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

Optimal. Leaf size=16 \[ \sinh ^{-1}\left (\frac{\sqrt{2 x+1}}{\sqrt{2}}\right ) \]

[Out]

ArcSinh[Sqrt[1 + 2*x]/Sqrt[2]]

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Rubi [A]  time = 0.0214437, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \sinh ^{-1}\left (\frac{\sqrt{2 x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcSinh[Sqrt[1 + 2*x]/Sqrt[2]]

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Rubi in Sympy [A]  time = 3.49911, size = 15, normalized size = 0.94 \[ \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{2 x + 1}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(sqrt(2)*sqrt(2*x + 1)/2)

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Mathematica [A]  time = 0.0107111, size = 16, normalized size = 1. \[ \sinh ^{-1}\left (\frac{\sqrt{2 x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcSinh[Sqrt[1 + 2*x]/Sqrt[2]]

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Maple [B]  time = 0.01, size = 57, normalized size = 3.6 \[{\frac{\sqrt{4}}{4}\sqrt{ \left ( 1+2\,x \right ) \left ( 3+2\,x \right ) }\ln \left ({\frac{ \left ( 4+4\,x \right ) \sqrt{4}}{4}}+\sqrt{4\,{x}^{2}+8\,x+3} \right ){\frac{1}{\sqrt{1+2\,x}}}{\frac{1}{\sqrt{3+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49294, size = 30, normalized size = 1.88 \[ \frac{1}{2} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 8 \, x + 3} + 8\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229493, size = 31, normalized size = 1.94 \[ -\frac{1}{2} \, \log \left (\sqrt{2 \, x + 3} \sqrt{2 \, x + 1} - 2 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.76516, size = 27, normalized size = 1.69 \[ \begin{cases} \operatorname{acosh}{\left (\sqrt{x + \frac{3}{2}} \right )} & \text{for}\: \left |{x + \frac{3}{2}}\right | > 1 \\- i \operatorname{asin}{\left (\sqrt{x + \frac{3}{2}} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((acosh(sqrt(x + 3/2)), Abs(x + 3/2) > 1), (-I*asin(sqrt(x + 3/2)), Tru
e))

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GIAC/XCAS [A]  time = 0.247608, size = 28, normalized size = 1.75 \[ -{\rm ln}\left ({\left | -\sqrt{2 \, x + 3} + \sqrt{2 \, x + 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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