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])

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Rubi [A]  time = 0.0162064, 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, Rules used = {1958, 50, 54, 215} \[ \frac{1}{2} \sqrt{x+1} \sqrt{2 x+3}-\frac{\sinh ^{-1}\left (\sqrt{2} \sqrt{x+1}\right )}{2 \sqrt{2}} \]

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])

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \sqrt{\frac{1+x}{3+2 x}} \, dx &=\int \frac{\sqrt{1+x}}{\sqrt{3+2 x}} \, dx\\ &=\frac{1}{2} \sqrt{1+x} \sqrt{3+2 x}-\frac{1}{4} \int \frac{1}{\sqrt{1+x} \sqrt{3+2 x}} \, dx\\ &=\frac{1}{2} \sqrt{1+x} \sqrt{3+2 x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+2 x^2}} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{2} \sqrt{1+x} \sqrt{3+2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} \sqrt{1+x}\right )}{2 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0358298, size = 71, normalized size = 1.61 \[ \frac{2 (x+1) \sqrt{2 x+3}-\sqrt{2} \sqrt{x+1} \sinh ^{-1}\left (\sqrt{2} \sqrt{x+1}\right )}{4 \sqrt{\frac{x+1}{2 x+3}} \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 + x)/(3 + 2*x)],x]

[Out]

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

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Maple [B]  time = 0.01, size = 75, normalized size = 1.7 \begin{align*} -{\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 ) }}}} \end{align*}

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)

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Maxima [B]  time = 1.40605, size = 108, normalized size = 2.45 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - 2 \, \sqrt{\frac{x + 1}{2 \, x + 3}}}{\sqrt{2} + 2 \, \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83084, size = 151, normalized size = 3.43 \begin{align*} \frac{1}{2} \,{\left (2 \, x + 3\right )} \sqrt{\frac{x + 1}{2 \, x + 3}} + \frac{1}{8} \, \sqrt{2} \log \left (2 \, \sqrt{2}{\left (2 \, x + 3\right )} \sqrt{\frac{x + 1}{2 \, x + 3}} - 4 \, x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))^(1/2),x, algorithm="fricas")

[Out]

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

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

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)

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Giac [A]  time = 1.10489, size = 82, normalized size = 1.86 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 5 \, x + 3}\right )} - 5 \right |}\right ) \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{1}{2} \, \sqrt{2 \, x^{2} + 5 \, x + 3} \mathrm{sgn}\left (2 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))^(1/2),x, algorithm="giac")

[Out]

1/8*sqrt(2)*log(abs(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + 5*x + 3)) - 5))*sgn(2*x + 3) + 1/2*sqrt(2*x^2 + 5*x +
3)*sgn(2*x + 3)