3.3063 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx\)

Optimal. Leaf size=135 \[ -\frac{\left (4 a c-3 b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{5/2}}-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{a} \]

[Out]

(-3*b*d*Sqrt[a + b*Sqrt[d/x] + c/x])/(2*a^2*Sqrt[d/x]) + (Sqrt[a + b*Sqrt[d/x] + c/x]*x)/a - ((4*a*c - 3*b^2*d
)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(4*a^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.180083, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1969, 1357, 744, 806, 724, 206} \[ -\frac{\left (4 a c-3 b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{5/2}}-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

(-3*b*d*Sqrt[a + b*Sqrt[d/x] + c/x])/(2*a^2*Sqrt[d/x]) + (Sqrt[a + b*Sqrt[d/x] + c/x]*x)/a - ((4*a*c - 3*b^2*d
)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(4*a^(5/2))

Rule 1969

Int[((a_.) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d, Subst[Int[(a + b*x^n +
(c*x^(2*n))/d^(2*n))^p/x^2, x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n] && IntegerQ[2*n]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

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{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx &=-\left (d \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b \sqrt{x}+\frac{c x}{d}}} \, dx,x,\frac{d}{x}\right )\right )\\ &=-\left ((2 d) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{a}+\frac{d \operatorname{Subst}\left (\int \frac{\frac{3 b}{2}+\frac{c x}{d}}{x^2 \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{a}\\ &=-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{a}+\frac{\left (4 a c-3 b^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{4 a^2}\\ &=-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{a}-\frac{\left (4 a c-3 b^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \sqrt{\frac{d}{x}}}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{2 a^2}\\ &=-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{a}-\frac{\left (4 a c-3 b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{5/2}}\\ \end{align*}

Mathematica [F]  time = 0.318175, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

Integrate[1/Sqrt[a + b*Sqrt[d/x] + c/x], x]

________________________________________________________________________________________

Maple [A]  time = 0.14, size = 213, normalized size = 1.6 \begin{align*} -{\frac{1}{4}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 6\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}b-4\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}-3\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) da{b}^{2}+4\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{2}c \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(6*a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b-
4*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)-3*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2
)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d*a*b^2+4*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2
)+2*a*x^(1/2))/a^(1/2))*a^2*c)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(7/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*sqrt(d/x) + a + c/x), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*sqrt(d/x) + c/x), x)

________________________________________________________________________________________

Giac [A]  time = 1.75067, size = 232, normalized size = 1.72 \begin{align*} \frac{{\left (3 \, b^{2} d \log \left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) - 4 \, a c \log \left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) + 6 \, \sqrt{c d} \sqrt{a} b\right )} \mathrm{sgn}\left (x\right )}{4 \, a^{\frac{5}{2}}} - \frac{2 \, \sqrt{a d x + \sqrt{d x} b d + c d}{\left (\frac{3 \, b d}{a^{2}} - \frac{2 \, \sqrt{d x}}{a}\right )} + \frac{{\left (3 \, b^{2} d^{2} - 4 \, a c d\right )} \log \left ({\left | -b d - 2 \,{\left (\sqrt{d x} \sqrt{a} - \sqrt{a d x + \sqrt{d x} b d + c d}\right )} \sqrt{a} \right |}\right )}{a^{\frac{5}{2}}}}{4 \, d \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*b^2*d*log(abs(-b*d + 2*sqrt(c*d)*sqrt(a))) - 4*a*c*log(abs(-b*d + 2*sqrt(c*d)*sqrt(a))) + 6*sqrt(c*d)*s
qrt(a)*b)*sgn(x)/a^(5/2) - 1/4*(2*sqrt(a*d*x + sqrt(d*x)*b*d + c*d)*(3*b*d/a^2 - 2*sqrt(d*x)/a) + (3*b^2*d^2 -
4*a*c*d)*log(abs(-b*d - 2*(sqrt(d*x)*sqrt(a) - sqrt(a*d*x + sqrt(d*x)*b*d + c*d))*sqrt(a)))/a^(5/2))/(d*sgn(x
))