\(\int \frac {1}{x^6 (a+b x^2)^2 (c+d x^2)} \, dx\) [693]

Optimal result

Integrand size = 22, antiderivative size = 250 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac {7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac {b^{7/2} (7 b c-9 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac {d^{9/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^2} \] Output:

-1/10*(-2*a*d+7*b*c)/a^2/c/(-a*d+b*c)/x^5+1/6*(-2*a^2*d^2-2*a*b*c*d+7*b^2* 
c^2)/a^3/c^2/(-a*d+b*c)/x^3-1/2*(-2*a^3*d^3-2*a^2*b*c*d^2-2*a*b^2*c^2*d+7* 
b^3*c^3)/a^4/c^3/(-a*d+b*c)/x+1/2*b/a/(-a*d+b*c)/x^5/(b*x^2+a)-1/2*b^(7/2) 
*(-9*a*d+7*b*c)*arctan(b^(1/2)*x/a^(1/2))/a^(9/2)/(-a*d+b*c)^2-d^(9/2)*arc 
tan(d^(1/2)*x/c^(1/2))/c^(7/2)/(-a*d+b*c)^2
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {1}{5 a^2 c x^5}+\frac {2 b c+a d}{3 a^3 c^2 x^3}+\frac {-3 b^2 c^2-2 a b c d-a^2 d^2}{a^4 c^3 x}+\frac {b^4 x}{2 a^4 (-b c+a d) \left (a+b x^2\right )}+\frac {b^{7/2} (-7 b c+9 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} (-b c+a d)^2}-\frac {d^{9/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^2} \] Input:

Integrate[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]
 

Output:

-1/5*1/(a^2*c*x^5) + (2*b*c + a*d)/(3*a^3*c^2*x^3) + (-3*b^2*c^2 - 2*a*b*c 
*d - a^2*d^2)/(a^4*c^3*x) + (b^4*x)/(2*a^4*(-(b*c) + a*d)*(a + b*x^2)) + ( 
b^(7/2)*(-7*b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(-(b*c) + 
 a*d)^2) - (d^(9/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^2)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {374, 25, 445, 27, 445, 27, 445, 397, 218}

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.

\(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 374

\(\displaystyle \frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}-\frac {\int -\frac {7 b d x^2+7 b c-2 a d}{x^6 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 a (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {7 b d x^2+7 b c-2 a d}{x^6 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {-\frac {\int \frac {5 \left (7 b^2 c^2-2 a b d c-2 a^2 d^2+b d (7 b c-2 a d) x^2\right )}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{5 a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {7 b^2 c^2-2 a b d c-2 a^2 d^2+b d (7 b c-2 a d) x^2}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {-\frac {-\frac {\int \frac {3 \left (7 b^3 c^3-2 a b^2 d c^2-2 a^2 b d^2 c-2 a^3 d^3+b d \left (7 b^2 c^2-2 a b d c-2 a^2 d^2\right ) x^2\right )}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{3 a c}-\frac {\frac {7 b^2 c}{a}-\frac {2 a d^2}{c}-2 b d}{3 x^3}}{a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {7 b^3 c^3-2 a b^2 d c^2-2 a^2 b d^2 c-2 a^3 d^3+b d \left (7 b^2 c^2-2 a b d c-2 a^2 d^2\right ) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {\frac {7 b^2 c}{a}-\frac {2 a d^2}{c}-2 b d}{3 x^3}}{a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {7 b^4 c^4-2 a b^3 d c^3-2 a^2 b^2 d^2 c^2-2 a^3 b d^3 c-2 a^4 d^4+b d \left (7 b^3 c^3-2 a b^2 d c^2-2 a^2 b d^2 c-2 a^3 d^3\right ) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{a c x}}{a c}-\frac {\frac {7 b^2 c}{a}-\frac {2 a d^2}{c}-2 b d}{3 x^3}}{a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {2 a^4 d^5 \int \frac {1}{d x^2+c}dx}{b c-a d}+\frac {b^4 c^3 (7 b c-9 a d) \int \frac {1}{b x^2+a}dx}{b c-a d}}{a c}-\frac {-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{a c x}}{a c}-\frac {\frac {7 b^2 c}{a}-\frac {2 a d^2}{c}-2 b d}{3 x^3}}{a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {2 a^4 d^{9/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}+\frac {b^{7/2} c^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (7 b c-9 a d)}{\sqrt {a} (b c-a d)}}{a c}-\frac {-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{a c x}}{a c}-\frac {\frac {7 b^2 c}{a}-\frac {2 a d^2}{c}-2 b d}{3 x^3}}{a c}-\frac {7 b c-2 a d}{5 a c x^5}}{2 a (b c-a d)}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)}\)

Input:

Int[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]
 

Output:

b/(2*a*(b*c - a*d)*x^5*(a + b*x^2)) + (-1/5*(7*b*c - 2*a*d)/(a*c*x^5) - (- 
1/3*((7*b^2*c)/a - 2*b*d - (2*a*d^2)/c)/x^3 - (-((7*b^3*c^3 - 2*a*b^2*c^2* 
d - 2*a^2*b*c*d^2 - 2*a^3*d^3)/(a*c*x)) - ((b^(7/2)*c^3*(7*b*c - 9*a*d)*Ar 
cTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)) + (2*a^4*d^(9/2)*ArcTan[( 
Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)))/(a*c))/(a*c))/(a*c))/(2*a*(b*c 
 - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.64

method	result	size
default	\(\frac {b^{4} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (9 a d -7 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4} \left (a d -b c \right )^{2}}-\frac {1}{5 a^{2} c \,x^{5}}-\frac {-a d -2 b c}{3 x^{3} a^{3} c^{2}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{a^{4} c^{3} x}-\frac {d^{5} \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{c^{3} \left (a d -b c \right )^{2} \sqrt {c d}}\)	\(161\)
risch	\(\text {Expression too large to display}\)	\(2059\)

Input:

int(1/x^6/(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

b^4/a^4/(a*d-b*c)^2*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(9*a*d-7*b*c)/(a*b) 
^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/5/a^2/c/x^5-1/3*(-a*d-2*b*c)/x^3/a^3/c^2 
-(a^2*d^2+2*a*b*c*d+3*b^2*c^2)/a^4/c^3/x-1/c^3*d^5/(a*d-b*c)^2/(c*d)^(1/2) 
*arctan(x*d/(c*d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 1489, normalized size of antiderivative = 5.96 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^6/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")
 

Output:

[-1/60*(12*a^3*b^2*c^4 - 24*a^4*b*c^3*d + 12*a^5*c^2*d^2 + 30*(7*b^5*c^4 - 
 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 20*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^ 
4*b*c*d^3 + 3*a^5*d^4)*x^4 - 4*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*b* 
c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^ 
4*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a 
)/(b*x^2 + a)) - 30*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(-d/c)*log((d*x^2 - 
2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6* 
b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6*b*c^4*d + a^7*c^3*d^2)*x^5), -1/60*( 
12*a^3*b^2*c^4 - 24*a^4*b*c^3*d + 12*a^5*c^2*d^2 + 30*(7*b^5*c^4 - 9*a*b^4 
*c^3*d + 2*a^4*b*d^4)*x^6 + 20*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^ 
3 + 3*a^5*d^4)*x^4 - 4*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 
+ 5*a^5*c*d^3)*x^2 + 60*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(d/c)*arctan(x*s 
qrt(d/c)) + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4*c^4 - 9*a^2*b^3 
*c^3*d)*x^5)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/( 
(a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6 
*b*c^4*d + a^7*c^3*d^2)*x^5), -1/30*(6*a^3*b^2*c^4 - 12*a^4*b*c^3*d + 6*a^ 
5*c^2*d^2 + 15*(7*b^5*c^4 - 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 10*(7*a*b^4 
*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 2*(7*a^2*b^3*c^4 - 
 9*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9 
*a*b^4*c^3*d)*x^7 + (7*a*b^4*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(b/a)*arct...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x**6/(b*x**2+a)**2/(d*x**2+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {d^{5} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt {c d}} - \frac {{\left (7 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} \sqrt {a b}} - \frac {6 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + 15 \, {\left (7 \, b^{4} c^{3} - 2 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{6} + 10 \, {\left (7 \, a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d - 2 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x^{4} - 2 \, {\left (7 \, a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d - 5 \, a^{4} c d^{2}\right )} x^{2}}{30 \, {\left ({\left (a^{4} b^{2} c^{4} - a^{5} b c^{3} d\right )} x^{7} + {\left (a^{5} b c^{4} - a^{6} c^{3} d\right )} x^{5}\right )}} \] Input:

integrate(1/x^6/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")
 

Output:

-d^5*arctan(d*x/sqrt(c*d))/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(c*d 
)) - 1/2*(7*b^5*c - 9*a*b^4*d)*arctan(b*x/sqrt(a*b))/((a^4*b^2*c^2 - 2*a^5 
*b*c*d + a^6*d^2)*sqrt(a*b)) - 1/30*(6*a^3*b*c^3 - 6*a^4*c^2*d + 15*(7*b^4 
*c^3 - 2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^6 + 10*(7*a*b^3*c^ 
3 - 2*a^2*b^2*c^2*d - 2*a^3*b*c*d^2 - 3*a^4*d^3)*x^4 - 2*(7*a^2*b^2*c^3 - 
2*a^3*b*c^2*d - 5*a^4*c*d^2)*x^2)/((a^4*b^2*c^4 - a^5*b*c^3*d)*x^7 + (a^5* 
b*c^4 - a^6*c^3*d)*x^5)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {d^{5} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt {c d}} - \frac {b^{4} x}{2 \, {\left (a^{4} b c - a^{5} d\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (7 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} \sqrt {a b}} - \frac {45 \, b^{2} c^{2} x^{4} + 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} - 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, a^{4} c^{3} x^{5}} \] Input:

integrate(1/x^6/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")
 

Output:

-d^5*arctan(d*x/sqrt(c*d))/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(c*d 
)) - 1/2*b^4*x/((a^4*b*c - a^5*d)*(b*x^2 + a)) - 1/2*(7*b^5*c - 9*a*b^4*d) 
*arctan(b*x/sqrt(a*b))/((a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*sqrt(a*b)) - 
 1/15*(45*b^2*c^2*x^4 + 30*a*b*c*d*x^4 + 15*a^2*d^2*x^4 - 10*a*b*c^2*x^2 - 
 5*a^2*c*d*x^2 + 3*a^2*c^2)/(a^4*c^3*x^5)
 

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 2737, normalized size of antiderivative = 10.95 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x)
 

Output:

(atan((a^9*d*x*(-c^7*d^9)^(3/2)*4i + b^9*c^16*d*x*(-c^7*d^9)^(1/2)*49i + a 
^2*b^7*c^14*d^3*x*(-c^7*d^9)^(1/2)*81i - a*b^8*c^15*d^2*x*(-c^7*d^9)^(1/2) 
*126i)/(4*a^9*c^11*d^14 - 49*b^9*c^20*d^5 + 126*a*b^8*c^19*d^6 - 81*a^2*b^ 
7*c^18*d^7))*(-c^7*d^9)^(1/2)*1i)/(b^2*c^9 + a^2*c^7*d^2 - 2*a*b*c^8*d) - 
(1/(5*a*c) - (x^2*(5*a*d + 7*b*c))/(15*a^2*c^2) + (x^4*(3*a^2*d^2 + 7*b^2* 
c^2 + 5*a*b*c*d))/(3*a^3*c^3) + (x^6*(2*a^3*b*d^3 - 7*b^4*c^3 + 2*a^2*b^2* 
c*d^2 + 2*a*b^3*c^2*d))/(2*a^4*c^3*(a*d - b*c)))/(a*x^5 + b*x^7) - (atan(( 
((x*(784*a^12*b^14*c^20*d^3 - 4368*a^13*b^13*c^19*d^4 + 9696*a^14*b^12*c^1 
8*d^5 - 10720*a^15*b^11*c^17*d^6 + 5904*a^16*b^10*c^16*d^7 - 1296*a^17*b^9 
*c^15*d^8 + 64*a^20*b^6*c^12*d^11 - 192*a^21*b^5*c^11*d^12 + 192*a^22*b^4* 
c^10*d^13 - 64*a^23*b^3*c^9*d^14) + ((9*a*d - 7*b*c)*(-a^9*b^7)^(1/2)*(281 
6*a^17*b^11*c^21*d^3 - 448*a^16*b^12*c^22*d^2 - 7360*a^18*b^10*c^20*d^4 + 
10240*a^19*b^9*c^19*d^5 - 8000*a^20*b^8*c^18*d^6 + 3200*a^21*b^7*c^17*d^7 
+ 64*a^22*b^6*c^16*d^8 - 1280*a^23*b^5*c^15*d^9 + 1280*a^24*b^4*c^14*d^10 
- 640*a^25*b^3*c^13*d^11 + 128*a^26*b^2*c^12*d^12 + (x*(9*a*d - 7*b*c)*(-a 
^9*b^7)^(1/2)*(256*a^20*b^10*c^23*d^2 - 1536*a^21*b^9*c^22*d^3 + 3584*a^22 
*b^8*c^21*d^4 - 3584*a^23*b^7*c^20*d^5 + 3584*a^25*b^5*c^18*d^7 - 3584*a^2 
6*b^4*c^17*d^8 + 1536*a^27*b^3*c^16*d^9 - 256*a^28*b^2*c^15*d^10))/(4*(a^1 
1*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d))))/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10 
*b*c*d)))*(9*a*d - 7*b*c)*(-a^9*b^7)^(1/2)*1i)/(4*(a^11*d^2 + a^9*b^2*c...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.70 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {135 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} c^{4} d \,x^{5}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} c^{5} x^{5}+135 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} c^{4} d \,x^{7}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} c^{5} x^{7}-30 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{6} d^{4} x^{5}-30 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{5} b \,d^{4} x^{7}-6 a^{6} c^{3} d^{2}+10 a^{6} c^{2} d^{3} x^{2}-30 a^{6} c \,d^{4} x^{4}+12 a^{5} b \,c^{4} d -6 a^{5} b \,c^{3} d^{2} x^{2}+10 a^{5} b \,c^{2} d^{3} x^{4}-30 a^{5} b c \,d^{4} x^{6}-6 a^{4} b^{2} c^{5}-18 a^{4} b^{2} c^{4} d \,x^{2}+14 a^{3} b^{3} c^{5} x^{2}+90 a^{3} b^{3} c^{4} d \,x^{4}-70 a^{2} b^{4} c^{5} x^{4}+135 a^{2} b^{4} c^{4} d \,x^{6}-105 a \,b^{5} c^{5} x^{6}}{30 a^{5} c^{4} x^{5} \left (a^{2} b \,d^{2} x^{2}-2 a \,b^{2} c d \,x^{2}+b^{3} c^{2} x^{2}+a^{3} d^{2}-2 a^{2} b c d +a \,b^{2} c^{2}\right )} \] Input:

int(1/x^6/(b*x^2+a)^2/(d*x^2+c),x)
 

Output:

(135*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**4*d*x**5 - 
 105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**5*x**5 + 135* 
sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**4*d*x**7 - 105*sqr 
t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*c**5*x**7 - 30*sqrt(d)*sqr 
t(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**6*d**4*x**5 - 30*sqrt(d)*sqrt(c)*ata 
n((d*x)/(sqrt(d)*sqrt(c)))*a**5*b*d**4*x**7 - 6*a**6*c**3*d**2 + 10*a**6*c 
**2*d**3*x**2 - 30*a**6*c*d**4*x**4 + 12*a**5*b*c**4*d - 6*a**5*b*c**3*d** 
2*x**2 + 10*a**5*b*c**2*d**3*x**4 - 30*a**5*b*c*d**4*x**6 - 6*a**4*b**2*c* 
*5 - 18*a**4*b**2*c**4*d*x**2 + 14*a**3*b**3*c**5*x**2 + 90*a**3*b**3*c**4 
*d*x**4 - 70*a**2*b**4*c**5*x**4 + 135*a**2*b**4*c**4*d*x**6 - 105*a*b**5* 
c**5*x**6)/(30*a**5*c**4*x**5*(a**3*d**2 - 2*a**2*b*c*d + a**2*b*d**2*x**2 
 + a*b**2*c**2 - 2*a*b**2*c*d*x**2 + b**3*c**2*x**2))