3.1.8 \(\int \frac {1}{(\sqrt {-1+x}+\sqrt {x})^2 \sqrt {-1+x}} \, dx\) [8]

Optimal. Leaf size=30 \[ 2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}-\frac {4 x^{3/2}}{3} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6821, 45} \begin {gather*} -\frac {4 x^{3/2}}{3}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 6821

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a,
b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (\sqrt {-1+x}+\sqrt {x}\right )^2 \sqrt {-1+x}} \, dx &=\int \left (-\frac {1}{\sqrt {-1+x}}-2 \sqrt {x}+\frac {2 x}{\sqrt {-1+x}}\right ) \, dx\\ &=-2 \sqrt {-1+x}-\frac {4 x^{3/2}}{3}+2 \int \frac {x}{\sqrt {-1+x}} \, dx\\ &=-2 \sqrt {-1+x}-\frac {4 x^{3/2}}{3}+2 \int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx\\ &=2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}-\frac {4 x^{3/2}}{3}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 26, normalized size = 0.87 \begin {gather*} -\frac {4 x^{3/2}}{3}+\frac {2}{3} \sqrt {-1+x} (1+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [A]
time = 2.09, size = 32, normalized size = 1.07 \begin {gather*} \frac {2 \left (-2 \sqrt {x}-\sqrt {-1+x}\right )}{-3+6 \sqrt {x} \sqrt {-1+x}+6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2 (-2 Sqrt[x] - Sqrt[-1 + x]) / (3 (-1 + 2 Sqrt[x] Sqrt[-1 + x] + 2 x))

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Maple [A]
time = 0.08, size = 21, normalized size = 0.70

method	result	size
default	\(\frac {4 \left (-1+x \right )^{\frac {3}{2}}}{3}-\frac {4 x^{\frac {3}{2}}}{3}+2 \sqrt {-1+x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x - 1)*(sqrt(x - 1) + sqrt(x))^2), x)

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Fricas [A]
time = 0.31, size = 18, normalized size = 0.60 \begin {gather*} \frac {2}{3} \, {\left (2 \, x + 1\right )} \sqrt {x - 1} - \frac {4}{3} \, x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\)
time = 0.31, size = 53, normalized size = 1.77 \begin {gather*} - \frac {4 \sqrt {x}}{6 \sqrt {x} \sqrt {x - 1} + 6 x - 3} - \frac {2 \sqrt {x - 1}}{6 \sqrt {x} \sqrt {x - 1} + 6 x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(x)/(6*sqrt(x)*sqrt(x - 1) + 6*x - 3) - 2*sqrt(x - 1)/(6*sqrt(x)*sqrt(x - 1) + 6*x - 3)

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Giac [A]
time = 0.00, size = 53, normalized size = 1.77 \begin {gather*} 2 \left (\frac {2}{3} \sqrt {x-1} \left (x-1\right )+\sqrt {x-1}+2 \left (-\frac 1{3}-\frac {1}{3} \sqrt {x-1} \sqrt {x-1}\right ) \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/((-1+x)^(1/2)+x^(1/2))^2,x)

[Out]

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

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Mupad [B]
time = 0.38, size = 21, normalized size = 0.70 \begin {gather*} \frac {4\,x\,\sqrt {x-1}}{3}+\frac {2\,\sqrt {x-1}}{3}-\frac {4\,x^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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