Integrand size = 24, antiderivative size = 241 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{2 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(4 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{5/2}}-\frac {b^{5/2} (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{5/2}} \]
1/2*(3*a*d+4*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^3/c^(5/2)-1/2*b^(5/2) *(-7*a*d+4*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a* d+b*c)^(5/2)-1/2*d*(3*a^2*d^2-2*a*b*c*d+2*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(d *x^2+c)^(1/2)-1/2*b*(-a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^(1/2 )-1/2/a/c/x^2/(b*x^2+a)/(d*x^2+c)^(1/2)
Time = 0.89 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\frac {a \left (2 b^3 c^2 x^2 \left (c+d x^2\right )+a^3 d^2 \left (c+3 d x^2\right )+a b^2 c \left (c^2-c d x^2-2 d^2 x^4\right )+a^2 b d \left (-2 c^2-c d x^2+3 d^2 x^4\right )\right )}{c^2 (b c-a d)^2 x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {b^{5/2} (4 b c-7 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}+\frac {(4 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{5/2}}}{2 a^3} \]
(-((a*(2*b^3*c^2*x^2*(c + d*x^2) + a^3*d^2*(c + 3*d*x^2) + a*b^2*c*(c^2 - c*d*x^2 - 2*d^2*x^4) + a^2*b*d*(-2*c^2 - c*d*x^2 + 3*d^2*x^4)))/(c^2*(b*c - a*d)^2*x^2*(a + b*x^2)*Sqrt[c + d*x^2])) + (b^(5/2)*(4*b*c - 7*a*d)*ArcT an[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2) + ( (4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(5/2))/(2*a^3)
Time = 0.44 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {354, 114, 27, 168, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {5 b d x^2+4 b c+3 a d}{2 x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{3/2}}dx^2}{a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {5 b d x^2+4 b c+3 a d}{x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{3/2}}dx^2}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\int \frac {3 b d (2 b c-a d) x^2+(b c-a d) (4 b c+3 a d)}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}-\frac {2 \int -\frac {(4 b c+3 a d) (b c-a d)^2+b d \left (2 b^2 c^2-2 a b d c+3 a^2 d^2\right ) x^2}{2 x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\int \frac {(4 b c+3 a d) (b c-a d)^2+b d \left (2 b^2 c^2-2 a b d c+3 a^2 d^2\right ) x^2}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{c (b c-a d)}+\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {(b c-a d)^2 (3 a d+4 b c) \int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {b^3 c^2 (4 b c-7 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a}}{c (b c-a d)}+\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {2 (b c-a d)^2 (3 a d+4 b c) \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{a d}-\frac {2 b^3 c^2 (4 b c-7 a d) \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{a d}}{c (b c-a d)}+\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}+\frac {\frac {2 b^{5/2} c^2 (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d)^2 (3 a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a \sqrt {c}}}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}\right )\) |
(-(1/(a*c*x^2*(a + b*x^2)*Sqrt[c + d*x^2])) - ((2*b*(2*b*c - a*d))/(a*(b*c - a*d)*(a + b*x^2)*Sqrt[c + d*x^2]) + ((2*d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^ 2*d^2))/(c*(b*c - a*d)*Sqrt[c + d*x^2]) + ((-2*(b*c - a*d)^2*(4*b*c + 3*a* d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(5/2)*c^2*(4*b*c - 7*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(c*(b*c - a*d)))/(a*(b*c - a*d)))/(2*a*c))/2
3.8.74.3.1 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] && ILtQ[m, -1]
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 = 3.33 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(-\frac {-4 \left (b \,x^{2}+a \right ) x^{2} c^{\frac {9}{2}} \left (b c -\frac {7 a d}{4}\right ) b^{3} \sqrt {d \,x^{2}+c}\, \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (-3 \left (b \,x^{2}+a \right ) x^{2} \left (a d +\frac {4 b c}{3}\right ) \sqrt {d \,x^{2}+c}\, c^{2} \left (a d -b c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+c^{\frac {5}{2}} \left (\left (2 c^{2} d \,x^{4}+2 c^{3} x^{2}\right ) b^{3}+b^{2} c \left (d \,x^{2}+c \right ) \left (-2 d \,x^{2}+c \right ) a -2 d \left (-d \,x^{2}+c \right ) \left (\frac {3 d \,x^{2}}{2}+c \right ) a^{2} b +a^{3} d^{2} \left (3 d \,x^{2}+c \right )\right ) a \right )}{2 \sqrt {d \,x^{2}+c}\, \sqrt {\left (a d -b c \right ) b}\, a^{3} \left (b \,x^{2}+a \right ) \left (a d -b c \right )^{2} x^{2} c^{\frac {9}{2}}}\) | \(269\) |
| risch | \(\text {Expression too large to display}\) | \(1305\) |
| default | \(\text {Expression too large to display}\) | \(2042\) |
-1/2/(d*x^2+c)^(1/2)*(-4*(b*x^2+a)*x^2*c^(9/2)*(b*c-7/4*a*d)*b^3*(d*x^2+c) ^(1/2)*arctan(b*(d*x^2+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*( -3*(b*x^2+a)*x^2*(a*d+4/3*b*c)*(d*x^2+c)^(1/2)*c^2*(a*d-b*c)^2*arctanh((d* x^2+c)^(1/2)/c^(1/2))+c^(5/2)*((2*c^2*d*x^4+2*c^3*x^2)*b^3+b^2*c*(d*x^2+c) *(-2*d*x^2+c)*a-2*d*(-d*x^2+c)*(3/2*d*x^2+c)*a^2*b+a^3*d^2*(3*d*x^2+c))*a) )/((a*d-b*c)*b)^(1/2)/a^3/(b*x^2+a)/(a*d-b*c)^2/x^2/c^(9/2)
Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (209) = 418\).
Time = 3.15 (sec) , antiderivative size = 2554, normalized size of antiderivative = 10.60 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/8*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^6 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^4 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*d)*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*((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^6 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x ^4 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^2)* sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 4*(a^2*b^2*c ^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3* a^3*b*c*d^3)*x^4 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c* d^3)*x^2)*sqrt(d*x^2 + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3 *d^3)*x^6 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^ 4 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2), -1/8*(4*((4*b^4*c^3* d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^6 + (4*b^4*c^4 - a* b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^4 + (4*a*b^3*c^ 4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^2)*sqrt(-c)*arctan( sqrt(-c)/sqrt(d*x^2 + c)) + ((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^6 + (4*b^4* c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^4 + (4*a*b^3*c^5 - 7*a^2*b^2*c^ 4*d)*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 ...
\[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.52 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{2} + c\right )}^{2} b^{3} c^{2} d - 2 \, {\left (d x^{2} + c\right )} b^{3} c^{3} d - 2 \, {\left (d x^{2} + c\right )}^{2} a b^{2} c d^{2} + 3 \, {\left (d x^{2} + c\right )} a b^{2} c^{2} d^{2} + 3 \, {\left (d x^{2} + c\right )}^{2} a^{2} b d^{3} - 7 \, {\left (d x^{2} + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \, {\left (d x^{2} + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {5}{2}} b - 2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b c + \sqrt {d x^{2} + c} b c^{2} + {\left (d x^{2} + c\right )}^{\frac {3}{2}} a d - \sqrt {d x^{2} + c} a c d\right )}} - \frac {{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c} c^{2}} \]
1/2*(4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/( (a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt(-b^2*c + a*b*d)) - 1/2*(2*(d*x^ 2 + c)^2*b^3*c^2*d - 2*(d*x^2 + c)*b^3*c^3*d - 2*(d*x^2 + c)^2*a*b^2*c*d^2 + 3*(d*x^2 + c)*a*b^2*c^2*d^2 + 3*(d*x^2 + c)^2*a^2*b*d^3 - 7*(d*x^2 + c) *a^2*b*c*d^3 + 2*a^2*b*c^2*d^3 + 3*(d*x^2 + c)*a^3*d^4 - 2*a^3*c*d^4)/((a^ 2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*((d*x^2 + c)^(5/2)*b - 2*(d*x^2 + c)^(3/2)*b*c + sqrt(d*x^2 + c)*b*c^2 + (d*x^2 + c)^(3/2)*a*d - sqrt(d*x^2 + c)*a*c*d)) - 1/2*(4*b*c + 3*a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3* sqrt(-c)*c^2)
Time = 9.08 (sec) , antiderivative size = 4286, normalized size of antiderivative = 17.78 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
(atan((((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*((c + d*x^2)^(1/2)*(512 *a^6*b^15*c^18*d^2 - 4608*a^7*b^14*c^17*d^3 + 17824*a^8*b^13*c^16*d^4 - 38 144*a^9*b^12*c^15*d^5 + 47680*a^10*b^11*c^14*d^6 - 31808*a^11*b^10*c^13*d^ 7 + 4624*a^12*b^9*c^12*d^8 + 8032*a^13*b^8*c^11*d^9 - 3536*a^14*b^7*c^10*d ^10 - 2560*a^15*b^6*c^9*d^11 + 2896*a^16*b^5*c^8*d^12 - 1056*a^17*b^4*c^7* d^13 + 144*a^18*b^3*c^6*d^14) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c )*(128*a^10*b^13*c^19*d^3 - 1216*a^11*b^12*c^18*d^4 + 4800*a^12*b^11*c^17* d^5 - 9792*a^13*b^10*c^16*d^6 + 9216*a^14*b^9*c^15*d^7 + 2688*a^15*b^8*c^1 4*d^8 - 18816*a^16*b^7*c^13*d^9 + 24960*a^17*b^6*c^12*d^10 - 18048*a^18*b^ 5*c^11*d^11 + 7744*a^19*b^4*c^10*d^12 - 1856*a^20*b^3*c^9*d^13 + 192*a^21* b^2*c^8*d^14 - ((-b^5*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 4*b* c)*(512*a^12*b^13*c^21*d^2 - 5376*a^13*b^12*c^20*d^3 + 25600*a^14*b^11*c^1 9*d^4 - 72960*a^15*b^10*c^18*d^5 + 138240*a^16*b^9*c^17*d^6 - 182784*a^17* b^8*c^16*d^7 + 172032*a^18*b^7*c^15*d^8 - 115200*a^19*b^6*c^14*d^9 + 53760 *a^20*b^5*c^13*d^10 - 16640*a^21*b^4*c^12*d^11 + 3072*a^22*b^3*c^11*d^12 - 256*a^23*b^2*c^10*d^13))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10 *a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^ 3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5* a^7*b*c*d^4)))*1i)/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^ 3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) + ((-b^5*(a*d - b*c)^5...