Integrand size = 26, antiderivative size = 155 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\frac {b \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{4 c^2}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{3 c}+\frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{8 c^{5/2}} \]
1/8*b*(-b^2*d+4*a*c)*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^(5/2)-2/3*(a+c/x+b*(d/x)^(1/2))^(3/2)/ c+1/4*b*(b*d+2*c*(d/x)^(1/2))*(a+c/x+b*(d/x)^(1/2))^(1/2)/c^2
Time = 0.85 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (-\frac {2 \sqrt {c} \left (8 c^2-3 b^2 d x+2 c \left (4 a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}+\frac {3 b d \left (-4 a c+b^2 d\right ) \log \left (c^2 \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{24 c^{5/2}} \]
(Sqrt[a + b*Sqrt[d/x] + c/x]*((-2*Sqrt[c]*(8*c^2 - 3*b^2*d*x + 2*c*(4*a + b*Sqrt[d/x])*x))/x + (3*b*d*(-4*a*c + b^2*d)*Log[c^2*(b*d + 2*c*Sqrt[d/x] - 2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x])])/Sqrt[(d*(c + (a + b*S qrt[d/x])*x))/x]))/(24*c^(5/2))
Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2066, 1680, 1160, 1087, 1092, 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.
| \(\displaystyle \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx\) |
| \(\Big \downarrow \) 2066 |
| \(\displaystyle -\frac {\int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}d\frac {d}{x}}{d}\) |
| \(\Big \downarrow \) 1680 |
| \(\displaystyle -\frac {2 \int \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{d}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle -\frac {2 \left (\frac {d \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 c}-\frac {b d \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}d\sqrt {\frac {d}{x}}}{2 c}\right )}{d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle -\frac {2 \left (\frac {d \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 c}-\frac {b d \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{2 c}\right )}{d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {2 \left (\frac {d \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 c}-\frac {b d \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\frac {4 c}{d}-\frac {d^2}{x^2}}d\frac {2 \sqrt {\frac {d}{x}} c+b d}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{2 c}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \left (\frac {d \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 c}-\frac {b d \left (\frac {\sqrt {d} \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{8 c^{3/2}}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{2 c}\right )}{d}\) |
(-2*((d*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/(3*c) - (b*d*(((b*d + 2*c*Sqr t[d/x])*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])/(4*c) + (Sqrt[d]*(4*a*c - b^2*d )*ArcTanh[d^(3/2)/(2*Sqrt[c]*x)])/(8*c^(3/2))))/(2*c)))/d
3.31.57.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k* n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && Fr actionQ[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. \(330\) vs. \(2(123)=246\).
Time = 0.24 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.14
| method | result | size |
| default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (-3 \sqrt {c}\, \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3}+6 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3}+6 a \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d \,x^{2} b^{2}+12 a \,c^{\frac {3}{2}} \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {\frac {d}{x}}\, x^{2} b -6 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d x \,b^{2}-12 a \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{2} b c +12 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x b c -16 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{2}\right )}{24 x \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{3}}\) | \(331\) |
1/24*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x*(-3*c^(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))*(d/x)^(3/2)*x^3*b^3+6 *(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^3*b^3+6*a*(b*(d/x)^(1/2)*x+a* x+c)^(1/2)*d*x^2*b^2+12*a*c^(3/2)*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))*(d/x)^(1/2)*x^2*b-6*(b*(d/x)^(1/2)*x+a*x +c)^(3/2)*d*x*b^2-12*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^2*b*c+1 2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x*b*c-16*(b*(d/x)^(1/2)*x+a*x+ c)^(3/2)*c^2)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^3
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x^{2}}\, dx \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{2}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx=\int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x^2} \,d x \]