\(\int \frac {1}{x^3 (a+b x^2)^2 (c+d x^2)^3} \, dx\) [711]

Optimal result

Integrand size = 22, antiderivative size = 215 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {1}{2 a^2 c^3 x^2}-\frac {b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4} \] Output:

-1/2/a^2/c^3/x^2-1/2*b^4/a^2/(-a*d+b*c)^3/(b*x^2+a)-1/4*d^3/c^2/(-a*d+b*c) 
^2/(d*x^2+c)^2-d^3*(-a*d+2*b*c)/c^3/(-a*d+b*c)^3/(d*x^2+c)-(3*a*d+2*b*c)*l 
n(x)/a^3/c^4+1/2*b^4*(-5*a*d+2*b*c)*ln(b*x^2+a)/a^3/(-a*d+b*c)^4+1/2*d^3*( 
3*a^2*d^2-10*a*b*c*d+10*b^2*c^2)*ln(d*x^2+c)/c^4/(-a*d+b*c)^4
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {1}{4} \left (-\frac {2}{a^2 c^3 x^2}+\frac {2 b^4}{a^2 (-b c+a d)^3 \left (a+b x^2\right )}-\frac {d^3}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {4 d^3 (-2 b c+a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {4 (2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}+\frac {2 d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}\right ) \] Input:

Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]
 

Output:

(-2/(a^2*c^3*x^2) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - d^3/(c^2* 
(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*( 
c + d*x^2)) - (4*(2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (2*b^4*(2*b*c - 5*a*d 
)*Log[a + b*x^2])/(a^3*(b*c - a*d)^4) + (2*d^3*(10*b^2*c^2 - 10*a*b*c*d + 
3*a^2*d^2)*Log[c + d*x^2])/(c^4*(b*c - a*d)^4))/4
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 99, 2009}

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.

\(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 99

\(\displaystyle \frac {1}{2} \int \left (-\frac {(5 a d-2 b c) b^5}{a^3 (a d-b c)^4 \left (b x^2+a\right )}-\frac {b^5}{a^2 (a d-b c)^3 \left (b x^2+a\right )^2}+\frac {d^4 \left (10 b^2 c^2-10 a b d c+3 a^2 d^2\right )}{c^4 (b c-a d)^4 \left (d x^2+c\right )}+\frac {-2 b c-3 a d}{a^3 c^4 x^2}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 \left (d x^2+c\right )^2}+\frac {d^4}{c^2 (b c-a d)^2 \left (d x^2+c\right )^3}+\frac {1}{a^2 c^3 x^4}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}-\frac {\log \left (x^2\right ) (3 a d+2 b c)}{a^3 c^4}-\frac {b^4}{a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}-\frac {1}{a^2 c^3 x^2}-\frac {2 d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {d^3}{2 c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right )\)

Input:

Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]
 

Output:

(-(1/(a^2*c^3*x^2)) - b^4/(a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(2*c^2*(b* 
c - a*d)^2*(c + d*x^2)^2) - (2*d^3*(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + 
d*x^2)) - ((2*b*c + 3*a*d)*Log[x^2])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[ 
a + b*x^2])/(a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^ 
2)*Log[c + d*x^2])/(c^4*(b*c - a*d)^4))/2
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.04

method	result	size
default	\(-\frac {b^{5} \left (-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}+\frac {\left (5 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{3} \left (a d -b c \right )^{4}}-\frac {1}{2 a^{2} c^{3} x^{2}}+\frac {\left (-3 a d -2 b c \right ) \ln \left (x \right )}{a^{3} c^{4}}+\frac {d^{4} \left (-\frac {2 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{d \left (x^{2} d +c \right )}+\frac {\left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \ln \left (x^{2} d +c \right )}{d}-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (x^{2} d +c \right )^{2}}\right )}{2 c^{4} \left (a d -b c \right )^{4}}\)	\(223\)
norman	\(\frac {-\frac {1}{2 a c}+\frac {\left (6 a^{5} d^{5}-12 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+a \,b^{4} c^{4} d -2 c^{5} b^{5}\right ) x^{4}}{2 c^{3} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (9 a^{5} d^{5}-7 a^{4} b c \,d^{4}-18 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}+4 a \,b^{4} c^{4} d -8 c^{5} b^{5}\right ) x^{6}}{4 c^{4} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{2} b \left (9 d^{4} a^{4}-19 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d -4 c^{4} b^{4}\right ) x^{8}}{4 c^{4} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{x^{2} \left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )^{2}}-\frac {\left (3 a d +2 b c \right ) \ln \left (x \right )}{a^{3} c^{4}}-\frac {b^{4} \left (5 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {d^{3} \left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \ln \left (x^{2} d +c \right )}{2 c^{4} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(536\)
risch	\(\frac {-\frac {b \,d^{2} \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{6}}{2 a^{2} c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (6 d^{4} a^{4}-5 a^{3} b c \,d^{3}-15 a^{2} b^{2} c^{2} d^{2}+10 a \,b^{3} c^{3} d -8 c^{4} b^{4}\right ) x^{4}}{4 a^{2} c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (9 d^{4} a^{4}-19 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d -4 c^{4} b^{4}\right ) x^{2}}{4 a^{2} c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {1}{2 a c}}{x^{2} \left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )^{2}}-\frac {3 \ln \left (x \right ) d}{a^{2} c^{4}}-\frac {2 \ln \left (x \right ) b}{a^{3} c^{3}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right ) d}{2 a^{2} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {b^{5} \ln \left (b \,x^{2}+a \right ) c}{a^{3} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {3 d^{5} \ln \left (-x^{2} d -c \right ) a^{2}}{2 c^{4} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}-\frac {5 d^{4} \ln \left (-x^{2} d -c \right ) a b}{c^{3} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {5 d^{3} \ln \left (-x^{2} d -c \right ) b^{2}}{c^{2} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(699\)
parallelrisch	\(\text {Expression too large to display}\)	\(1376\)

Input:

int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*b^5/a^3/(a*d-b*c)^4*(-(a*d-b*c)*a/b/(b*x^2+a)+(5*a*d-2*b*c)/b*ln(b*x^ 
2+a))-1/2/a^2/c^3/x^2+(-3*a*d-2*b*c)/a^3/c^4*ln(x)+1/2*d^4/c^4/(a*d-b*c)^4 
*(-2*c*(a^2*d^2-3*a*b*c*d+2*b^2*c^2)/d/(d*x^2+c)+(3*a^2*d^2-10*a*b*c*d+10* 
b^2*c^2)/d*ln(d*x^2+c)-1/2*c^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d/(d*x^2+c)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1227 vs. \(2 (205) = 410\).

Time = 16.03 (sec) , antiderivative size = 1227, normalized size of antiderivative = 5.71 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

-1/4*(2*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d + 12*a^4*b^2*c^5*d^2 - 8*a^5*b*c^4*d 
^3 + 2*a^6*c^3*d^4 + 2*(2*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c 
^3*d^4 - 10*a^4*b^2*c^2*d^5 + 3*a^5*b*c*d^6)*x^6 + (8*a*b^5*c^6*d - 18*a^2 
*b^4*c^5*d^2 + 25*a^3*b^3*c^4*d^3 - 10*a^4*b^2*c^3*d^4 - 11*a^5*b*c^2*d^5 
+ 6*a^6*c*d^6)*x^4 + (4*a*b^5*c^7 - 6*a^2*b^4*c^6*d - 4*a^3*b^3*c^5*d^2 + 
25*a^4*b^2*c^4*d^3 - 28*a^5*b*c^3*d^4 + 9*a^6*c^2*d^5)*x^2 - 2*((2*b^6*c^5 
*d^2 - 5*a*b^5*c^4*d^3)*x^8 + (4*b^6*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c 
^4*d^3)*x^6 + (2*b^6*c^7 - a*b^5*c^6*d - 10*a^2*b^4*c^5*d^2)*x^4 + (2*a*b^ 
5*c^7 - 5*a^2*b^4*c^6*d)*x^2)*log(b*x^2 + a) - 2*((10*a^3*b^3*c^2*d^5 - 10 
*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (20*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c^2*d 
^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^6 + (10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3 
*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (10*a^4*b^2*c^4*d^3 - 10*a^5* 
b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(d*x^2 + c) + 4*((2*b^6*c^5*d^2 - 5*a*b 
^5*c^4*d^3 + 10*a^3*b^3*c^2*d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (4 
*b^6*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^3*d^4 - 10 
*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^6 + (2*b^6*c^7 - a*b^5*c^6 
*d - 10*a^2*b^4*c^5*d^2 + 10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5 
*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d + 10*a^4*b^ 
2*c^4*d^3 - 10*a^5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(x))/((a^3*b^5*c^8*d 
^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (205) = 410\).

Time = 0.07 (sec) , antiderivative size = 651, normalized size of antiderivative = 3.03 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac {2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} + {\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} + {\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \, {\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} + {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \] Input:

integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

1/2*(2*b^5*c - 5*a*b^4*d)*log(b*x^2 + a)/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 
6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4) + 1/2*(10*b^2*c^2*d^3 - 10*a* 
b*c*d^4 + 3*a^2*d^5)*log(d*x^2 + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c 
^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4) - 1/4*(2*a*b^3*c^5 - 6*a^2*b^2*c^4 
*d + 6*a^3*b*c^3*d^2 - 2*a^4*c^2*d^3 + 2*(2*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 
+ 7*a^2*b^2*c*d^4 - 3*a^3*b*d^5)*x^6 + (8*b^4*c^4*d - 10*a*b^3*c^3*d^2 + 1 
5*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d^4 - 6*a^4*d^5)*x^4 + (4*b^4*c^5 - 2*a*b^3* 
c^4*d - 6*a^2*b^2*c^3*d^2 + 19*a^3*b*c^2*d^3 - 9*a^4*c*d^4)*x^2)/((a^2*b^4 
*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^8 + (2 
*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a 
^6*c^3*d^5)*x^6 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5 
*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^4 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b 
*c^6*d^2 - a^6*c^5*d^3)*x^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/(a^3*c^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (205) = 410\).

Time = 0.12 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.97 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac {10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \, {\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )} {\left (b x^{4} + a x^{2}\right )}} - \frac {30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \] Input:

integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

1/2*(2*b^6*c - 5*a*b^5*d)*log(abs(b*x^2 + a))/(a^3*b^5*c^4 - 4*a^4*b^4*c^3 
*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4) + 1/2*(10*b^2*c^2*d^ 
4 - 10*a*b*c*d^5 + 3*a^2*d^6)*log(abs(d*x^2 + c))/(b^4*c^8*d - 4*a*b^3*c^7 
*d^2 + 6*a^2*b^2*c^6*d^3 - 4*a^3*b*c^5*d^4 + a^4*c^4*d^5) + 1/4*(10*a^2*b^ 
3*c^2*d^3*x^4 - 10*a^3*b^2*c*d^4*x^4 + 3*a^4*b*d^5*x^4 - 4*b^5*c^5*x^2 + 1 
0*a*b^4*c^4*d*x^2 - 12*a^2*b^3*c^3*d^2*x^2 + 18*a^3*b^2*c^2*d^3*x^2 - 12*a 
^4*b*c*d^4*x^2 + 3*a^5*d^5*x^2 - 2*a*b^4*c^5 + 8*a^2*b^3*c^4*d - 12*a^3*b^ 
2*c^3*d^2 + 8*a^4*b*c^2*d^3 - 2*a^5*c*d^4)/((a^2*b^4*c^8 - 4*a^3*b^3*c^7*d 
 + 6*a^4*b^2*c^6*d^2 - 4*a^5*b*c^5*d^3 + a^6*c^4*d^4)*(b*x^4 + a*x^2)) - 1 
/4*(30*b^2*c^2*d^5*x^4 - 30*a*b*c*d^6*x^4 + 9*a^2*d^7*x^4 + 68*b^2*c^3*d^4 
*x^2 - 72*a*b*c^2*d^5*x^2 + 22*a^2*c*d^6*x^2 + 39*b^2*c^4*d^3 - 44*a*b*c^3 
*d^4 + 14*a^2*c^2*d^5)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a 
^3*b*c^5*d^3 + a^4*c^4*d^4)*(d*x^2 + c)^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/ 
(a^3*c^4)
 

Mupad [B] (verification not implemented)

Time = 2.98 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (2\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^4-8\,a^6\,b\,c\,d^3+12\,a^5\,b^2\,c^2\,d^2-8\,a^4\,b^3\,c^3\,d+2\,a^3\,b^4\,c^4}-\frac {\frac {1}{2\,a\,c}-\frac {x^4\,\left (-6\,a^4\,d^5+5\,a^3\,b\,c\,d^4+15\,a^2\,b^2\,c^2\,d^3-10\,a\,b^3\,c^3\,d^2+8\,b^4\,c^4\,d\right )}{4\,a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x^2\,\left (9\,a^4\,d^4-19\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d-4\,b^4\,c^4\right )}{4\,a^2\,c^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d^2\,x^6\,\left (3\,a^3\,d^3-7\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^4\,\left (b\,c^2+2\,a\,d\,c\right )+x^6\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^2+b\,d^2\,x^8}+\frac {\ln \left (d\,x^2+c\right )\,\left (3\,a^2\,d^5-10\,a\,b\,c\,d^4+10\,b^2\,c^2\,d^3\right )}{2\,a^4\,c^4\,d^4-8\,a^3\,b\,c^5\,d^3+12\,a^2\,b^2\,c^6\,d^2-8\,a\,b^3\,c^7\,d+2\,b^4\,c^8}-\frac {\ln \left (x\right )\,\left (3\,a\,d+2\,b\,c\right )}{a^3\,c^4} \] Input:

int(1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x)
 

Output:

(log(a + b*x^2)*(2*b^5*c - 5*a*b^4*d))/(2*a^7*d^4 + 2*a^3*b^4*c^4 - 8*a^4* 
b^3*c^3*d + 12*a^5*b^2*c^2*d^2 - 8*a^6*b*c*d^3) - (1/(2*a*c) - (x^4*(8*b^4 
*c^4*d - 6*a^4*d^5 - 10*a*b^3*c^3*d^2 + 15*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d^4 
))/(4*a^2*c^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x^2* 
(9*a^4*d^4 - 4*b^4*c^4 + 6*a^2*b^2*c^2*d^2 + 2*a*b^3*c^3*d - 19*a^3*b*c*d^ 
3))/(4*a^2*c^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*d 
^2*x^6*(3*a^3*d^3 - 2*b^3*c^3 + 3*a*b^2*c^2*d - 7*a^2*b*c*d^2))/(2*a^2*c^3 
*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(x^4*(b*c^2 + 2*a*c 
*d) + x^6*(a*d^2 + 2*b*c*d) + a*c^2*x^2 + b*d^2*x^8) + (log(c + d*x^2)*(3* 
a^2*d^5 + 10*b^2*c^2*d^3 - 10*a*b*c*d^4))/(2*b^4*c^8 + 2*a^4*c^4*d^4 - 8*a 
^3*b*c^5*d^3 + 12*a^2*b^2*c^6*d^2 - 8*a*b^3*c^7*d) - (log(x)*(3*a*d + 2*b* 
c))/(a^3*c^4)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1909, normalized size of antiderivative = 8.88 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x)
 

Output:

( - 10*log(a + b*x**2)*a**3*b**4*c**6*d**2*x**2 - 20*log(a + b*x**2)*a**3* 
b**4*c**5*d**3*x**4 - 10*log(a + b*x**2)*a**3*b**4*c**4*d**4*x**6 - 16*log 
(a + b*x**2)*a**2*b**5*c**7*d*x**2 - 42*log(a + b*x**2)*a**2*b**5*c**6*d** 
2*x**4 - 36*log(a + b*x**2)*a**2*b**5*c**5*d**3*x**6 - 10*log(a + b*x**2)* 
a**2*b**5*c**4*d**4*x**8 + 8*log(a + b*x**2)*a*b**6*c**8*x**2 - 24*log(a + 
 b*x**2)*a*b**6*c**6*d**2*x**6 - 16*log(a + b*x**2)*a*b**6*c**5*d**3*x**8 
+ 8*log(a + b*x**2)*b**7*c**8*x**4 + 16*log(a + b*x**2)*b**7*c**7*d*x**6 + 
 8*log(a + b*x**2)*b**7*c**6*d**2*x**8 + 6*log(c + d*x**2)*a**7*c**2*d**6* 
x**2 + 12*log(c + d*x**2)*a**7*c*d**7*x**4 + 6*log(c + d*x**2)*a**7*d**8*x 
**6 - 8*log(c + d*x**2)*a**6*b*c**3*d**5*x**2 - 10*log(c + d*x**2)*a**6*b* 
c**2*d**6*x**4 + 4*log(c + d*x**2)*a**6*b*c*d**7*x**6 + 6*log(c + d*x**2)* 
a**6*b*d**8*x**8 - 20*log(c + d*x**2)*a**5*b**2*c**4*d**4*x**2 - 48*log(c 
+ d*x**2)*a**5*b**2*c**3*d**5*x**4 - 36*log(c + d*x**2)*a**5*b**2*c**2*d** 
6*x**6 - 8*log(c + d*x**2)*a**5*b**2*c*d**7*x**8 + 40*log(c + d*x**2)*a**4 
*b**3*c**5*d**3*x**2 + 60*log(c + d*x**2)*a**4*b**3*c**4*d**4*x**4 - 20*lo 
g(c + d*x**2)*a**4*b**3*c**2*d**6*x**8 + 40*log(c + d*x**2)*a**3*b**4*c**5 
*d**3*x**4 + 80*log(c + d*x**2)*a**3*b**4*c**4*d**4*x**6 + 40*log(c + d*x* 
*2)*a**3*b**4*c**3*d**5*x**8 - 12*log(x)*a**7*c**2*d**6*x**2 - 24*log(x)*a 
**7*c*d**7*x**4 - 12*log(x)*a**7*d**8*x**6 + 16*log(x)*a**6*b*c**3*d**5*x* 
*2 + 20*log(x)*a**6*b*c**2*d**6*x**4 - 8*log(x)*a**6*b*c*d**7*x**6 - 12...