3.610 \(\int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0087786, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[2 + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0148595, size = 43, normalized size = 1. \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[2 + b*x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 62, normalized size = 1.4 \begin{align*}{\frac{1}{b}\sqrt{x}\sqrt{bx+2}}-{\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(b*x+2)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83451, size = 239, normalized size = 5.56 \begin{align*} \left [\frac{\sqrt{b x + 2} b \sqrt{x} + \sqrt{b} \log \left (b x - \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right )}{b^{2}}, \frac{\sqrt{b x + 2} b \sqrt{x} + 2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(b*x + 2)*b*sqrt(x) + sqrt(b)*log(b*x - sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1))/b^2, (sqrt(b*x + 2)*b*sqrt(x
) + 2*sqrt(-b)*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))))/b^2]

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Sympy [A]  time = 2.25833, size = 54, normalized size = 1.26 \begin{align*} \frac{x^{\frac{3}{2}}}{\sqrt{b x + 2}} + \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(b*x+2)**(1/2),x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError