Integrand size = 24, antiderivative size = 176 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}} \]
b^3*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(5/2)/(-a*d+b*c)^ (3/2)-d/c/(-a*d+b*c)/x^3/(d*x^2+c)^(1/2)-1/3*(-4*a*d+b*c)*(d*x^2+c)^(1/2)/ a/c^2/(-a*d+b*c)/x^3+1/3*(-4*a*d+3*b*c)*(2*a*d+b*c)*(d*x^2+c)^(1/2)/a^2/c^ 3/(-a*d+b*c)/x
Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {3 b^2 c^2 x^2 \left (c+d x^2\right )+a^2 d \left (c^2-4 c d x^2-8 d^2 x^4\right )+a b c \left (-c^2+c d x^2+2 d^2 x^4\right )}{3 a^2 c^3 (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {b^3 \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2} (b c-a d)^{3/2}} \]
(3*b^2*c^2*x^2*(c + d*x^2) + a^2*d*(c^2 - 4*c*d*x^2 - 8*d^2*x^4) + a*b*c*( -c^2 + c*d*x^2 + 2*d^2*x^4))/(3*a^2*c^3*(b*c - a*d)*x^3*Sqrt[c + d*x^2]) - (b^3*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt [b*c - a*d])])/(a^(5/2)*(b*c - a*d)^(3/2))
Time = 0.36 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {374, 445, 445, 27, 291, 218}
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.
| \(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\int \frac {-4 b d x^2+b c-4 a d}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{c (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int \frac {2 b d (b c-4 a d) x^2+(3 b c-4 a d) (b c+2 a d)}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{c (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {3 b^3 c^3}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{a c x}}{3 a c}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{c (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 b^3 c^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{a c x}}{3 a c}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{c (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {-\frac {-\frac {3 b^3 c^2 \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{a}-\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{a c x}}{3 a c}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{c (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {-\frac {3 b^3 c^2 \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{a c x}}{3 a c}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{c (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)}\) |
-(d/(c*(b*c - a*d)*x^3*Sqrt[c + d*x^2])) + (-1/3*((b*c - 4*a*d)*Sqrt[c + d *x^2])/(a*c*x^3) - (-(((3*b*c - 4*a*d)*(b*c + 2*a*d)*Sqrt[c + d*x^2])/(a*c *x)) - (3*b^3*c^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/( a^(3/2)*Sqrt[b*c - a*d]))/(3*a*c))/(c*(b*c - a*d))
3.8.21.3.1 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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Time = 3.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.71
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (-5 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 x^{3} a^{2}}+\frac {d^{3} x}{\left (a d -b c \right ) \sqrt {d \,x^{2}+c}}-\frac {b^{3} c^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\left (a d -b c \right ) a^{2} \sqrt {\left (a d -b c \right ) a}}}{c^{3}}\) | \(125\) |
| risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-5 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 c^{3} a^{2} x^{3}}-\frac {b \,d^{3} \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{2 c^{3} \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {b^{4} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b \,d^{3} \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 c^{3} \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {b^{4} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}\) | \(641\) |
| default | \(\frac {-\frac {1}{3 c \,x^{3} \sqrt {d \,x^{2}+c}}-\frac {4 d \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )}{3 c}}{a}-\frac {b \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )}{a^{2}}+\frac {b^{2} \left (-\frac {b}{\left (a d -b c \right ) \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 a^{2} \sqrt {-a b}}-\frac {b^{2} \left (-\frac {b}{\left (a d -b c \right ) \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 a^{2} \sqrt {-a b}}\) | \(847\) |
(-1/3*(d*x^2+c)^(1/2)*(-5*a*d*x^2-3*b*c*x^2+a*c)/x^3/a^2+d^3/(a*d-b*c)/(d* x^2+c)^(1/2)*x-1/(a*d-b*c)/a^2*b^3*c^3/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+ c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)))/c^3
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (156) = 312\).
Time = 0.42 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.01 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c^{2} d^{2} - 10 \, a^{3} b c d^{3} + 8 \, a^{4} d^{4}\right )} x^{4} - {\left (3 \, a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d - 5 \, a^{3} b c^{2} d^{2} + 4 \, a^{4} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{3} b^{2} c^{5} d - 2 \, a^{4} b c^{4} d^{2} + a^{5} c^{3} d^{3}\right )} x^{5} + {\left (a^{3} b^{2} c^{6} - 2 \, a^{4} b c^{5} d + a^{5} c^{4} d^{2}\right )} x^{3}\right )}}, \frac {3 \, {\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c^{2} d^{2} - 10 \, a^{3} b c d^{3} + 8 \, a^{4} d^{4}\right )} x^{4} - {\left (3 \, a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d - 5 \, a^{3} b c^{2} d^{2} + 4 \, a^{4} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{3} b^{2} c^{5} d - 2 \, a^{4} b c^{4} d^{2} + a^{5} c^{3} d^{3}\right )} x^{5} + {\left (a^{3} b^{2} c^{6} - 2 \, a^{4} b c^{5} d + a^{5} c^{4} d^{2}\right )} x^{3}\right )}}\right ] \]
[1/12*(3*(b^3*c^3*d*x^5 + b^3*c^4*x^3)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4 *((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^ 4 + 2*a*b*x^2 + a^2)) - 4*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 - (3* a*b^3*c^3*d - a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 8*a^4*d^4)*x^4 - (3*a*b^3 *c^4 - 2*a^2*b^2*c^3*d - 5*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*x^2)*sqrt(d*x^2 + c))/((a^3*b^2*c^5*d - 2*a^4*b*c^4*d^2 + a^5*c^3*d^3)*x^5 + (a^3*b^2*c^6 - 2*a^4*b*c^5*d + a^5*c^4*d^2)*x^3), 1/6*(3*(b^3*c^3*d*x^5 + b^3*c^4*x^3)*sq rt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c) *sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*(a ^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 - (3*a*b^3*c^3*d - a^2*b^2*c^2*d^ 2 - 10*a^3*b*c*d^3 + 8*a^4*d^4)*x^4 - (3*a*b^3*c^4 - 2*a^2*b^2*c^3*d - 5*a ^3*b*c^2*d^2 + 4*a^4*c*d^3)*x^2)*sqrt(d*x^2 + c))/((a^3*b^2*c^5*d - 2*a^4* b*c^4*d^2 + a^5*c^3*d^3)*x^5 + (a^3*b^2*c^6 - 2*a^4*b*c^5*d + a^5*c^4*d^2) *x^3)]
\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Time = 0.86 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{3} \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c - a^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {d^{3} x}{{\left (b c^{4} - a c^{3} d\right )} \sqrt {d x^{2} + c}} - \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d} + 5 \, a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} c^{2}} \]
b^3*sqrt(d)*arctan(-1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/ sqrt(a*b*c*d - a^2*d^2))/((a^2*b*c - a^3*d)*sqrt(a*b*c*d - a^2*d^2)) - d^3 *x/((b*c^4 - a*c^3*d)*sqrt(d*x^2 + c)) - 2/3*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c*sqrt(d) + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d^(3/2) - 6*(sqrt( d)*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) - 12*(sqrt(d)*x - sqrt(d*x^2 + c)) ^2*a*c*d^(3/2) + 3*b*c^3*sqrt(d) + 5*a*c^2*d^(3/2))/(((sqrt(d)*x - sqrt(d* x^2 + c))^2 - c)^3*a^2*c^2)
Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^4\,\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]