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

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]

________________________________________________________________________________________

Rubi [A]  time = 0.006951, 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, Rules used = {50, 54, 215} \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]

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]

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

Mathematica [A]  time = 0.0188439, size = 49, normalized size = 1.4 \[ \frac{3 \sqrt{x+1} (3 x+2)-\sqrt{9 x+6} \sinh ^{-1}\left (\sqrt{3 x+2}\right )}{3 \sqrt{3 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 + x]*(2 + 3*x) - Sqrt[6 + 9*x]*ArcSinh[Sqrt[2 + 3*x]])/(3*Sqrt[2 + 3*x])

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 67, normalized size = 1.9 \begin{align*} \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}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.61212, size = 55, normalized size = 1.57 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(1+x)^(1/2),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)

________________________________________________________________________________________

Fricas [A]  time = 1.48094, size = 157, normalized size = 4.49 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (-4 \, \sqrt{3}{\left (6 \, x + 5\right )} \sqrt{3 \, x + 2} \sqrt{x + 1} + 72 \, x^{2} + 120 \, x + 49\right ) + \sqrt{3 \, x + 2} \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.61436, size = 97, normalized size = 2.77 \begin{align*} \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} \end{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.08706, size = 53, normalized size = 1.51 \begin{align*} \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3 \, x + 3} \sqrt{3 \, x + 2} + \log \left (\sqrt{3 \, x + 3} - \sqrt{3 \, x + 2}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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