3.594 \(\int \frac{x}{(1+x) \sqrt{\frac{2+x}{3+x}}} \, dx\)

Optimal. Leaf size=54 \[ \sqrt{x+2} \sqrt{x+3}-\sinh ^{-1}\left (\sqrt{x+2}\right )+2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{x+2}}{\sqrt{x+3}}\right ) \]

[Out]

Sqrt[2 + x]*Sqrt[3 + x] - ArcSinh[Sqrt[2 + x]] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt
[2 + x])/Sqrt[3 + x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.176261, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \sqrt{x+2} \sqrt{x+3}-\sinh ^{-1}\left (\sqrt{x+2}\right )+2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{x+2}}{\sqrt{x+3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/((1 + x)*Sqrt[(2 + x)/(3 + x)]),x]

[Out]

Sqrt[2 + x]*Sqrt[3 + x] - ArcSinh[Sqrt[2 + x]] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt
[2 + x])/Sqrt[3 + x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.85097, size = 48, normalized size = 0.89 \[ \sqrt{x + 2} \sqrt{x + 3} - \operatorname{asinh}{\left (\sqrt{x + 2} \right )} + 2 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x + 2}}{\sqrt{x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(1+x)/((2+x)/(3+x))**(1/2),x)

[Out]

sqrt(x + 2)*sqrt(x + 3) - asinh(sqrt(x + 2)) + 2*sqrt(2)*atanh(sqrt(2)*sqrt(x +
2)/sqrt(x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.102448, size = 101, normalized size = 1.87 \[ \sqrt{\frac{x+2}{x+3}} x+3 \sqrt{\frac{x+2}{x+3}}-\sqrt{2} \log (x+1)-\frac{1}{2} \log \left (2 x+2 \sqrt{x+2} \sqrt{x+3}+5\right )+\sqrt{2} \log \left (3 x+2 \sqrt{2} \sqrt{x+2} \sqrt{x+3}+7\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((1 + x)*Sqrt[(2 + x)/(3 + x)]),x]

[Out]

3*Sqrt[(2 + x)/(3 + x)] + x*Sqrt[(2 + x)/(3 + x)] - Sqrt[2]*Log[1 + x] - Log[5 +
2*x + 2*Sqrt[2 + x]*Sqrt[3 + x]]/2 + Sqrt[2]*Log[7 + 3*x + 2*Sqrt[2]*Sqrt[2 + x
]*Sqrt[3 + x]]

_______________________________________________________________________________________

Maple [A]  time = 0.026, size = 79, normalized size = 1.5 \[ -{\frac{2+x}{2} \left ( -2\,\sqrt{2}{\it Artanh} \left ( 1/4\,{\frac{ \left ( 3\,x+7 \right ) \sqrt{2}}{\sqrt{{x}^{2}+5\,x+6}}} \right ) +\ln \left ({\frac{5}{2}}+x+\sqrt{{x}^{2}+5\,x+6} \right ) -2\,\sqrt{{x}^{2}+5\,x+6} \right ){\frac{1}{\sqrt{{\frac{2+x}{3+x}}}}}{\frac{1}{\sqrt{ \left ( 3+x \right ) \left ( 2+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+x)/((2+x)/(3+x))^(1/2),x)

[Out]

-1/2*(2+x)*(-2*2^(1/2)*arctanh(1/4*(3*x+7)*2^(1/2)/(x^2+5*x+6)^(1/2))+ln(5/2+x+(
x^2+5*x+6)^(1/2))-2*(x^2+5*x+6)^(1/2))/((2+x)/(3+x))^(1/2)/((3+x)*(2+x))^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.832791, size = 139, normalized size = 2.57 \[ -\sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - 2 \, \sqrt{\frac{x + 2}{x + 3}}\right )}}{2 \, \sqrt{2} + 4 \, \sqrt{\frac{x + 2}{x + 3}}}\right ) - \frac{\sqrt{\frac{x + 2}{x + 3}}}{\frac{x + 2}{x + 3} - 1} - \frac{1}{2} \, \log \left (\sqrt{\frac{x + 2}{x + 3}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 2}{x + 3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 1)*sqrt((x + 2)/(x + 3))),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.295916, size = 112, normalized size = 2.07 \[{\left (x + 3\right )} \sqrt{\frac{x + 2}{x + 3}} + \sqrt{2} \log \left (\frac{2 \, \sqrt{2}{\left (x + 3\right )} \sqrt{\frac{x + 2}{x + 3}} + 3 \, x + 7}{x + 1}\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{x + 2}{x + 3}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 2}{x + 3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 1)*sqrt((x + 2)/(x + 3))),x, algorithm="fricas")

[Out]

(x + 3)*sqrt((x + 2)/(x + 3)) + sqrt(2)*log((2*sqrt(2)*(x + 3)*sqrt((x + 2)/(x +
3)) + 3*x + 7)/(x + 1)) - 1/2*log(sqrt((x + 2)/(x + 3)) + 1) + 1/2*log(sqrt((x
+ 2)/(x + 3)) - 1)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\frac{x + 2}{x + 3}} \left (x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+x)/((2+x)/(3+x))**(1/2),x)

[Out]

Integral(x/(sqrt((x + 2)/(x + 3))*(x + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2846, size = 144, normalized size = 2.67 \[ -\sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \sqrt{\frac{x + 2}{x + 3}} \right |}}{2 \,{\left (\sqrt{2} + 2 \, \sqrt{\frac{x + 2}{x + 3}}\right )}}\right ) - \frac{\sqrt{\frac{x + 2}{x + 3}}}{\frac{x + 2}{x + 3} - 1} - \frac{1}{2} \,{\rm ln}\left (\sqrt{\frac{x + 2}{x + 3}} + 1\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | \sqrt{\frac{x + 2}{x + 3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 1)*sqrt((x + 2)/(x + 3))),x, algorithm="giac")

[Out]

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