Integrand size = 24, antiderivative size = 170 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {(2 b c-a d) \sqrt {c+d x^2}}{2 a^2 \left (a+b x^2\right )}-\frac {c \sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b}} \]
1/2*(-3*a*d+4*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))*c^(1/2)/a^3-1/2*(-a*d+ 4*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^(1/2)/ a^3/b^(1/2)-1/2*(-a*d+2*b*c)*(d*x^2+c)^(1/2)/a^2/(b*x^2+a)-1/2*c*(d*x^2+c) ^(1/2)/a/x^2/(b*x^2+a)
Time = 0.55 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\frac {a \sqrt {c+d x^2} \left (-a c-2 b c x^2+a d x^2\right )}{x^2 \left (a+b x^2\right )}+\frac {\left (4 b^2 c^2-5 a b c d+a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}}+\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3} \]
((a*Sqrt[c + d*x^2]*(-(a*c) - 2*b*c*x^2 + a*d*x^2))/(x^2*(a + b*x^2)) + (( 4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b *c) + a*d]])/(Sqrt[b]*Sqrt[-(b*c) + a*d]) + Sqrt[c]*(4*b*c - 3*a*d)*ArcTan h[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3)
Time = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {354, 109, 27, 168, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (d x^2+c\right )^{3/2}}{x^4 \left (b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {d (3 b c-2 a d) x^2+c (4 b c-3 a d)}{2 x^2 \left (b x^2+a\right )^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {c \sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {d (3 b c-2 a d) x^2+c (4 b c-3 a d)}{x^2 \left (b x^2+a\right )^2 \sqrt {d x^2+c}}dx^2}{2 a}-\frac {c \sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\int \frac {d (b c-a d) (2 b c-a d) x^2+c (4 b c-3 a d) (b c-a d)}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a (b c-a d)}+\frac {2 \sqrt {c+d x^2} (2 b c-a d)}{a \left (a+b x^2\right )}}{2 a}-\frac {c \sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {c (4 b c-3 a d) (b c-a d) \int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{a}-\frac {(b c-a d)^2 (4 b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{a}}{a (b c-a d)}+\frac {2 \sqrt {c+d x^2} (2 b c-a d)}{a \left (a+b x^2\right )}}{2 a}-\frac {c \sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 c (4 b c-3 a d) (b c-a d) \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{a d}-\frac {2 (b c-a d)^2 (4 b c-a d) \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{a d}}{a (b c-a d)}+\frac {2 \sqrt {c+d x^2} (2 b c-a d)}{a \left (a+b x^2\right )}}{2 a}-\frac {c \sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\frac {2 (b c-a d)^{3/2} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {2 \sqrt {c} (4 b c-3 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}}{a (b c-a d)}+\frac {2 \sqrt {c+d x^2} (2 b c-a d)}{a \left (a+b x^2\right )}}{2 a}-\frac {c \sqrt {c+d x^2}}{a x^2 \left (a+b x^2\right )}\right )\) |
(-((c*Sqrt[c + d*x^2])/(a*x^2*(a + b*x^2))) - ((2*(2*b*c - a*d)*Sqrt[c + d *x^2])/(a*(a + b*x^2)) + ((-2*Sqrt[c]*(4*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[ Sqrt[c + d*x^2]/Sqrt[c]])/a + (2*(b*c - a*d)^(3/2)*(4*b*c - a*d)*ArcTanh[( Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(a*(b*c - a*d)))/( 2*a))/2
3.8.47.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[(((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 = 3.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(\frac {\frac {x^{2} \left (b \,x^{2}+a \right ) \left (a d -b c \right ) \left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2}+2 \sqrt {\left (a d -b c \right ) b}\, \left (\left (b \,x^{2}+a \right ) \left (c^{\frac {3}{2}} b -\frac {3 a d \sqrt {c}}{4}\right ) x^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )-\frac {\left (2 c b \,x^{2}+a \left (-d \,x^{2}+c \right )\right ) a \sqrt {d \,x^{2}+c}}{4}\right )}{x^{2} \sqrt {\left (a d -b c \right ) b}\, \left (b \,x^{2}+a \right ) a^{3}}\) | \(165\) |
| risch | \(-\frac {c \sqrt {d \,x^{2}+c}}{2 a^{2} x^{2}}-\frac {\frac {\sqrt {c}\, \left (3 a d -4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a}-\frac {2 c \left (a d -b c \right ) \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 \sqrt {-\frac {a d -b c}{b}}}-\frac {2 c \left (a d -b c \right ) \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 \sqrt {-\frac {a d -b c}{b}}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a 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 b \sqrt {-a b}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a 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 b \sqrt {-a b}}}{2 a^{2}}\) | \(939\) |
| default | \(\text {Expression too large to display}\) | \(3521\) |
2/((a*d-b*c)*b)^(1/2)*(1/4*x^2*(b*x^2+a)*(a*d-b*c)*(a*d-4*b*c)*arctan(b*(d *x^2+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*((b*x^2+a)*(c^(3/2) *b-3/4*a*d*c^(1/2))*x^2*arctanh((d*x^2+c)^(1/2)/c^(1/2))-1/4*(2*c*b*x^2+a* (-d*x^2+c))*a*(d*x^2+c)^(1/2)))/x^2/(b*x^2+a)/a^3
Time = 0.44 (sec) , antiderivative size = 1034, normalized size of antiderivative = 6.08 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 4 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {4 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \]
[-1/8*(((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt((b*c - a*d)/b) *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*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a* d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*((4*b^2*c - 3*a*b*d)*x^4 + (4*a*b* c - 3*a^2*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x ^2) + 4*(a^2*c + (2*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^4 + a^4* x^2), -1/8*(4*((4*b^2*c - 3*a*b*d)*x^4 + (4*a*b*c - 3*a^2*d)*x^2)*sqrt(-c) *arctan(sqrt(-c)/sqrt(d*x^2 + c)) + ((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^ 2*d)*x^2)*sqrt((b*c - a*d)/b)*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*(b^2*d*x^2 + 2*b^2*c - a*b*d)*s qrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(a^2* c + (2*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^4 + a^4*x^2), -1/4*(( (4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt(-(b*c - a*d)/b)*arctan (-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/b)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + ((4*b^2*c - 3*a*b*d)*x^4 + (4*a*b*c - 3* a^2*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(a^2*c + (2*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^4 + a^4*x^2), -1/4*(((4*b^2*c - a*b*d)*x^4 + (4*a*b*c - a^2*d)*x^2)*sqrt(-(b*c - a*d)/b) *arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/b)/ (b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + 2*((4*b^2*c - 3*a*b*d)*x^4 + (...
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{x^{3} \left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.27 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b c d - 2 \, \sqrt {d x^{2} + c} b c^{2} d - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a d^{2} + 2 \, \sqrt {d x^{2} + c} a c d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )} a^{2}} \]
1/2*(4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3) - 1/2*(4*b*c^2 - 3*a*c*d)*arctan(sqr t(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)) - 1/2*(2*(d*x^2 + c)^(3/2)*b*c*d - 2 *sqrt(d*x^2 + c)*b*c^2*d - (d*x^2 + c)^(3/2)*a*d^2 + 2*sqrt(d*x^2 + c)*a*c *d^2)/(((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)*a^2)
Time = 7.57 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.59 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b^2\,c^2\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{\frac {a^2\,b\,c\,d^7}{4}-\frac {5\,a\,b^2\,c^2\,d^6}{4}+b^3\,c^3\,d^5}-\frac {b\,c\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{4\,\left (\frac {a\,b\,c\,d^7}{4}-\frac {5\,b^2\,c^2\,d^6}{4}+\frac {b^3\,c^3\,d^5}{a}\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-4\,b\,c\right )}{2\,a^3\,b}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {3\,b\,\sqrt {c}\,d^7\,\sqrt {d\,x^2+c}}{4\,\left (\frac {3\,b\,c\,d^7}{4}-\frac {7\,b^2\,c^2\,d^6}{4\,a}+\frac {b^3\,c^3\,d^5}{a^2}\right )}-\frac {7\,b^2\,c^{3/2}\,d^6\,\sqrt {d\,x^2+c}}{4\,\left (\frac {3\,a\,b\,c\,d^7}{4}-\frac {7\,b^2\,c^2\,d^6}{4}+\frac {b^3\,c^3\,d^5}{a}\right )}+\frac {b^3\,c^{5/2}\,d^5\,\sqrt {d\,x^2+c}}{\frac {3\,a^2\,b\,c\,d^7}{4}-\frac {7\,a\,b^2\,c^2\,d^6}{4}+b^3\,c^3\,d^5}\right )\,\left (3\,a\,d-4\,b\,c\right )}{2\,a^3}-\frac {\frac {\left (a\,c\,d^2-b\,c^2\,d\right )\,\sqrt {d\,x^2+c}}{a^2}-\frac {d\,{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-2\,b\,c\right )}{2\,a^2}}{\left (d\,x^2+c\right )\,\left (a\,d-2\,b\,c\right )+b\,{\left (d\,x^2+c\right )}^2+b\,c^2-a\,c\,d} \]
(atanh((b^2*c^2*d^5*(c + d*x^2)^(1/2)*(b^2*c - a*b*d)^(1/2))/(b^3*c^3*d^5 - (5*a*b^2*c^2*d^6)/4 + (a^2*b*c*d^7)/4) - (b*c*d^6*(c + d*x^2)^(1/2)*(b^2 *c - a*b*d)^(1/2))/(4*((a*b*c*d^7)/4 - (5*b^2*c^2*d^6)/4 + (b^3*c^3*d^5)/a )))*(-b*(a*d - b*c))^(1/2)*(a*d - 4*b*c))/(2*a^3*b) - (c^(1/2)*atanh((3*b* c^(1/2)*d^7*(c + d*x^2)^(1/2))/(4*((3*b*c*d^7)/4 - (7*b^2*c^2*d^6)/(4*a) + (b^3*c^3*d^5)/a^2)) - (7*b^2*c^(3/2)*d^6*(c + d*x^2)^(1/2))/(4*((3*a*b*c* d^7)/4 - (7*b^2*c^2*d^6)/4 + (b^3*c^3*d^5)/a)) + (b^3*c^(5/2)*d^5*(c + d*x ^2)^(1/2))/(b^3*c^3*d^5 - (7*a*b^2*c^2*d^6)/4 + (3*a^2*b*c*d^7)/4))*(3*a*d - 4*b*c))/(2*a^3) - (((a*c*d^2 - b*c^2*d)*(c + d*x^2)^(1/2))/a^2 - (d*(c + d*x^2)^(3/2)*(a*d - 2*b*c))/(2*a^2))/((c + d*x^2)*(a*d - 2*b*c) + b*(c + d*x^2)^2 + b*c^2 - a*c*d)