Integrand size = 24, antiderivative size = 248 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}} \]
1/64*(35*b^4*d^2-120*a*b^2*c*d+48*a^2*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^(9/2)-7/12*b*d^2*(a+c/x+b*(d/x)^(1 /2))^(1/2)/a^2/(d/x)^(3/2)-1/48*(-35*b^2*d+36*a*c)*x*(a+c/x+b*(d/x)^(1/2)) ^(1/2)/a^3+1/2*x^2*(a+c/x+b*(d/x)^(1/2))^(1/2)/a+5/96*b*d*(-21*b^2*d+44*a* c)*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^4/(d/x)^(1/2)
Time = 1.54 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.12 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\frac {\sqrt {a} d \left (-105 b^3 d \left (b d+c \sqrt {\frac {d}{x}}\right )+48 a^4 x^2-8 a^3 x \left (3 c+b \sqrt {\frac {d}{x}} x\right )+a^2 \left (-72 c^2+14 b^2 d x+92 b c \sqrt {\frac {d}{x}} x\right )-5 a b \left (-58 b c d-44 c^2 \sqrt {\frac {d}{x}}+7 b^2 d \sqrt {\frac {d}{x}} x\right )\right )-3 \sqrt {d} \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{96 a^{9/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
(Sqrt[a]*d*(-105*b^3*d*(b*d + c*Sqrt[d/x]) + 48*a^4*x^2 - 8*a^3*x*(3*c + b *Sqrt[d/x]*x) + a^2*(-72*c^2 + 14*b^2*d*x + 92*b*c*Sqrt[d/x]*x) - 5*a*b*(- 58*b*c*d - 44*c^2*Sqrt[d/x] + 7*b^2*d*Sqrt[d/x]*x)) - 3*Sqrt[d]*(48*a^2*c^ 2 - 120*a*b^2*c*d + 35*b^4*d^2)*Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]*ArcT anh[(Sqrt[c]*Sqrt[d/x] - Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x])/(Sqrt[a]*S qrt[d])])/(96*a^(9/2)*d*Sqrt[a + b*Sqrt[d/x] + c/x])
Time = 0.58 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2066, 1693, 1167, 27, 1237, 27, 1237, 27, 1228, 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.
| \(\displaystyle \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\) |
| \(\Big \downarrow \) 2066 |
| \(\displaystyle -d^2 \int \frac {x^3}{d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}d\frac {d}{x}\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle -2 d^2 \int \frac {x^5}{d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle -2 d^2 \left (-\frac {\int \frac {\left (6 \sqrt {\frac {d}{x}} c+7 b d\right ) x^4}{2 d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{4 a}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^2 \left (-\frac {\int \frac {\left (6 \sqrt {\frac {d}{x}} c+7 b d\right ) x^4}{d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -2 d^2 \left (-\frac {-\frac {\int -\frac {\left (-35 d b^2-28 c \sqrt {\frac {d}{x}} b+36 a c\right ) x^3}{2 d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{3 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^2 \left (-\frac {\frac {\int \frac {\left (-35 d b^2-28 c \sqrt {\frac {d}{x}} b+36 a c\right ) x^3}{d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {\int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (36 a c-35 b^2 d\right )+5 b d \left (44 a c-21 b^2 d\right )\right ) x^2}{2 d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {\int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (36 a c-35 b^2 d\right )+5 b d \left (44 a c-21 b^2 d\right )\right ) x^2}{d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {\frac {3 \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {5 b x \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {-\frac {3 \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \int \frac {1}{4 a-\frac {d^2}{x^2}}d\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{a}-\frac {5 b x \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {-\frac {3 \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )}{2 a^{3/2}}-\frac {5 b x \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\) |
-2*d^2*(-1/4*(Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^4)/(a*d^4) - ((-7*b*Sqrt [a + b*Sqrt[d/x] + (c*d)/x^2]*x^3)/(3*a*d^2) + (-1/2*((36*a*c - 35*b^2*d)* Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^2)/(a*d^2) - ((-5*b*(44*a*c - 21*b^2*d )*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x)/a - (3*(48*a^2*c^2 - 120*a*b^2*c*d + 35*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])])/(2*a^(3/2)))/(4*a*d))/(6*a))/(8*a*d))
3.31.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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/(((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[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] + Simp[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[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(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] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Time = 0.25 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (96 x^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}}-112 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b -210 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}-144 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} c \sqrt {x}+440 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b c +140 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d \,a^{\frac {5}{2}} \sqrt {x}\, b^{2}+105 d^{2} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{4}-360 d \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{2} c +144 \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} c^{2}\right )}{192 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {11}{2}}}\) | \(398\) |
1/192*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(96*x^(3/2)*(b*(d/x)^(1/2) *x+a*x+c)^(1/2)*a^(9/2)-112*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(7/2)*(d/x)^(1 /2)*x^(3/2)*b-210*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(3/2)*(d/x)^(3/2)*x^(3/2 )*b^3-144*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(7/2)*c*x^(1/2)+440*(b*(d/x)^(1/ 2)*x+a*x+c)^(1/2)*a^(5/2)*(d/x)^(1/2)*x^(1/2)*b*c+140*(b*(d/x)^(1/2)*x+a*x +c)^(1/2)*d*a^(5/2)*x^(1/2)*b^2+105*d^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*b^4-360*d*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*b^2*c+144*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^3*c^2)/(b*(d/x)^(1/2)*x+a* x+c)^(1/2)/a^(11/2)
Timed out. \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\text {Timed out} \]
\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\int \frac {x}{\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \]
\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\int { \frac {x}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}} \,d x } \]
Time = 0.42 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.46 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=-\frac {2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {d x}}{a d}\right )} - \frac {35 \, a b^{2} d^{2} - 36 \, a^{2} c d}{a^{4} d}\right )} + \frac {5 \, {\left (21 \, b^{3} d^{3} - 44 \, a b c d^{2}\right )}}{a^{4} d}\right )} + \frac {3 \, {\left (35 \, b^{4} d^{4} - 120 \, a b^{2} c d^{3} + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a^{4}} - \frac {105 \, b^{4} d^{4} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 360 \, a b^{2} c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 144 \, a^{2} c^{2} d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 210 \, \sqrt {c d^{2}} \sqrt {a d} b^{3} d^{2} - 440 \, \sqrt {c d^{2}} \sqrt {a d} a b c d}{\sqrt {a d} a^{4}}}{192 \, d^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
-1/192*(2*sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2)*(2*sqrt(d*x)*(4*sqrt(d*x )*(7*b/a^2 - 6*sqrt(d*x)/(a*d)) - (35*a*b^2*d^2 - 36*a^2*c*d)/(a^4*d)) + 5 *(21*b^3*d^3 - 44*a*b*c*d^2)/(a^4*d)) + 3*(35*b^4*d^4 - 120*a*b^2*c*d^3 + 48*a^2*c^2*d^2)*log(abs(-b*d^2 - 2*sqrt(a*d)*(sqrt(a*d)*sqrt(d*x) - sqrt(a *d^2*x + sqrt(d*x)*b*d^2 + c*d^2))))/(sqrt(a*d)*a^4) - (105*b^4*d^4*log(ab s(-b*d^2 + 2*sqrt(c*d^2)*sqrt(a*d))) - 360*a*b^2*c*d^3*log(abs(-b*d^2 + 2* sqrt(c*d^2)*sqrt(a*d))) + 144*a^2*c^2*d^2*log(abs(-b*d^2 + 2*sqrt(c*d^2)*s qrt(a*d))) + 210*sqrt(c*d^2)*sqrt(a*d)*b^3*d^2 - 440*sqrt(c*d^2)*sqrt(a*d) *a*b*c*d)/(sqrt(a*d)*a^4))/(d^(3/2)*sgn(x))
Timed out. \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\int \frac {x}{\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \]