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]

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

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Rubi [A]  time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.05, 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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fricas [A]  time = 0.57, size = 57, normalized size = 0.55 \[ \frac {1}{24} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {5}{16} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \]

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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giac [A]  time = 0.23, size = 76, normalized size = 0.73 \[ \frac {1}{24} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (\sqrt {x} + 1\right )} {\left (\sqrt {x} - 4\right )} + 45\right )} {\left (\sqrt {x} + 1\right )} - 55\right )} {\left (\sqrt {x} + 1\right )} + 85\right )} {\left (\sqrt {x} + 1\right )} - 33\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {5}{4} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \]

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]

1/24*((2*((4*(sqrt(x) + 1)*(sqrt(x) - 4) + 45)*(sqrt(x) + 1) - 55)*(sqrt(x) + 1) + 85)*(sqrt(x) + 1) - 33)*sqr
t(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 5/4*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))

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maple [A]  time = 0.06, size = 65, normalized size = 0.62 \[ \frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (8 \sqrt {x -1}\, x^{\frac {5}{2}}+10 \sqrt {x -1}\, x^{\frac {3}{2}}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )+15 \sqrt {x -1}\, \sqrt {x}\right )}{24 \sqrt {x -1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 47, normalized size = 0.45 \[ \frac {1}{3} \, \sqrt {x - 1} x^{\frac {5}{2}} + \frac {5}{12} \, \sqrt {x - 1} x^{\frac {3}{2}} + \frac {5}{8} \, \sqrt {x - 1} \sqrt {x} + \frac {5}{8} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]

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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mupad [B]  time = 27.09, size = 632, normalized size = 6.08 \[ \frac {5\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )}{2}-\frac {-\frac {175\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {311\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {8361\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {42259\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {25295\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {25295\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}+\frac {42259\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}+\frac {8361\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{17}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{17}}+\frac {311\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{19}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{19}}-\frac {175\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{21}}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{21}}+\frac {5\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{23}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{23}}+\frac {5\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{2\,\left (\sqrt {\sqrt {x}+1}-1\right )}}{1+\frac {66\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {220\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {495\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {792\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {924\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {792\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {495\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {220\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{18}}+\frac {66\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{20}}-\frac {12\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{22}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{24}}-\frac {12\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1)))/2 - ((311*((x^(1/2) - 1)^(1/2) - 1i)^5)/(2*((x
^(1/2) + 1)^(1/2) - 1)^5) - (175*((x^(1/2) - 1)^(1/2) - 1i)^3)/(6*((x^(1/2) + 1)^(1/2) - 1)^3) + (8361*((x^(1/
2) - 1)^(1/2) - 1i)^7)/(2*((x^(1/2) + 1)^(1/2) - 1)^7) + (42259*((x^(1/2) - 1)^(1/2) - 1i)^9)/(3*((x^(1/2) + 1
)^(1/2) - 1)^9) + (25295*((x^(1/2) - 1)^(1/2) - 1i)^11)/((x^(1/2) + 1)^(1/2) - 1)^11 + (25295*((x^(1/2) - 1)^(
1/2) - 1i)^13)/((x^(1/2) + 1)^(1/2) - 1)^13 + (42259*((x^(1/2) - 1)^(1/2) - 1i)^15)/(3*((x^(1/2) + 1)^(1/2) -
1)^15) + (8361*((x^(1/2) - 1)^(1/2) - 1i)^17)/(2*((x^(1/2) + 1)^(1/2) - 1)^17) + (311*((x^(1/2) - 1)^(1/2) - 1
i)^19)/(2*((x^(1/2) + 1)^(1/2) - 1)^19) - (175*((x^(1/2) - 1)^(1/2) - 1i)^21)/(6*((x^(1/2) + 1)^(1/2) - 1)^21)
+ (5*((x^(1/2) - 1)^(1/2) - 1i)^23)/(2*((x^(1/2) + 1)^(1/2) - 1)^23) + (5*((x^(1/2) - 1)^(1/2) - 1i))/(2*((x^
(1/2) + 1)^(1/2) - 1)))/((66*((x^(1/2) - 1)^(1/2) - 1i)^4)/((x^(1/2) + 1)^(1/2) - 1)^4 - (12*((x^(1/2) - 1)^(1
/2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (220*((x^(1/2) - 1)^(1/2) - 1i)^6)/((x^(1/2) + 1)^(1/2) - 1)^6 + (4
95*((x^(1/2) - 1)^(1/2) - 1i)^8)/((x^(1/2) + 1)^(1/2) - 1)^8 - (792*((x^(1/2) - 1)^(1/2) - 1i)^10)/((x^(1/2) +
1)^(1/2) - 1)^10 + (924*((x^(1/2) - 1)^(1/2) - 1i)^12)/((x^(1/2) + 1)^(1/2) - 1)^12 - (792*((x^(1/2) - 1)^(1/
2) - 1i)^14)/((x^(1/2) + 1)^(1/2) - 1)^14 + (495*((x^(1/2) - 1)^(1/2) - 1i)^16)/((x^(1/2) + 1)^(1/2) - 1)^16 -
(220*((x^(1/2) - 1)^(1/2) - 1i)^18)/((x^(1/2) + 1)^(1/2) - 1)^18 + (66*((x^(1/2) - 1)^(1/2) - 1i)^20)/((x^(1/
2) + 1)^(1/2) - 1)^20 - (12*((x^(1/2) - 1)^(1/2) - 1i)^22)/((x^(1/2) + 1)^(1/2) - 1)^22 + ((x^(1/2) - 1)^(1/2)
- 1i)^24/((x^(1/2) + 1)^(1/2) - 1)^24 + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {5}{2}}}{\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \]

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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