Integrand size = 26, antiderivative size = 145 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=-2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-\frac {b \sqrt {d} \text {arctanh}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {c}} \]
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)-b*arctanh(1/2*(b*d+2*c*(d/x)^(1/2))/c^(1/2)/d^(1/2)/(a+c/x+b*(d/x)^(1 /2))^(1/2))*d^(1/2)/c^(1/2)-2*(a+c/x+b*(d/x)^(1/2))^(1/2)
Time = 0.61 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}+4 \sqrt {a} \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {-\sqrt {c} \sqrt {\frac {d}{x}}+\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )+b d \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )\right )}{\sqrt {c} \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}} \]
(Sqrt[a + b*Sqrt[d/x] + c/x]*(-2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x) )/x] + 4*Sqrt[a]*Sqrt[c]*Sqrt[d]*ArcTanh[(-(Sqrt[c]*Sqrt[d/x]) + Sqrt[(d*( c + a*x + b*Sqrt[d/x]*x))/x])/(Sqrt[a]*Sqrt[d])] + b*d*Log[b*d + 2*c*Sqrt[ d/x] - 2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x]]))/(Sqrt[c]*Sqrt[(d *(c + (a + b*Sqrt[d/x])*x))/x])
Time = 0.39 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.80, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2066, 1693, 1162, 25, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx\) |
| \(\Big \downarrow \) 2066 |
| \(\displaystyle -\int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{d}d\frac {d}{x}\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle -2 \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x}{d}d\sqrt {\frac {d}{x}}\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle -2 \left (\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}-\frac {1}{2} \int -\frac {\left (2 a+b \sqrt {\frac {d}{x}}\right ) x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\frac {1}{2} \int \frac {\left (2 a+b \sqrt {\frac {d}{x}}\right ) x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}+\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -2 \left (\frac {1}{2} \left (b \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}+2 a \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}\right )+\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -2 \left (\frac {1}{2} \left (2 b \int \frac {1}{\frac {4 c}{d}-\frac {d^2}{x^2}}d\frac {2 \sqrt {\frac {d}{x}} c+b d}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}+2 a \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}\right )+\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 \left (\frac {1}{2} \left (2 a \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}+\frac {b \sqrt {d} \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{\sqrt {c}}\right )+\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -2 \left (\frac {1}{2} \left (\frac {b \sqrt {d} \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{\sqrt {c}}-4 a \int \frac {1}{4 a-\frac {d^2}{x^2}}d\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )+\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 \left (\frac {1}{2} \left (\frac {b \sqrt {d} \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{\sqrt {c}}-2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )\right )+\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}\right )\) |
-2*(Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2] + (-2*Sqrt[a]*ArcTanh[(2*a + b*Sqrt[ d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])] + (b*Sqrt[d]*ArcTanh[ d^(3/2)/(2*Sqrt[c]*x)])/Sqrt[c])/2)
3.31.56.3.1 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[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x _Symbol] :> Simp[-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]
Leaf count of result is larger than twice the leaf count of optimal. \(236\) vs. \(2(113)=226\).
Time = 0.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(-\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (\ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {c}\, a^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x b -2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} x -2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x b +2 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {3}{2}}-2 \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} c \sqrt {x}\right )}{\sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c \,a^{\frac {3}{2}}}\) | \(237\) |
-((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*( d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(1/2)*a^(3/2)*(d/x)^(1/2)*x*b-2*(b*( d/x)^(1/2)*x+a*x+c)^(1/2)*a^(5/2)*x-2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(3/2 )*(d/x)^(1/2)*x*b+2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(3/2)-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^2*c*x^(1/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c/a^(3/2)
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=\int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x}\, dx \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx=\int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x} \,d x \]