Integrand size = 24, antiderivative size = 185 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=\frac {8 b^2 c^2-5 a d (8 b c-7 a d)}{24 c^3 \left (c+d x^2\right )^{3/2}}-\frac {a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac {a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac {8 b^2 c^2-5 a d (8 b c-7 a d)}{8 c^4 \sqrt {c+d x^2}}-\frac {\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{8 c^{9/2}} \]
1/24*(8*b^2*c^2-5*a*d*(-7*a*d+8*b*c))/c^3/(d*x^2+c)^(3/2)-1/4*a^2/c/x^4/(d *x^2+c)^(3/2)-1/8*a*(-7*a*d+8*b*c)/c^2/x^2/(d*x^2+c)^(3/2)-1/8*(8*b^2*c^2- 5*a*d*(-7*a*d+8*b*c))*arctanh((d*x^2+c)^(1/2)/c^(1/2))/c^(9/2)+1/8*(8*b^2* c^2-5*a*d*(-7*a*d+8*b*c))/c^4/(d*x^2+c)^(1/2)
Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=\frac {8 b^2 c^2 x^4 \left (4 c+3 d x^2\right )-8 a b c x^2 \left (3 c^2+20 c d x^2+15 d^2 x^4\right )+a^2 \left (-6 c^3+21 c^2 d x^2+140 c d^2 x^4+105 d^3 x^6\right )}{24 c^4 x^4 \left (c+d x^2\right )^{3/2}}-\frac {\left (8 b^2 c^2-40 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{8 c^{9/2}} \]
(8*b^2*c^2*x^4*(4*c + 3*d*x^2) - 8*a*b*c*x^2*(3*c^2 + 20*c*d*x^2 + 15*d^2* x^4) + a^2*(-6*c^3 + 21*c^2*d*x^2 + 140*c*d^2*x^4 + 105*d^3*x^6))/(24*c^4* x^4*(c + d*x^2)^(3/2)) - ((8*b^2*c^2 - 40*a*b*c*d + 35*a^2*d^2)*ArcTanh[Sq rt[c + d*x^2]/Sqrt[c]])/(8*c^(9/2))
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {354, 100, 27, 87, 61, 61, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^2}{x^6 \left (d x^2+c\right )^{5/2}}dx^2\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {4 b^2 c x^2+a (8 b c-7 a d)}{2 x^4 \left (d x^2+c\right )^{5/2}}dx^2}{2 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {4 b^2 c x^2+a (8 b c-7 a d)}{x^4 \left (d x^2+c\right )^{5/2}}dx^2}{4 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \int \frac {1}{x^2 \left (d x^2+c\right )^{5/2}}dx^2}{2 c}-\frac {a (8 b c-7 a d)}{c x^2 \left (c+d x^2\right )^{3/2}}}{4 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \left (\frac {\int \frac {1}{x^2 \left (d x^2+c\right )^{3/2}}dx^2}{c}+\frac {2}{3 c \left (c+d x^2\right )^{3/2}}\right )}{2 c}-\frac {a (8 b c-7 a d)}{c x^2 \left (c+d x^2\right )^{3/2}}}{4 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \left (\frac {\frac {\int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{c}+\frac {2}{c \sqrt {c+d x^2}}}{c}+\frac {2}{3 c \left (c+d x^2\right )^{3/2}}\right )}{2 c}-\frac {a (8 b c-7 a d)}{c x^2 \left (c+d x^2\right )^{3/2}}}{4 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \left (\frac {\frac {2 \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{c d}+\frac {2}{c \sqrt {c+d x^2}}}{c}+\frac {2}{3 c \left (c+d x^2\right )^{3/2}}\right )}{2 c}-\frac {a (8 b c-7 a d)}{c x^2 \left (c+d x^2\right )^{3/2}}}{4 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \left (\frac {\frac {2}{c \sqrt {c+d x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{c}+\frac {2}{3 c \left (c+d x^2\right )^{3/2}}\right )}{2 c}-\frac {a (8 b c-7 a d)}{c x^2 \left (c+d x^2\right )^{3/2}}}{4 c}-\frac {a^2}{2 c x^4 \left (c+d x^2\right )^{3/2}}\right )\) |
(-1/2*a^2/(c*x^4*(c + d*x^2)^(3/2)) + (-((a*(8*b*c - 7*a*d))/(c*x^2*(c + d *x^2)^(3/2))) + ((8*b^2*c^2 - 5*a*d*(8*b*c - 7*a*d))*(2/(3*c*(c + d*x^2)^( 3/2)) + (2/(c*Sqrt[c + d*x^2]) - (2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(3 /2))/c))/(2*c))/(4*c))/2
3.7.69.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 3.00 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {35 x^{4} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} d^{2}-\frac {8}{7} a b c d +\frac {8}{35} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{8}+\frac {7 d \,x^{2} \left (\frac {8}{7} b^{2} x^{4}-\frac {160}{21} a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}}{8}+\frac {7 \left (\frac {32}{21} b^{2} x^{4}-\frac {8}{7} a b \,x^{2}-\frac {2}{7} a^{2}\right ) c^{\frac {7}{2}}}{8}+\frac {35 x^{4} \left (\left (-\frac {8 b \,x^{2}}{7}+\frac {4 a}{3}\right ) c^{\frac {3}{2}}+a \sqrt {c}\, d \,x^{2}\right ) d^{2} a}{8}}{c^{\frac {9}{2}} x^{4} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(154\) |
| default | \(b^{2} \left (\frac {1}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}}{c}\right )+a^{2} \left (-\frac {1}{4 c \,x^{4} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {7 d \left (-\frac {1}{2 c \,x^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {5 d \left (\frac {1}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}}{c}\right )}{2 c}\right )}{4 c}\right )+2 a b \left (-\frac {1}{2 c \,x^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {5 d \left (\frac {1}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}}{c}\right )}{2 c}\right )\) | \(269\) |
| risch | \(-\frac {\sqrt {d \,x^{2}+c}\, a \left (-11 a d \,x^{2}+8 c b \,x^{2}+2 a c \right )}{8 c^{4} x^{4}}-\frac {35 \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a^{2} d^{2}}{8 c^{\frac {9}{2}}}+\frac {5 \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a b d}{c^{\frac {7}{2}}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) b^{2}}{c^{\frac {5}{2}}}-\frac {19 \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a^{2} d^{2}}{12 c^{4} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {13 \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a b d}{6 c^{3} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {7 \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c^{2} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {19 \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a^{2} d^{2}}{12 c^{4} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {13 \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a b d}{6 c^{3} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {7 \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c^{2} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {d \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 c^{4} \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a b}{6 c^{3} \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c^{2} d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {d \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 c^{4} \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a b}{6 c^{3} \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c^{2} d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}\) | \(933\) |
7/8/(d*x^2+c)^(3/2)*(-5*x^4*(d*x^2+c)^(3/2)*(a^2*d^2-8/7*a*b*c*d+8/35*b^2* c^2)*arctanh((d*x^2+c)^(1/2)/c^(1/2))+d*x^2*(8/7*b^2*x^4-160/21*a*b*x^2+a^ 2)*c^(5/2)+(32/21*b^2*x^4-8/7*a*b*x^2-2/7*a^2)*c^(7/2)+5*x^4*((-8/7*b*x^2+ 4/3*a)*c^(3/2)+a*c^(1/2)*d*x^2)*d^2*a)/c^(9/2)/x^4
Time = 0.27 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.90 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + 2 \, {\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} + {\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} - 6 \, a^{2} c^{4} + 4 \, {\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} - 3 \, {\left (8 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, {\left (c^{5} d^{2} x^{8} + 2 \, c^{6} d x^{6} + c^{7} x^{4}\right )}}, \frac {3 \, {\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + 2 \, {\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} + {\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, {\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} - 6 \, a^{2} c^{4} + 4 \, {\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} - 3 \, {\left (8 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (c^{5} d^{2} x^{8} + 2 \, c^{6} d x^{6} + c^{7} x^{4}\right )}}\right ] \]
[1/48*(3*((8*b^2*c^2*d^2 - 40*a*b*c*d^3 + 35*a^2*d^4)*x^8 + 2*(8*b^2*c^3*d - 40*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^6 + (8*b^2*c^4 - 40*a*b*c^3*d + 35*a^2 *c^2*d^2)*x^4)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(3*(8*b^2*c^3*d - 40*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^6 - 6*a^2*c^4 + 4* (8*b^2*c^4 - 40*a*b*c^3*d + 35*a^2*c^2*d^2)*x^4 - 3*(8*a*b*c^4 - 7*a^2*c^3 *d)*x^2)*sqrt(d*x^2 + c))/(c^5*d^2*x^8 + 2*c^6*d*x^6 + c^7*x^4), 1/24*(3*( (8*b^2*c^2*d^2 - 40*a*b*c*d^3 + 35*a^2*d^4)*x^8 + 2*(8*b^2*c^3*d - 40*a*b* c^2*d^2 + 35*a^2*c*d^3)*x^6 + (8*b^2*c^4 - 40*a*b*c^3*d + 35*a^2*c^2*d^2)* x^4)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) + (3*(8*b^2*c^3*d - 40*a*b* c^2*d^2 + 35*a^2*c*d^3)*x^6 - 6*a^2*c^4 + 4*(8*b^2*c^4 - 40*a*b*c^3*d + 35 *a^2*c^2*d^2)*x^4 - 3*(8*a*b*c^4 - 7*a^2*c^3*d)*x^2)*sqrt(d*x^2 + c))/(c^5 *d^2*x^8 + 2*c^6*d*x^6 + c^7*x^4)]
\[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{5} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {b^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{c^{\frac {5}{2}}} + \frac {5 \, a b d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{c^{\frac {7}{2}}} - \frac {35 \, a^{2} d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{8 \, c^{\frac {9}{2}}} + \frac {b^{2}}{\sqrt {d x^{2} + c} c^{2}} + \frac {b^{2}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c} - \frac {5 \, a b d}{\sqrt {d x^{2} + c} c^{3}} - \frac {5 \, a b d}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {35 \, a^{2} d^{2}}{8 \, \sqrt {d x^{2} + c} c^{4}} + \frac {35 \, a^{2} d^{2}}{24 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3}} - \frac {a b}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c x^{2}} + \frac {7 \, a^{2} d}{8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2} x^{2}} - \frac {a^{2}}{4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c x^{4}} \]
-b^2*arcsinh(c/(sqrt(c*d)*abs(x)))/c^(5/2) + 5*a*b*d*arcsinh(c/(sqrt(c*d)* abs(x)))/c^(7/2) - 35/8*a^2*d^2*arcsinh(c/(sqrt(c*d)*abs(x)))/c^(9/2) + b^ 2/(sqrt(d*x^2 + c)*c^2) + 1/3*b^2/((d*x^2 + c)^(3/2)*c) - 5*a*b*d/(sqrt(d* x^2 + c)*c^3) - 5/3*a*b*d/((d*x^2 + c)^(3/2)*c^2) + 35/8*a^2*d^2/(sqrt(d*x ^2 + c)*c^4) + 35/24*a^2*d^2/((d*x^2 + c)^(3/2)*c^3) - a*b/((d*x^2 + c)^(3 /2)*c*x^2) + 7/8*a^2*d/((d*x^2 + c)^(3/2)*c^2*x^2) - 1/4*a^2/((d*x^2 + c)^ (3/2)*c*x^4)
Time = 0.29 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left (8 \, b^{2} c^{2} - 40 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{8 \, \sqrt {-c} c^{4}} + \frac {3 \, {\left (d x^{2} + c\right )} b^{2} c^{2} + b^{2} c^{3} - 12 \, {\left (d x^{2} + c\right )} a b c d - 2 \, a b c^{2} d + 9 \, {\left (d x^{2} + c\right )} a^{2} d^{2} + a^{2} c d^{2}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{4}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c d - 8 \, \sqrt {d x^{2} + c} a b c^{2} d - 11 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2} + 13 \, \sqrt {d x^{2} + c} a^{2} c d^{2}}{8 \, c^{4} d^{2} x^{4}} \]
1/8*(8*b^2*c^2 - 40*a*b*c*d + 35*a^2*d^2)*arctan(sqrt(d*x^2 + c)/sqrt(-c)) /(sqrt(-c)*c^4) + 1/3*(3*(d*x^2 + c)*b^2*c^2 + b^2*c^3 - 12*(d*x^2 + c)*a* b*c*d - 2*a*b*c^2*d + 9*(d*x^2 + c)*a^2*d^2 + a^2*c*d^2)/((d*x^2 + c)^(3/2 )*c^4) - 1/8*(8*(d*x^2 + c)^(3/2)*a*b*c*d - 8*sqrt(d*x^2 + c)*a*b*c^2*d - 11*(d*x^2 + c)^(3/2)*a^2*d^2 + 13*sqrt(d*x^2 + c)*a^2*c*d^2)/(c^4*d^2*x^4)
Time = 6.08 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{3\,c}+\frac {\left (d\,x^2+c\right )\,\left (7\,a^2\,d^2-8\,a\,b\,c\,d+b^2\,c^2\right )}{3\,c^2}-\frac {5\,{\left (d\,x^2+c\right )}^2\,\left (35\,a^2\,d^2-40\,a\,b\,c\,d+8\,b^2\,c^2\right )}{24\,c^3}+\frac {{\left (d\,x^2+c\right )}^3\,\left (35\,a^2\,d^2-40\,a\,b\,c\,d+8\,b^2\,c^2\right )}{8\,c^4}}{{\left (d\,x^2+c\right )}^{7/2}-2\,c\,{\left (d\,x^2+c\right )}^{5/2}+c^2\,{\left (d\,x^2+c\right )}^{3/2}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (35\,a^2\,d^2-40\,a\,b\,c\,d+8\,b^2\,c^2\right )}{8\,c^{9/2}} \]
((a^2*d^2 + b^2*c^2 - 2*a*b*c*d)/(3*c) + ((c + d*x^2)*(7*a^2*d^2 + b^2*c^2 - 8*a*b*c*d))/(3*c^2) - (5*(c + d*x^2)^2*(35*a^2*d^2 + 8*b^2*c^2 - 40*a*b *c*d))/(24*c^3) + ((c + d*x^2)^3*(35*a^2*d^2 + 8*b^2*c^2 - 40*a*b*c*d))/(8 *c^4))/((c + d*x^2)^(7/2) - 2*c*(c + d*x^2)^(5/2) + c^2*(c + d*x^2)^(3/2)) - (atanh((c + d*x^2)^(1/2)/c^(1/2))*(35*a^2*d^2 + 8*b^2*c^2 - 40*a*b*c*d) )/(8*c^(9/2))