3.105 \(\int \frac{1}{\sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=56 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.059945, antiderivative size = 56, 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 = {2010, 2013, 620, 206} \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

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

Rule 2010

Int[1/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(-2*Sqrt[a*x^j + b*x^n])/(b*(n - 2)*x^(n -
1)), x] - Dist[(a*(2*n - j - 2))/(b*(n - 2)), Int[1/(x^(n - j)*Sqrt[a*x^j + b*x^n]), x], x] /; FreeQ[{a, b}, x
] && LtQ[2*(n - 1), j, n]

Rule 2013

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[(a*x^Simplify[j/n]
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j
/n]] && EqQ[Simplify[m - n + 1], 0]

Rule 620

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0479315, size = 88, normalized size = 1.57 \[ \frac{2 \sqrt{a} \sqrt{x} \left (a \sqrt{x}+b\right )-2 b^{3/2} \sqrt [4]{x} \sqrt{\frac{a \sqrt{x}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} \sqrt [4]{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 83, normalized size = 1.5 \begin{align*} -{\sqrt{b\sqrt{x}+ax} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ) -2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b \sqrt{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*x + b*sqrt(x)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*x + b*sqrt(x)), x)

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Giac [A]  time = 1.3343, size = 73, normalized size = 1.3 \begin{align*} \frac{b \log \left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{a^{\frac{3}{2}}} + \frac{2 \, \sqrt{a x + b \sqrt{x}}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/a^(3/2) + 2*sqrt(a*x + b*sqrt(x))/a