+3 \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Sqrt[x]],x]

[Out]

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

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Maple [A]
time = 0.08, size = 24, normalized size = 0.69

method	result	size
derivativedivides	\(-\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {5}{2}}}{5}\)	\(24\)
default	\(-\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {5}{2}}}{5}\)	\(24\)
meijerg	\(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-\sqrt {x}\right )^{\frac {3}{2}} \left (3 \sqrt {x}+2\right )}{15}}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.38, size = 23, normalized size = 0.66 \begin {gather*} \frac {4}{5} \, {\left (-\sqrt {x} + 1\right )}^{\frac {5}{2}} - \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.58, size = 21, normalized size = 0.60 \begin {gather*} \frac {4}{15} \, {\left (3 \, x - \sqrt {x} - 2\right )} \sqrt {-\sqrt {x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 306, normalized size = 8.74 \begin {gather*} \begin {cases} - \frac {12 i x^{\frac {7}{2}} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {4 i x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {16 i x^{3} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 i x^{2} \sqrt {\sqrt {x} - 1}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {8 x^{2}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} & \text {for}\: \left |{\sqrt {x}}\right | > 1 \\- \frac {12 x^{\frac {7}{2}} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {4 x^{\frac {5}{2}} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {16 x^{3} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} - \frac {8 x^{2} \sqrt {1 - \sqrt {x}}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} + \frac {8 x^{2}}{- 15 x^{\frac {5}{2}} + 15 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x**(1/2))**(1/2),x)

[Out]

Piecewise((-12*I*x**(7/2)*sqrt(sqrt(x) - 1)/(-15*x**(5/2) + 15*x**2) + 4*I*x**(5/2)*sqrt(sqrt(x) - 1)/(-15*x**
(5/2) + 15*x**2) - 8*x**(5/2)/(-15*x**(5/2) + 15*x**2) + 16*I*x**3*sqrt(sqrt(x) - 1)/(-15*x**(5/2) + 15*x**2)
- 8*I*x**2*sqrt(sqrt(x) - 1)/(-15*x**(5/2) + 15*x**2) + 8*x**2/(-15*x**(5/2) + 15*x**2), Abs(sqrt(x)) > 1), (-
12*x**(7/2)*sqrt(1 - sqrt(x))/(-15*x**(5/2) + 15*x**2) + 4*x**(5/2)*sqrt(1 - sqrt(x))/(-15*x**(5/2) + 15*x**2)
- 8*x**(5/2)/(-15*x**(5/2) + 15*x**2) + 16*x**3*sqrt(1 - sqrt(x))/(-15*x**(5/2) + 15*x**2) - 8*x**2*sqrt(1 -
sqrt(x))/(-15*x**(5/2) + 15*x**2) + 8*x**2/(-15*x**(5/2) + 15*x**2), True))

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Giac [A]
time = 0.50, size = 30, normalized size = 0.86 \begin {gather*} \frac {4}{5} \, {\left (\sqrt {x} - 1\right )}^{2} \sqrt {-\sqrt {x} + 1} - \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.19, size = 10, normalized size = 0.29 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},2;\ 3;\ \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^(1/2))^(1/2),x)

[Out]

x*hypergeom([-1/2, 2], 3, x^(1/2))

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Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.31 \begin {gather*} -\frac {4 \left (1-\sqrt {x}\right )^{\frac {5}{2}}}{5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((1-x^(1/2))^(1/2),x)

[Out]

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

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