Integrand size = 24, antiderivative size = 304 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{7/2}}-\frac {b^{7/2} (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}} \]
-1/6*d*(5*a^2*d^2-6*a*b*c*d+6*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(d*x^2+c)^(3/2 )-1/2*b*(-a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^(3/2)-1/2/a/c/x^ 2/(b*x^2+a)/(d*x^2+c)^(3/2)+1/2*(5*a*d+4*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1 /2))/a^3/c^(7/2)-1/2*b^(7/2)*(-9*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)^(7/2)-1/2*d*(-a*d+2*b*c)*(5*a^2*d^2-a*b *c*d+b^2*c^2)/a^2/c^3/(-a*d+b*c)^3/(d*x^2+c)^(1/2)
Time = 1.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {a \left (-6 b^4 c^3 x^2 \left (c+d x^2\right )^2-3 a b^3 c^2 \left (c-3 d x^2\right ) \left (c+d x^2\right )^2+a^4 d^3 \left (3 c^2+20 c d x^2+15 d^2 x^4\right )+a^2 b^2 c d \left (9 c^3+9 c^2 d x^2-35 c d^2 x^4-33 d^3 x^6\right )+a^3 b d^2 \left (-9 c^3-41 c^2 d x^2-13 c d^2 x^4+15 d^3 x^6\right )\right )}{c^3 (b c-a d)^3 x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {3 b^{7/2} (4 b c-9 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}+\frac {3 (4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{7/2}}}{6 a^3} \]
((a*(-6*b^4*c^3*x^2*(c + d*x^2)^2 - 3*a*b^3*c^2*(c - 3*d*x^2)*(c + d*x^2)^ 2 + a^4*d^3*(3*c^2 + 20*c*d*x^2 + 15*d^2*x^4) + a^2*b^2*c*d*(9*c^3 + 9*c^2 *d*x^2 - 35*c*d^2*x^4 - 33*d^3*x^6) + a^3*b*d^2*(-9*c^3 - 41*c^2*d*x^2 - 1 3*c*d^2*x^4 + 15*d^3*x^6)))/(c^3*(b*c - a*d)^3*x^2*(a + b*x^2)*(c + d*x^2) ^(3/2)) - (3*b^(7/2)*(4*b*c - 9*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt [-(b*c) + a*d]])/(-(b*c) + a*d)^(7/2) + (3*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(7/2))/(6*a^3)
Time = 0.55 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {354, 114, 27, 168, 169, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{5/2}}dx^2\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {7 b d x^2+4 b c+5 a d}{2 x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{5/2}}dx^2}{a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {7 b d x^2+4 b c+5 a d}{x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{5/2}}dx^2}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\int \frac {5 b d (2 b c-a d) x^2+(b c-a d) (4 b c+5 a d)}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^{5/2}}dx^2}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {2 \int -\frac {3 \left ((4 b c+5 a d) (b c-a d)^2+b d \left (6 b^2 c^2-6 a b d c+5 a^2 d^2\right ) x^2\right )}{2 x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2}{3 c (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\int \frac {(4 b c+5 a d) (b c-a d)^2+b d \left (6 b^2 c^2-6 a b d c+5 a^2 d^2\right ) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}-\frac {2 \int -\frac {(4 b c+5 a d) (b c-a d)^3+b d (2 b c-a d) \left (b^2 c^2-a b d c+5 a^2 d^2\right ) x^2}{2 x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{c (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {\int \frac {(4 b c+5 a d) (b c-a d)^3+b d (2 b c-a d) \left (b^2 c^2-a b d c+5 a^2 d^2\right ) x^2}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{c (b c-a d)}+\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {\frac {(b c-a d)^3 (5 a d+4 b c) \int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {b^4 c^3 (4 b c-9 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a}}{c (b c-a d)}+\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {\frac {2 (b c-a d)^3 (5 a d+4 b c) \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{a d}-\frac {2 b^4 c^3 (4 b c-9 a d) \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{a d}}{c (b c-a d)}+\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x^2} (b c-a d)}+\frac {\frac {2 b^{7/2} c^3 (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d)^3 (5 a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a \sqrt {c}}}{c (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}\right )\) |
(-(1/(a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2))) - ((2*b*(2*b*c - a*d))/(a*(b *c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((2*d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) + ((2*d*(2*b*c - a*d)*(b^2 *c^2 - a*b*c*d + 5*a^2*d^2))/(c*(b*c - a*d)*Sqrt[c + d*x^2]) + ((-2*(b*c - a*d)^3*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(a*Sqrt[c]) + (2 *b^(7/2)*c^3*(4*b*c - 9*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)))/(c*(b*c - a*d)))/(a*(b*c - a* d)))/(2*a*c))/2
3.8.83.3.1 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 = 3.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(d^{3} \left (\frac {\sqrt {d \,x^{2}+c}\, b^{4}}{2 a^{2} d^{3} \left (b \,x^{2}+a \right ) \left (a d -b c \right )^{3}}+\frac {9 \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) b^{4}}{2 \sqrt {\left (a d -b c \right ) b}\, a^{2} d^{2} \left (a d -b c \right )^{3}}-\frac {2 \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) b^{5} c}{\sqrt {\left (a d -b c \right ) b}\, a^{3} d^{3} \left (a d -b c \right )^{3}}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right ) a d \,x^{2}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right ) b c \,x^{2}-\sqrt {d \,x^{2}+c}\, a \sqrt {c}}{2 x^{2} c^{\frac {7}{2}} a^{3} d^{3}}-\frac {1}{3 c^{2} \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 \left (a d -2 b c \right )}{c^{3} \left (a d -b c \right )^{3} \sqrt {d \,x^{2}+c}}\right )\) | \(288\) |
| risch | \(\text {Expression too large to display}\) | \(2418\) |
| default | \(\text {Expression too large to display}\) | \(3641\) |
d^3*(1/2*(d*x^2+c)^(1/2)*b^4/a^2/d^3/(b*x^2+a)/(a*d-b*c)^3+9/2/((a*d-b*c)* b)^(1/2)*arctan(b*(d*x^2+c)^(1/2)/((a*d-b*c)*b)^(1/2))/a^2/d^2*b^4/(a*d-b* c)^3-2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x^2+c)^(1/2)/((a*d-b*c)*b)^(1/2))*b ^5*c/a^3/d^3/(a*d-b*c)^3+1/2*(5*arctanh((d*x^2+c)^(1/2)/c^(1/2))*a*d*x^2+4 *arctanh((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 ^(7/2)/a^3/d^3-1/3/c^2/(a*d-b*c)^2/(d*x^2+c)^(3/2)-2*(a*d-2*b*c)/c^3/(a*d- b*c)^3/(d*x^2+c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (268) = 536\).
Time = 7.96 (sec) , antiderivative size = 4115, normalized size of antiderivative = 13.54 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/24*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4* c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c ^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*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)) + 6*((4*b^5* c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b *d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3* b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17 *a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^ 4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2* c)/x^2) - 4*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3 *d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4 *b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c ))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d ^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b *c^5*d^4 - a^7*c^4*d^5)*x^6 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*...
\[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.66 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{4} c^{3} d - 2 \, \sqrt {d x^{2} + c} b^{4} c^{4} d - 3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{2} + 4 \, \sqrt {d x^{2} + c} a b^{3} c^{3} d^{2} + 3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{3} - 6 \, \sqrt {d x^{2} + c} a^{2} b^{2} c^{2} d^{3} - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{3} b d^{4} + 4 \, \sqrt {d x^{2} + c} a^{3} b c d^{4} - \sqrt {d x^{2} + c} a^{4} d^{5}}{2 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )} {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac {12 \, {\left (d x^{2} + c\right )} b c d^{3} + b c^{2} d^{3} - 6 \, {\left (d x^{2} + c\right )} a d^{4} - a c d^{4}}{3 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c} c^{3}} \]
1/2*(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/( (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*sqrt(-b^2*c + a* b*d)) - 1/2*(2*(d*x^2 + c)^(3/2)*b^4*c^3*d - 2*sqrt(d*x^2 + c)*b^4*c^4*d - 3*(d*x^2 + c)^(3/2)*a*b^3*c^2*d^2 + 4*sqrt(d*x^2 + c)*a*b^3*c^3*d^2 + 3*( d*x^2 + c)^(3/2)*a^2*b^2*c*d^3 - 6*sqrt(d*x^2 + c)*a^2*b^2*c^2*d^3 - (d*x^ 2 + c)^(3/2)*a^3*b*d^4 + 4*sqrt(d*x^2 + c)*a^3*b*c*d^4 - sqrt(d*x^2 + c)*a ^4*d^5)/((a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*( (d*x^2 + c)^2*b - 2*(d*x^2 + c)*b*c + b*c^2 + (d*x^2 + c)*a*d - a*c*d)) - 1/3*(12*(d*x^2 + c)*b*c*d^3 + b*c^2*d^3 - 6*(d*x^2 + c)*a*d^4 - a*c*d^4)/( (b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(d*x^2 + c)^(3/2 )) - 1/2*(4*b*c + 5*a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^ 3)
Time = 11.32 (sec) , antiderivative size = 5800, normalized size of antiderivative = 19.08 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
((5*d^3*(c + d*x^2)*(a*d - 2*b*c))/(3*(b*c^2 - a*c*d)^2) - d^3/(3*(b*c^2 - a*c*d)) + (d*(c + d*x^2)^2*(15*a^4*d^4 + 6*b^4*c^4 + 64*a^2*b^2*c^2*d^2 - 12*a*b^3*c^3*d - 58*a^3*b*c*d^3))/(6*a^2*(b*c^2 - a*c*d)^3) + (d*(c + d*x ^2)^3*(a*d - 2*b*c)*(b^3*c^2 + 5*a^2*b*d^2 - a*b^2*c*d))/(2*a^2*(b*c^2 - a *c*d)^3))/(b*(c + d*x^2)^(7/2) + (c + d*x^2)^(3/2)*(b*c^2 - a*c*d) + (c + d*x^2)^(5/2)*(a*d - 2*b*c)) - (atan((a^19*c^15*d^19*(c + d*x^2)^(1/2)*125i + a^3*b^16*c^31*d^3*(c + d*x^2)^(1/2)*420i - a^4*b^15*c^30*d^4*(c + d*x^2 )^(1/2)*4515i + a^5*b^14*c^29*d^5*(c + d*x^2)^(1/2)*20916i - a^6*b^13*c^28 *d^6*(c + d*x^2)^(1/2)*52836i + a^7*b^12*c^27*d^7*(c + d*x^2)^(1/2)*71070i - a^8*b^11*c^26*d^8*(c + d*x^2)^(1/2)*19530i - a^9*b^10*c^25*d^9*(c + d*x ^2)^(1/2)*107740i + a^10*b^9*c^24*d^10*(c + d*x^2)^(1/2)*212608i - a^11*b^ 8*c^23*d^11*(c + d*x^2)^(1/2)*184563i + a^12*b^7*c^22*d^12*(c + d*x^2)^(1/ 2)*40965i + a^13*b^6*c^21*d^13*(c + d*x^2)^(1/2)*91560i - a^14*b^5*c^20*d^ 14*(c + d*x^2)^(1/2)*126720i + a^15*b^4*c^19*d^15*(c + d*x^2)^(1/2)*87276i - a^16*b^3*c^18*d^16*(c + d*x^2)^(1/2)*37776i + a^17*b^2*c^17*d^17*(c + d *x^2)^(1/2)*10440i - a^18*b*c^16*d^18*(c + d*x^2)^(1/2)*1700i)/(c^7*(c^7)^ (1/2)*(c^7*(c^7*(212608*a^10*b^9*d^10 - 107740*a^9*b^10*c*d^9 + 420*a^3*b^ 16*c^7*d^3 - 4515*a^4*b^15*c^6*d^4 + 20916*a^5*b^14*c^5*d^5 - 52836*a^6*b^ 13*c^4*d^6 + 71070*a^7*b^12*c^3*d^7 - 19530*a^8*b^11*c^2*d^8) + 10440*a^17 *b^2*d^17 - 37776*a^16*b^3*c*d^16 - 184563*a^11*b^8*c^6*d^11 + 40965*a^...