3.1011 \(\int \frac{x^{5/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\)

Optimal. Leaf size=104 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

(5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/8 + (5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/12 + (Sq
rt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/3 + (5*ArcCosh[Sqrt[x]])/8

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Rubi [A]  time = 0.0452857, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {323, 330, 52} \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/8 + (5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/12 + (Sq
rt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/3 + (5*ArcCosh[Sqrt[x]])/8

Rule 323

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(2
*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a
1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x],
x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m +
2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 330

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x]
, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx &=\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{6} \int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{8} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=\frac{5}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{16} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=\frac{5}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0349079, size = 67, normalized size = 0.64 \[ \frac{1}{24} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left (8 x^2+10 x+15\right )+30 \tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(15 + 10*x + 8*x^2) + 30*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x
])]])/24

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Maple [A]  time = 0.013, size = 65, normalized size = 0.6 \begin{align*}{\frac{1}{24}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(8*x^(5/2)*(-1+x)^(1/2)+10*x^(3/2)*(-1+x)^(1/2)+15*x^(1/2)*(-1+x)^(1
/2)+15*ln(x^(1/2)+(-1+x)^(1/2)))/(-1+x)^(1/2)

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Maxima [A]  time = 0.936321, size = 63, normalized size = 0.61 \begin{align*} \frac{1}{3} \, \sqrt{x - 1} x^{\frac{5}{2}} + \frac{5}{12} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{x - 1} \sqrt{x} + \frac{5}{8} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(x - 1)*x^(5/2) + 5/12*sqrt(x - 1)*x^(3/2) + 5/8*sqrt(x - 1)*sqrt(x) + 5/8*log(2*sqrt(x - 1) + 2*sqrt(
x))

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Fricas [A]  time = 0.947352, size = 186, normalized size = 1.79 \begin{align*} \frac{1}{24} \,{\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - \frac{5}{16} \, \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(8*x^2 + 10*x + 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 5/16*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sq
rt(sqrt(x) - 1) - 2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{5}{2}}}{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError