Integrand size = 24, antiderivative size = 156 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d (b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \sqrt {c+d x^2}}+\frac {(2 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{5/2}}-\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 (b c-a d)^{3/2}} \]
1/2*(3*a*d+2*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^2/c^(5/2)-b^(5/2)*arc tanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/a^2/(-a*d+b*c)^(3/2)-1/2*d* (-3*a*d+b*c)/a/c^2/(-a*d+b*c)/(d*x^2+c)^(1/2)-1/2/a/c/x^2/(d*x^2+c)^(1/2)
Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {a \left (-b c \left (c+d x^2\right )+a d \left (c+3 d x^2\right )\right )}{c^2 (b c-a d) x^2 \sqrt {c+d x^2}}-\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(2 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{5/2}}}{2 a^2} \]
((a*(-(b*c*(c + d*x^2)) + a*d*(c + 3*d*x^2)))/(c^2*(b*c - a*d)*x^2*Sqrt[c + d*x^2]) - (2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d] ])/(-(b*c) + a*d)^(3/2) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c] ])/c^(5/2))/(2*a^2)
Time = 0.35 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {354, 114, 27, 169, 27, 174, 73, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {3 b d x^2+2 b c+3 a d}{2 x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2}{a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {3 b d x^2+2 b c+3 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2}{2 a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {2 d (b c-3 a d)}{c \sqrt {c+d x^2} (b c-a d)}-\frac {2 \int -\frac {b d (b c-3 a d) x^2+(b c-a d) (2 b c+3 a d)}{2 x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{c (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\int \frac {b d (b c-3 a d) x^2+(b c-a d) (2 b c+3 a d)}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{c (b c-a d)}+\frac {2 d (b c-3 a d)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {(b c-a d) (3 a d+2 b c) \int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {2 b^3 c^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a}}{c (b c-a d)}+\frac {2 d (b c-3 a d)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 (b c-a d) (3 a d+2 b c) \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{a d}-\frac {4 b^3 c^2 \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{a d}}{c (b c-a d)}+\frac {2 d (b c-3 a d)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {4 b^{5/2} c^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d) (3 a d+2 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a \sqrt {c}}}{c (b c-a d)}+\frac {2 d (b c-3 a d)}{c \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \sqrt {c+d x^2}}\right )\) |
(-(1/(a*c*x^2*Sqrt[c + d*x^2])) - ((2*d*(b*c - 3*a*d))/(c*(b*c - a*d)*Sqrt [c + d*x^2]) + ((-2*(b*c - a*d)*(2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sq rt[c]])/(a*Sqrt[c]) + (4*b^(5/2)*c^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqr t[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(c*(b*c - a*d)))/(2*a*c))/2
3.8.20.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 3.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(d^{2} \left (-\frac {b^{3} \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) a^{2} d^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right ) a d \,x^{2}+2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right ) b c \,x^{2}-\sqrt {d \,x^{2}+c}\, a \sqrt {c}}{2 x^{2} c^{\frac {5}{2}} a^{2} d^{2}}-\frac {1}{\left (a d -b c \right ) c^{2} \sqrt {d \,x^{2}+c}}\right )\) | \(159\) |
| risch | \(-\frac {\sqrt {d \,x^{2}+c}}{2 c^{2} a \,x^{2}}-\frac {-\frac {\left (3 a d +2 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a \sqrt {c}}-\frac {b \,d^{3} a \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{\left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {b^{3} d \,c^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {b \,d^{3} a \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{\left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {b^{3} d \,c^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \sqrt {-\frac {a d -b c}{b}}}}{2 a \,c^{2}}\) | \(673\) |
| default | \(\frac {-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}}{a}-\frac {b \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{a^{2}}+\frac {b \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}}+\frac {b \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}}\) | \(847\) |
d^2*(-1/(a*d-b*c)*b^3/a^2/d^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x^2+c)^(1/2) /((a*d-b*c)*b)^(1/2))+1/2*(3*arctanh((d*x^2+c)^(1/2)/c^(1/2))*a*d*x^2+2*ar ctanh((d*x^2+c)^(1/2)/c^(1/2))*b*c*x^2-(d*x^2+c)^(1/2)*a*c^(1/2))/x^2/c^(5 /2)/a^2/d^2-1/(a*d-b*c)/c^2/(d*x^2+c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (130) = 260\).
Time = 0.61 (sec) , antiderivative size = 1291, normalized size of antiderivative = 8.28 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/4*((b^2*c^3*d*x^4 + b^2*c^4*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(2*b ^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sq rt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - ((2*b^2*c^2*d + a*b*c*d^ 2 - 3*a^2*d^3)*x^4 + (2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^2)*x^2)*sqrt(c)*lo g(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(a*b*c^3 - a^2*c^2*d + (a*b*c^2*d - 3*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c))/((a^2*b*c^4*d - a^3*c^3 *d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2), -1/4*(2*((2*b^2*c^2*d + a*b*c*d^ 2 - 3*a^2*d^3)*x^4 + (2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^2)*x^2)*sqrt(-c)*a rctan(sqrt(-c)/sqrt(d*x^2 + c)) + (b^2*c^3*d*x^4 + b^2*c^4*x^2)*sqrt(b/(b* c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c* d - 3*a*b*d^2)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d ^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(a*b*c^3 - a^2*c^2*d + (a*b*c^2*d - 3*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c) )/((a^2*b*c^4*d - a^3*c^3*d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2), 1/4*(2* (b^2*c^3*d*x^4 + b^2*c^4*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2 *b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) + ((2*b^ 2*c^2*d + a*b*c*d^2 - 3*a^2*d^3)*x^4 + (2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^ 2)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(a *b*c^3 - a^2*c^2*d + (a*b*c^2*d - 3*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c))/((...
\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{3} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{2} b c - a^{3} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {{\left (d x^{2} + c\right )} b c d - 3 \, {\left (d x^{2} + c\right )} a d^{2} + 2 \, a c d^{2}}{2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} - \sqrt {d x^{2} + c} c\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c} c^{2}} \]
b^3*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b*c - a^3*d)*sqrt (-b^2*c + a*b*d)) - 1/2*((d*x^2 + c)*b*c*d - 3*(d*x^2 + c)*a*d^2 + 2*a*c*d ^2)/((a*b*c^3 - a^2*c^2*d)*((d*x^2 + c)^(3/2) - sqrt(d*x^2 + c)*c)) - 1/2* (2*b*c + 3*a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^2*sqrt(-c)*c^2)
Time = 6.76 (sec) , antiderivative size = 3025, normalized size of antiderivative = 19.39 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
(d^2/(b*c^2 - a*c*d) + (d*(c + d*x^2)*(3*a*d - b*c))/(2*a*c^2*(a*d - b*c)) )/(c*(c + d*x^2)^(1/2) - (c + d*x^2)^(3/2)) + (atan((((-b^5*(a*d - b*c)^3) ^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^10*c^13*d^2 - 320*a^4*b^9*c^12*d^3 + 16*a^5*b^8*c^11*d^4 + 496*a^6*b^7*c^10*d^5 - 160*a^7*b^6*c^9*d^6 - 544*a^ 8*b^5*c^8*d^7 + 528*a^9*b^4*c^7*d^8 - 144*a^10*b^3*c^6*d^9))/2 - ((-b^5*(a *d - b*c)^3)^(1/2)*(416*a^8*b^6*c^12*d^5 - 32*a^6*b^8*c^14*d^3 - 1024*a^9* b^5*c^11*d^6 + 1056*a^10*b^4*c^10*d^7 - 512*a^11*b^3*c^9*d^8 + 96*a^12*b^2 *c^8*d^9 + ((-b^5*(a*d - b*c)^3)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^8*c^16 *d^2 - 2816*a^8*b^7*c^15*d^3 + 6400*a^9*b^6*c^14*d^4 - 7680*a^10*b^5*c^13* d^5 + 5120*a^11*b^4*c^12*d^6 - 1792*a^12*b^3*c^11*d^7 + 256*a^13*b^2*c^10* d^8))/(4*a^2*(a*d - b*c)^3)))/(2*a^2*(a*d - b*c)^3))*1i)/(a^2*(a*d - b*c)^ 3) + ((-b^5*(a*d - b*c)^3)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^10*c^13*d^ 2 - 320*a^4*b^9*c^12*d^3 + 16*a^5*b^8*c^11*d^4 + 496*a^6*b^7*c^10*d^5 - 16 0*a^7*b^6*c^9*d^6 - 544*a^8*b^5*c^8*d^7 + 528*a^9*b^4*c^7*d^8 - 144*a^10*b ^3*c^6*d^9))/2 - ((-b^5*(a*d - b*c)^3)^(1/2)*(32*a^6*b^8*c^14*d^3 - 416*a^ 8*b^6*c^12*d^5 + 1024*a^9*b^5*c^11*d^6 - 1056*a^10*b^4*c^10*d^7 + 512*a^11 *b^3*c^9*d^8 - 96*a^12*b^2*c^8*d^9 + ((-b^5*(a*d - b*c)^3)^(1/2)*(c + d*x^ 2)^(1/2)*(512*a^7*b^8*c^16*d^2 - 2816*a^8*b^7*c^15*d^3 + 6400*a^9*b^6*c^14 *d^4 - 7680*a^10*b^5*c^13*d^5 + 5120*a^11*b^4*c^12*d^6 - 1792*a^12*b^3*c^1 1*d^7 + 256*a^13*b^2*c^10*d^8))/(4*a^2*(a*d - b*c)^3)))/(2*a^2*(a*d - b...