\(\int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx\) [3064]

Optimal result

Integrand size = 26, antiderivative size = 54 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\frac {2 \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1994, 1371, 738, 212} \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\frac {2 \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {a}} \]

[In]

Int[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x),x]

[Out]

(2*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/Sqrt[a]

Rule 212

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

Rule 738

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 1371

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 1994

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right ) \\ & = 4 \text {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 ) \\ & = \frac {2 \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 )}{\sqrt {a}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(54)=108\).

Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.13 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\frac {4 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \text {arctanh}\left (\frac {-\sqrt {c} \sqrt {\frac {d}{x}}+\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {a} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(42)=84\).

Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {2 \sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right )}{\sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}}\)	\(94\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\int \frac {1}{x \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\int { \frac {1}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x} \,d x } \]

[In]

integrate(1/x/(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), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (42) = 84\).

Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=-\frac {2 \, \sqrt {d} {\left (\frac {\sqrt {a d} \log \left ({\left | -\sqrt {a d} b d - 2 \, {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} a \right |}\right )}{a d} - \frac {\sqrt {a d} \log \left ({\left | -\sqrt {a d} b d + 2 \, \sqrt {c d^{2}} a \right |}\right )}{a d}\right )}}{\mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-2*sqrt(d)*(sqrt(a*d)*log(abs(-sqrt(a*d)*b*d - 2*(sqrt(a*d)*sqrt(d*x) - sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2
))*a))/(a*d) - sqrt(a*d)*log(abs(-sqrt(a*d)*b*d + 2*sqrt(c*d^2)*a))/(a*d))/sgn(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx=\int \frac {1}{x\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \]

[In]

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

[Out]

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