Integrand size = 21, antiderivative size = 177 \[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {(b+4 a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{8 c (b+a c)^2 x^2}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{4 c (b+a c) x^4}+\frac {b (b+4 a c) d^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{8 c^{3/2} (b+a c)^{5/2}} \]
1/8*b*(4*a*c+b)*d^2*arctanh(c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*c +b)^(1/2))/c^(3/2)/(a*c+b)^(5/2)+1/8*(4*a*c+b)*d*(d*x^2+c)*((a*d*x^2+a*c+b )/(d*x^2+c))^(1/2)/c/(a*c+b)^2/x^2-1/4*(d*x^2+c)^2*((a*d*x^2+a*c+b)/(d*x^2 +c))^(1/2)/c/(a*c+b)/x^4
Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (2 a c \left (c-d x^2\right )+b \left (2 c+d x^2\right )\right )}{8 c (b+a c)^2 x^4}-\frac {b (b+4 a c) d^2 \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{8 c^{3/2} (-b-a c)^{5/2}} \]
-1/8*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(2*a*c*(c - d*x^2) + b*(2*c + d*x^2)))/(c*(b + a*c)^2*x^4) - (b*(b + 4*a*c)*d^2*ArcTan[(Sqrt [c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(8*c^(3/2)*(-b - a*c)^(5/2))
Time = 0.38 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2057, 2053, 2052, 25, 27, 298, 215, 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.
| \(\displaystyle \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {1}{x^5 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int -\frac {d \left (a-x^4\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b d \int \frac {d \left (a-x^4\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b d^2 \int \frac {a-x^4}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle b d^2 \left (\frac {(4 a c+b) \int \frac {1}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c (a c+b)}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c (a c+b) \left (a c+b-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle b d^2 \left (\frac {(4 a c+b) \left (\frac {\int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 (a c+b)}+\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 (a c+b) \left (a c+b-c x^4\right )}\right )}{4 c (a c+b)}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c (a c+b) \left (a c+b-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle b d^2 \left (\frac {(4 a c+b) \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{2 \sqrt {c} (a c+b)^{3/2}}+\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 (a c+b) \left (a c+b-c x^4\right )}\right )}{4 c (a c+b)}-\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c (a c+b) \left (a c+b-c x^4\right )^2}\right )\) |
b*d^2*(-1/4*(b*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(c*(b + a*c)*(b + a* c - c*x^4)^2) + ((b + 4*a*c)*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/(2*(b + a*c)*(b + a*c - c*x^4)) + ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]]/(2*Sqrt[c]*(b + a*c)^(3/2))))/(4*c*(b + a*c)))
3.4.49.3.1 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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Time = 0.17 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(-\frac {\left (a d \,x^{2}+a c +b \right ) \left (-2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c \right )}{8 \left (a c +b \right )^{2} x^{4} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {d^{2} b \left (4 a c +b \right ) \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{16 \left (a c +b \right )^{2} c \sqrt {a \,c^{2}+b c}\, \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(236\) |
| default | \(-\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (12 a^{2} d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, x^{6} c \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-4 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a^{3} b \,c^{5} d^{2} x^{4}+2 a \,d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, x^{6} b \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-9 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a^{2} b^{2} c^{4} d^{2} x^{4}+20 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, a^{2} c^{2} d^{2} x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-6 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) a \,b^{3} c^{3} d^{2} x^{4}+12 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, a c \,d^{2} b \,x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-\ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) b^{4} c^{2} d^{2} x^{4}+2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, b^{2} d^{2} x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-12 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} a c d \,x^{2} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}-2 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} b d \,x^{2} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}+4 \left (a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} a \,c^{2}+4 \left (a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} b c \right )}{16 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \left (a c +b \right )^{3} c^{2} x^{4} \left (a \,c^{2}+b c \right )^{\frac {3}{2}}}\) | \(923\) |
-1/8*(a*d*x^2+a*c+b)*(-2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c)/(a*c+b)^2/x^4/c/ ((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/16*d^2*b*(4*a*c+b)/(a*c+b)^2/c/(a*c^2+ b*c)^(1/2)*ln((2*a*c^2+2*b*c+(2*a*c*d+b*d)*x^2+2*(a*c^2+b*c)^(1/2)*(a*c^2+ b*c+(2*a*c*d+b*d)*x^2+a*d^2*x^4)^(1/2))/x^2)/((a*d*x^2+a*c+b)/(d*x^2+c))^( 1/2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(d*x^2+c)
Time = 0.40 (sec) , antiderivative size = 593, normalized size of antiderivative = 3.35 \[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\left [\frac {{\left (4 \, a b c + b^{2}\right )} \sqrt {a c^{2} + b c} d^{2} x^{4} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + a b c^{2} - b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} + 3 \, {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} x^{4}}, -\frac {{\left (4 \, a b c + b^{2}\right )} \sqrt {-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + a b c^{2} - b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} + 3 \, {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} x^{4}}\right ] \]
[1/32*((4*a*b*c + b^2)*sqrt(a*c^2 + b*c)*d^2*x^4*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a* b*c^2 + b^2*c)*d*x^2 + 4*((2*a*c + b)*d^2*x^4 + 2*a*c^3 + (4*a*c^2 + 3*b*c )*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) )/x^4) - 4*(2*a^2*c^5 - (2*a^2*c^3 + a*b*c^2 - b^2*c)*d^2*x^4 + 4*a*b*c^4 + 2*b^2*c^3 + 3*(a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^5 + 3*a^2*b*c^4 + 3*a*b^2*c^3 + b^3*c^2)*x^4), -1/16*((4*a *b*c + b^2)*sqrt(-a*c^2 - b*c)*d^2*x^4*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a *c^2 + 2*b*c)*sqrt(-a*c^2 - b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^ 2*c^3 + 2*a*b*c^2 + (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) + 2*(2*a^2*c^5 - (2* a^2*c^3 + a*b*c^2 - b^2*c)*d^2*x^4 + 4*a*b*c^4 + 2*b^2*c^3 + 3*(a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^5 + 3*a^2*b *c^4 + 3*a*b^2*c^3 + b^3*c^2)*x^4)]
\[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^{5} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (157) = 314\).
Time = 0.30 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.03 \[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {{\left (4 \, a b c + b^{2}\right )} d^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{16 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + b^{2} c\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {{\left (4 \, a b c^{2} + b^{2} c\right )} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c^{2} + 3 \, a b^{2} c - b^{3}\right )} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{4} c^{5} + 4 \, a^{3} b c^{4} + 6 \, a^{2} b^{2} c^{3} + 4 \, a b^{3} c^{2} + b^{4} c + \frac {{\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {2 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \]
-1/16*(4*a*b*c + b^2)*d^2*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - s qrt((a*c + b)*c))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b )*c)))/((a^2*c^3 + 2*a*b*c^2 + b^2*c)*sqrt((a*c + b)*c)) - 1/8*((4*a*b*c^2 + b^2*c)*d^2*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) - (4*a^2*b*c^2 + 3*a *b^2*c - b^3)*d^2*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*c^5 + 4*a^3* b*c^4 + 6*a^2*b^2*c^3 + 4*a*b^3*c^2 + b^4*c + (a^2*c^5 + 2*a*b*c^4 + b^2*c ^3)*(a*d*x^2 + a*c + b)^2/(d*x^2 + c)^2 - 2*(a^3*c^5 + 3*a^2*b*c^4 + 3*a*b ^2*c^3 + b^3*c^2)*(a*d*x^2 + a*c + b)/(d*x^2 + c))
Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (157) = 314\).
Time = 0.41 (sec) , antiderivative size = 778, normalized size of antiderivative = 4.40 \[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\frac {{\left (4 \, a b c d^{2} + b^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{\sqrt {-a c^{2} - b c}}\right )}{{\left (a^{2} c^{3} + 2 \, a b c^{2} + b^{2} c\right )} \sqrt {-a c^{2} - b c}} - \frac {8 \, a^{\frac {7}{2}} c^{5} d {\left | d \right |} + 16 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{3} c^{4} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {5}{2}} c^{3} d {\left | d \right |} + 16 \, a^{\frac {5}{2}} b c^{4} d {\left | d \right |} + 28 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} b c^{3} d^{2} + 16 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} b c^{2} d {\left | d \right |} + 8 \, a^{\frac {3}{2}} b^{2} c^{3} d {\left | d \right |} + 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a b c d^{2} + 13 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b^{2} c^{2} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} \sqrt {a} b^{2} c d {\left | d \right |} + {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} b^{2} d^{2} + {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} b^{3} c d^{2}}{{\left (a^{2} c^{3} + 2 \, a b c^{2} + b^{2} c\right )} {\left (a c^{2} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} + b c\right )}^{2}}}{8 \, \mathrm {sgn}\left (d x^{2} + c\right )} \]
-1/8*((4*a*b*c*d^2 + b^2*d^2)*arctan(-(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))/sqrt(-a*c^2 - b*c))/((a^2*c^3 + 2*a* b*c^2 + b^2*c)*sqrt(-a*c^2 - b*c)) - (8*a^(7/2)*c^5*d*abs(d) + 16*(sqrt(a* d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^3*c^4* d^2 + 8*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^(5/2)*c^3*d*abs(d) + 16*a^(5/2)*b*c^4*d*abs(d) + 28*(sqrt(a*d^ 2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^2*b*c^3* d^2 + 16*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^(3/2)*b*c^2*d*abs(d) + 8*a^(3/2)*b^2*c^3*d*abs(d) + 4*(sqrt(a *d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*a*b*c *d^2 + 13*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^ 2 + b*c))*a*b^2*c^2*d^2 + 8*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^ 2 + b*d*x^2 + a*c^2 + b*c))^2*sqrt(a)*b^2*c*d*abs(d) + (sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*b^2*d^2 + (sqrt(a *d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*b^3*c*d ^2)/((a^2*c^3 + 2*a*b*c^2 + b^2*c)*(a*c^2 - (sqrt(a*d^2)*x^2 - sqrt(a*d^2* x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2 + b*c)^2))/sgn(d*x^2 + c)
Timed out. \[ \int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^5\,\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]