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

Optimal. Leaf size=44 \[ \frac{1}{2} \sqrt{x+1} \sqrt{2 x+3}-\frac{\sinh ^{-1}\left (\sqrt{2} \sqrt{x+1}\right )}{2 \sqrt{2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0331115, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2} \sqrt{x+1} \sqrt{2 x+3}-\frac{\sinh ^{-1}\left (\sqrt{2} \sqrt{x+1}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(1 + x)/(3 + 2*x)],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.92834, size = 48, normalized size = 1.09 \[ \frac{\sqrt{\frac{x + 1}{2 x + 3}}}{2 \left (- \frac{2 \left (x + 1\right )}{2 x + 3} + 1\right )} - \frac{\sqrt{2} \operatorname{atanh}{\left (\sqrt{2} \sqrt{\frac{x + 1}{2 x + 3}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((x + 1)/(2*x + 3))/(2*(-2*(x + 1)/(2*x + 3) + 1)) - sqrt(2)*atanh(sqrt(2)*s
qrt((x + 1)/(2*x + 3)))/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.0633515, size = 66, normalized size = 1.5 \[ \frac{\sqrt{\frac{x+1}{2 x+3}} \left (2 \sqrt{x+1} (2 x+3)-\sqrt{4 x+6} \sinh ^{-1}\left (\sqrt{2} \sqrt{x+1}\right )\right )}{4 \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(1 + x)/(3 + 2*x)]*(2*Sqrt[1 + x]*(3 + 2*x) - Sqrt[6 + 4*x]*ArcSinh[Sqrt[2
]*Sqrt[1 + x]]))/(4*Sqrt[1 + x])

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 75, normalized size = 1.7 \[ -{\frac{3+2\,x}{8}\sqrt{{\frac{1+x}{3+2\,x}}} \left ( \ln \left ({\frac{5\,\sqrt{2}}{4}}+x\sqrt{2}+\sqrt{2\,{x}^{2}+5\,x+3} \right ) \sqrt{2}-4\,\sqrt{2\,{x}^{2}+5\,x+3} \right ){\frac{1}{\sqrt{ \left ( 3+2\,x \right ) \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((1+x)/(3+2*x))^(1/2),x)

[Out]

-1/8*((1+x)/(3+2*x))^(1/2)*(3+2*x)*(ln(5/4*2^(1/2)+x*2^(1/2)+(2*x^2+5*x+3)^(1/2)
)*2^(1/2)-4*(2*x^2+5*x+3)^(1/2))/((3+2*x)*(1+x))^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51942, size = 108, normalized size = 2.45 \[ \frac{1}{8} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - 2 \, \sqrt{\frac{x + 1}{2 \, x + 3}}\right )}}{2 \, \sqrt{2} + 4 \, \sqrt{\frac{x + 1}{2 \, x + 3}}}\right ) - \frac{\sqrt{\frac{x + 1}{2 \, x + 3}}}{2 \,{\left (\frac{2 \,{\left (x + 1\right )}}{2 \, x + 3} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*log(-2*(sqrt(2) - 2*sqrt((x + 1)/(2*x + 3)))/((2*sqrt(2)) + 4*sqrt((
x + 1)/(2*x + 3)))) - 1/2*sqrt((x + 1)/(2*x + 3))/(2*(x + 1)/(2*x + 3) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.209959, size = 82, normalized size = 1.86 \[ \frac{1}{8} \, \sqrt{2}{\left (2 \, \sqrt{2}{\left (2 \, x + 3\right )} \sqrt{\frac{x + 1}{2 \, x + 3}} + \log \left (-\sqrt{2}{\left (4 \, x + 5\right )} + 4 \,{\left (2 \, x + 3\right )} \sqrt{\frac{x + 1}{2 \, x + 3}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(2*sqrt(2)*(2*x + 3)*sqrt((x + 1)/(2*x + 3)) + log(-sqrt(2)*(4*x + 5
) + 4*(2*x + 3)*sqrt((x + 1)/(2*x + 3))))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x + 1)/(2*x + 3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214876, size = 82, normalized size = 1.86 \[ \frac{1}{8} \, \sqrt{2}{\rm ln}\left ({\left | -2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 5 \, x + 3}\right )} - 5 \right |}\right ){\rm sign}\left (2 \, x + 3\right ) + \frac{1}{2} \, \sqrt{2 \, x^{2} + 5 \, x + 3}{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*ln(abs(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + 5*x + 3)) - 5))*sign(2*x
+ 3) + 1/2*sqrt(2*x^2 + 5*x + 3)*sign(2*x + 3)