Integrand size = 22, antiderivative size = 113 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\frac {\left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{2 a}+\frac {\left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{4 a^{3/2}} \]
1/4*(-b^2*d+4*a*c)*arctanh(1/2*(2*a+b*(d/x)^(1/2))/a^(1/2)/(a+c/x+b*(d/x)^ (1/2))^(1/2))/a^(3/2)+1/2*x*(2*a+b*(d/x)^(1/2))*(a+c/x+b*(d/x)^(1/2))^(1/2 )/a
Time = 0.81 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.35 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\sqrt {a} \left (2 a+b \sqrt {\frac {d}{x}}\right ) x+\frac {\sqrt {d} \left (-4 a c+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{2 a^{3/2}} \]
(Sqrt[a + b*Sqrt[d/x] + c/x]*(Sqrt[a]*(2*a + b*Sqrt[d/x])*x + (Sqrt[d]*(-4 *a*c + b^2*d)*ArcTanh[(Sqrt[c]*Sqrt[d/x] - Sqrt[(d*(c + a*x + b*Sqrt[d/x]* x))/x])/(Sqrt[a]*Sqrt[d])])/Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]))/(2*a^( 3/2))
Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2065, 1693, 1152, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx\) |
\(\Big \downarrow \) 2065 |
\(\displaystyle -d \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{d^2}d\frac {d}{x}\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle -2 d \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^3}{d^3}d\sqrt {\frac {d}{x}}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle -2 d \left (-\frac {\left (b^2-\frac {4 a c}{d}\right ) \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 a}-\frac {x^2 \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^2}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle -2 d \left (\frac {\left (b^2-\frac {4 a c}{d}\right ) \int \frac {1}{4 a-\frac {d^2}{x^2}}d\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 a}-\frac {x^2 \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 d \left (\frac {\left (b^2-\frac {4 a c}{d}\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )}{8 a^{3/2}}-\frac {x^2 \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^2}\right )\) |
-2*d*(-1/4*((2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^2)/(a* d^2) + ((b^2 - (4*a*c)/d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])])/(8*a^(3/2)))
3.31.55.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(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] && IntegerQ [Simplify[(m + 1)/n]]
Int[((a_.) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] : > Simp[-d Subst[Int[(a + b*x^n + (c/d^(2*n))*x^(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]
Leaf count of result is larger than twice the leaf count of optimal. \(212\) vs. \(2(91)=182\).
Time = 0.23 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (2 a^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, \sqrt {x}\, b -\ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) d a \,b^{2}+4 a^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {x}+4 \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} c \right )}{4 a^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}\) | \(213\) |
1/4*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(2*a^(3/2)*(b*(d/x)^(1/2)*x+ a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b-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))*d*a*b^2+4*a^(5/2)*(b* (d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)+4*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^2*c)/a^(5/2)/(b*( d/x)^(1/2)*x+a*x+c)^(1/2)
Timed out. \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\text {Timed out} \]
\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\int \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}\, dx \]
\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\int { \sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (90) = 180\).
Time = 0.38 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.88 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\frac {{\left (2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (\frac {b d}{a} + 2 \, \sqrt {d x}\right )} + \frac {{\left (b^{2} d^{3} - 4 \, a c d^{2}\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a} - \frac {b^{2} d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 4 \, a c d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 2 \, \sqrt {c d^{2}} \sqrt {a d} b d}{\sqrt {a d} a}\right )} \mathrm {sgn}\left (x\right )}{4 \, d^{\frac {3}{2}}} \]
1/4*(2*sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2)*(b*d/a + 2*sqrt(d*x)) + (b^ 2*d^3 - 4*a*c*d^2)*log(abs(-b*d^2 - 2*sqrt(a*d)*(sqrt(a*d)*sqrt(d*x) - sqr t(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2))))/(sqrt(a*d)*a) - (b^2*d^3*log(abs(- b*d^2 + 2*sqrt(c*d^2)*sqrt(a*d))) - 4*a*c*d^2*log(abs(-b*d^2 + 2*sqrt(c*d^ 2)*sqrt(a*d))) + 2*sqrt(c*d^2)*sqrt(a*d)*b*d)/(sqrt(a*d)*a))*sgn(x)/d^(3/2 )
Timed out. \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx=\int \sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}} \,d x \]