Integrand size = 21, antiderivative size = 163 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}-\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \left (a-b \sqrt {c}\right )^{3/2} \sqrt {c}}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \left (a+b \sqrt {c}\right )^{3/2} \sqrt {c}} \]
-1/2*b*d*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a-b*c^(1/2))^(1/2))/c^(1/2)/(a -b*c^(1/2))^(3/2)+1/2*b*d*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a+b*c^(1/2))^ (1/2))/c^(1/2)/(a+b*c^(1/2))^(3/2)-(a-b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2)) ^(1/2)/(-b^2*c+a^2)/x
Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {1}{2} \left (-\frac {2 \left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}+\frac {b d \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\left (-a-b \sqrt {c}\right )^{3/2} \sqrt {c}}-\frac {b d \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\left (-a+b \sqrt {c}\right )^{3/2} \sqrt {c}}\right ) \]
((-2*(a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/((a^2 - b^2*c)*x) + (b*d*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a - b*Sqrt[c]]])/((-a - b*Sqrt [c])^(3/2)*Sqrt[c]) - (b*d*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a + b*Sq rt[c]]])/((-a + b*Sqrt[c])^(3/2)*Sqrt[c]))/2
Time = 0.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {896, 1732, 561, 25, 27, 1492, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle d \int \frac {1}{d^2 x^2 \sqrt {a+b \sqrt {c+d x}}}d(c+d x)\) |
| \(\Big \downarrow \) 1732 |
| \(\displaystyle 2 d \int \frac {\sqrt {c+d x}}{d^2 x^2 \sqrt {a+b \sqrt {c+d x}}}d\sqrt {c+d x}\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {4 d \int -\frac {a-c-d x}{b \left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}d\sqrt {a+b \sqrt {c+d x}}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 d \int \frac {a-c-d x}{b \left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}d\sqrt {a+b \sqrt {c+d x}}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 d \int \frac {a-c-d x}{\left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}d\sqrt {a+b \sqrt {c+d x}}}{b^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {4 d \left (\frac {b^2 (2 a-c-d x) \sqrt {a+b \sqrt {c+d x}}}{4 \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}-\frac {b^4 \int -\frac {2 c (2 a-c-d x)}{b^2 \left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}d\sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right )}\right )}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 d \left (\frac {b^2 \int \frac {2 a-c-d x}{\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c}d\sqrt {a+b \sqrt {c+d x}}}{4 \left (a^2-b^2 c\right )}+\frac {b^2 (2 a-c-d x) \sqrt {a+b \sqrt {c+d x}}}{4 \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}\right )}{b^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {4 d \left (\frac {b^2 \left (-\frac {\left (\frac {a}{\sqrt {c}}+b\right ) \int \frac {1}{\frac {c+d x}{b^2}-\frac {a-b \sqrt {c}}{b^2}}d\sqrt {a+b \sqrt {c+d x}}}{2 b}-\frac {1}{2} \left (1-\frac {a}{b \sqrt {c}}\right ) \int \frac {1}{\frac {c+d x}{b^2}-\frac {a+b \sqrt {c}}{b^2}}d\sqrt {a+b \sqrt {c+d x}}\right )}{4 \left (a^2-b^2 c\right )}+\frac {b^2 (2 a-c-d x) \sqrt {a+b \sqrt {c+d x}}}{4 \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}\right )}{b^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {4 d \left (\frac {b^2 \left (\frac {b^2 \left (1-\frac {a}{b \sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}}}+\frac {b \left (\frac {a}{\sqrt {c}}+b\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}}}\right )}{4 \left (a^2-b^2 c\right )}+\frac {b^2 (2 a-c-d x) \sqrt {a+b \sqrt {c+d x}}}{4 \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}\right )}{b^2}\) |
(-4*d*((b^2*(2*a - c - d*x)*Sqrt[a + b*Sqrt[c + d*x]])/(4*(a^2 - b^2*c)*(a ^2/b^2 - c - (2*a*(c + d*x))/b^2 + (c + d*x)^2/b^2)) + (b^2*((b*(b + a/Sqr t[c])*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(2*Sqrt[a - b*Sqrt[c]]) + (b^2*(1 - a/(b*Sqrt[c]))*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/S qrt[a + b*Sqrt[c]]])/(2*Sqrt[a + b*Sqrt[c]])))/(4*(a^2 - b^2*c))))/b^2
3.7.51.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(d + e*x^(g* n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} , x] && EqQ[n2, 2*n] && FractionQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(125)=250\).
Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.59
| method | result | size |
| derivativedivides | \(4 d \,b^{2} \left (-\frac {\sqrt {b^{2} c}\, \left (\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (4 \sqrt {b^{2} c}-4 a \right ) \left (b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\left (4 \sqrt {b^{2} c}-4 a \right ) \sqrt {\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}+\frac {\sqrt {b^{2} c}\, \left (-\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \left (-b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \sqrt {-\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}\right )\) | \(259\) |
| default | \(4 d \,b^{2} \left (-\frac {\sqrt {b^{2} c}\, \left (\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (4 \sqrt {b^{2} c}-4 a \right ) \left (b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\left (4 \sqrt {b^{2} c}-4 a \right ) \sqrt {\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}+\frac {\sqrt {b^{2} c}\, \left (-\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \left (-b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \sqrt {-\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}\right )\) | \(259\) |
4*d*b^2*(-1/4*(b^2*c)^(1/2)/b^2/c*(2*(a+b*(d*x+c)^(1/2))^(1/2)/(4*(b^2*c)^ (1/2)-4*a)/(b*(d*x+c)^(1/2)+(b^2*c)^(1/2))+2/(4*(b^2*c)^(1/2)-4*a)/((b^2*c )^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2)) )+1/4*(b^2*c)^(1/2)/b^2/c*(-2*(a+b*(d*x+c)^(1/2))^(1/2)/(-4*(b^2*c)^(1/2)- 4*a)/(-b*(d*x+c)^(1/2)+(b^2*c)^(1/2))+2/(-4*(b^2*c)^(1/2)-4*a)/(-(b^2*c)^( 1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 2493 vs. \(2 (127) = 254\).
Time = 0.41 (sec) , antiderivative size = 2493, normalized size of antiderivative = 15.29 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\text {Too large to display} \]
1/4*((b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^2*b ^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)* d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8* b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2* c^2 - a^6*c))*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 + (2*( a*b^6*c^2 + 3*a^3*b^4*c)*d^2 - (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - a^8*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^1 0*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12 *c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6 *c))) - (b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^ 2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^ 6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a ^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b ^2*c^2 - a^6*c))*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 - ( 2*(a*b^6*c^2 + 3*a^3*b^4*c)*d^2 - (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - a^8*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2* b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^...
\[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b \sqrt {c + d x}}}\, dx \]
\[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\int { \frac {1}{\sqrt {\sqrt {d x + c} b + a} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (127) = 254\).
Time = 0.62 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.95 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {\frac {{\left ({\left (b^{3} c - a^{2} b\right )}^{2} b^{4} c d^{2} - 2 \, {\left (a b^{6} c^{\frac {3}{2}} - a^{3} b^{4} \sqrt {c}\right )} d^{2} {\left | -b^{3} c + a^{2} b \right |} + {\left (a^{2} b^{8} c^{2} - 2 \, a^{4} b^{6} c + a^{6} b^{4}\right )} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c - a^{3} + \sqrt {{\left (a b^{2} c - a^{3}\right )}^{2} + {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} {\left (b^{2} c - a^{2}\right )}}}{b^{2} c - a^{2}}}}\right )}{{\left (b^{5} c^{3} + a b^{4} c^{\frac {5}{2}} - 2 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c^{\frac {3}{2}} + a^{4} b c + a^{5} \sqrt {c}\right )} \sqrt {b \sqrt {c} - a} {\left | -b^{3} c + a^{2} b \right |}} + \frac {{\left ({\left (b^{3} c - a^{2} b\right )}^{2} b^{4} c d^{2} + 2 \, {\left (a b^{6} c^{\frac {3}{2}} - a^{3} b^{4} \sqrt {c}\right )} d^{2} {\left | -b^{3} c + a^{2} b \right |} + {\left (a^{2} b^{8} c^{2} - 2 \, a^{4} b^{6} c + a^{6} b^{4}\right )} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c - a^{3} - \sqrt {{\left (a b^{2} c - a^{3}\right )}^{2} + {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} {\left (b^{2} c - a^{2}\right )}}}{b^{2} c - a^{2}}}}\right )}{{\left (b^{5} c^{3} - a b^{4} c^{\frac {5}{2}} - 2 \, a^{2} b^{3} c^{2} + 2 \, a^{3} b^{2} c^{\frac {3}{2}} + a^{4} b c - a^{5} \sqrt {c}\right )} \sqrt {-b \sqrt {c} - a} {\left | -b^{3} c + a^{2} b \right |}} + \frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} d^{2} - 2 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} d^{2}\right )}}{{\left (b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{2} + 2 \, {\left (\sqrt {d x + c} b + a\right )} a - a^{2}\right )} {\left (b^{2} c - a^{2}\right )}}}{2 \, b^{2} d} \]
1/2*(((b^3*c - a^2*b)^2*b^4*c*d^2 - 2*(a*b^6*c^(3/2) - a^3*b^4*sqrt(c))*d^ 2*abs(-b^3*c + a^2*b) + (a^2*b^8*c^2 - 2*a^4*b^6*c + a^6*b^4)*d^2)*arctan( sqrt(sqrt(d*x + c)*b + a)/sqrt(-(a*b^2*c - a^3 + sqrt((a*b^2*c - a^3)^2 + (b^4*c^2 - 2*a^2*b^2*c + a^4)*(b^2*c - a^2)))/(b^2*c - a^2)))/((b^5*c^3 + a*b^4*c^(5/2) - 2*a^2*b^3*c^2 - 2*a^3*b^2*c^(3/2) + a^4*b*c + a^5*sqrt(c)) *sqrt(b*sqrt(c) - a)*abs(-b^3*c + a^2*b)) + ((b^3*c - a^2*b)^2*b^4*c*d^2 + 2*(a*b^6*c^(3/2) - a^3*b^4*sqrt(c))*d^2*abs(-b^3*c + a^2*b) + (a^2*b^8*c^ 2 - 2*a^4*b^6*c + a^6*b^4)*d^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-(a* b^2*c - a^3 - sqrt((a*b^2*c - a^3)^2 + (b^4*c^2 - 2*a^2*b^2*c + a^4)*(b^2* c - a^2)))/(b^2*c - a^2)))/((b^5*c^3 - a*b^4*c^(5/2) - 2*a^2*b^3*c^2 + 2*a ^3*b^2*c^(3/2) + a^4*b*c - a^5*sqrt(c))*sqrt(-b*sqrt(c) - a)*abs(-b^3*c + a^2*b)) + 2*((sqrt(d*x + c)*b + a)^(3/2)*b^4*d^2 - 2*sqrt(sqrt(d*x + c)*b + a)*a*b^4*d^2)/((b^2*c - (sqrt(d*x + c)*b + a)^2 + 2*(sqrt(d*x + c)*b + a )*a - a^2)*(b^2*c - a^2)))/(b^2*d)
Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {1}{x^2\,\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \]