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

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

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Rubi [A]  time = 0.149613, 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 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]

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

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Rubi in Sympy [A]  time = 16.0636, size = 94, normalized size = 0.9 \[ \frac{x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{3} + \frac{5 x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{12} + \frac{5 \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{8} + \frac{5 \operatorname{acosh}{\left (\sqrt{x} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

x**(5/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/3 + 5*x**(3/2)*sqrt(sqrt(x) - 1)*sq
rt(sqrt(x) + 1)/12 + 5*sqrt(x)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/8 + 5*acosh(s
qrt(x))/8

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Mathematica [A]  time = 0.0471159, size = 76, normalized size = 0.73 \[ \frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left (8 x^2+10 x+15\right )+\frac{5}{8} \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[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))/24 + (5*Log[S
qrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]] + Sqrt[x]])/8

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Maple [A]  time = 0.014, size = 65, normalized size = 0.6 \[{\frac{1}{24}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]

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 = 1.42741, size = 63, normalized size = 0.61 \[ \frac{1}{3} \, \sqrt{x - 1} x^{\frac{5}{2}} + \frac{5}{12} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{x - 1} \sqrt{x} + \frac{5}{8} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),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 = 0.213126, size = 261, normalized size = 2.51 \[ -\frac{2048 \, x^{6} - 1536 \, x^{5} + 1152 \, x^{4} - 3840 \, x^{3} - 2 \,{\left (1024 \, x^{5} - 256 \, x^{4} + 576 \, x^{3} - 1600 \, x^{2} + 448 \, x + 51\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 2304 \, x^{2} + 60 \,{\left (32 \, x^{3} - 2 \,{\left (16 \, x^{2} - 16 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 48 \, x^{2} + 18 \, x - 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) - 54 \, x - 37}{192 \,{\left (32 \, x^{3} - 2 \,{\left (16 \, x^{2} - 16 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 48 \, x^{2} + 18 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="fricas")

[Out]

-1/192*(2048*x^6 - 1536*x^5 + 1152*x^4 - 3840*x^3 - 2*(1024*x^5 - 256*x^4 + 576*
x^3 - 1600*x^2 + 448*x + 51)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 2304*
x^2 + 60*(32*x^3 - 2*(16*x^2 - 16*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x)
- 1) - 48*x^2 + 18*x - 1)*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*
x + 1) - 54*x - 37)/(32*x^3 - 2*(16*x^2 - 16*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sq
rt(sqrt(x) - 1) - 48*x^2 + 18*x - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="giac")

[Out]

Exception raised: TypeError