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

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

[Out]

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

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Rubi [A]  time = 0.0824455, antiderivative size = 30, normalized size of antiderivative = 1., 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 = {6689, 43} \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]

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 6689

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]

Rule 43

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])

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*}

Mathematica [A]  time = 0.0413616, size = 30, normalized size = 1. \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]

Antiderivative was successfully verified.

[In]

Integrate[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

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Maple [A]  time = 0.003, size = 21, normalized size = 0.7 \begin{align*}{\frac{4}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}}}-{\frac{4}{3}{x}^{{\frac{3}{2}}}}+2\,\sqrt{-1+x} \end{align*}

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)

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1}{\left (\sqrt{x - 1} + \sqrt{x}\right )}^{2}}\,{d x} \end{align*}

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 = 2.00777, size = 55, normalized size = 1.83 \begin{align*} \frac{2}{3} \,{\left (2 \, x + 1\right )} \sqrt{x - 1} - \frac{4}{3} \, x^{\frac{3}{2}} \end{align*}

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]  time = 0.818293, size = 53, normalized size = 1.77 \begin{align*} - \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{align*}

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 = 1.07121, size = 27, normalized size = 0.9 \begin{align*} \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{4}{3} \, x^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \end{align*}

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="giac")

[Out]

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