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

Optimal. Leaf size=35 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0294906, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.045, size = 31, normalized size = 0.89 \[ \sqrt{x + 1} \sqrt{3 x + 2} - \frac{\sqrt{3} \operatorname{asinh}{\left (\sqrt{3 x + 2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0253311, size = 45, normalized size = 1.29 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\log \left (3 \sqrt{x+1}+\sqrt{9 x+6}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 + x]*Sqrt[2 + 3*x] - Log[3*Sqrt[1 + x] + Sqrt[6 + 9*x]]/Sqrt[3]

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Maple [B]  time = 0.008, size = 67, normalized size = 1.9 \[ \sqrt{1+x}\sqrt{2+3\,x}-{\frac{\sqrt{3}}{6}\sqrt{ \left ( 1+x \right ) \left ( 2+3\,x \right ) }\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ){\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.759216, size = 55, normalized size = 1.57 \[ -\frac{1}{6} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + sqrt(3*x^2 + 5*x +
2)

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Fricas [A]  time = 0.27843, size = 78, normalized size = 2.23 \[ \frac{1}{12} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x + 2} \sqrt{x + 1} + \log \left (-12 \,{\left (6 \, x + 5\right )} \sqrt{3 \, x + 2} \sqrt{x + 1} + \sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*(4*sqrt(3)*sqrt(3*x + 2)*sqrt(x + 1) + log(-12*(6*x + 5)*sqrt(3*x +
2)*sqrt(x + 1) + sqrt(3)*(72*x^2 + 120*x + 49)))

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Sympy [A]  time = 5.777, size = 97, normalized size = 2.77 \[ \begin{cases} \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{3 x + 2}} - \frac{\sqrt{x + 1}}{\sqrt{3 x + 2}} - \frac{\sqrt{3} \operatorname{acosh}{\left (\sqrt{3} \sqrt{x + 1} \right )}}{3} & \text{for}\: 3 \left |{x + 1}\right | > 1 \\i \sqrt{- 3 x - 2} \sqrt{x + 1} + \frac{\sqrt{3} i \operatorname{asin}{\left (\sqrt{3} \sqrt{x + 1} \right )}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268893, size = 53, normalized size = 1.51 \[ \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3 \, x + 3} \sqrt{3 \, x + 2} +{\rm ln}\left (\sqrt{3 \, x + 3} - \sqrt{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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