\(\int \frac {x}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [707]

Optimal result

Integrand size = 20, antiderivative size = 126 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {b^2}{2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {b d}{(b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac {3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {\frac {2 b^2 (b c-a d)}{a+b x^2}+\frac {d (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {4 b d (b c-a d)}{c+d x^2}+6 b^2 d \log \left (a+b x^2\right )-6 b^2 d \log \left (c+d x^2\right )}{4 (b c-a d)^4} \] Input:

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

Output:

-1/4*((2*b^2*(b*c - a*d))/(a + b*x^2) + (d*(b*c - a*d)^2)/(c + d*x^2)^2 + 
(4*b*d*(b*c - a*d))/(c + d*x^2) + 6*b^2*d*Log[a + b*x^2] - 6*b^2*d*Log[c + 
 d*x^2])/(b*c - a*d)^4
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {353, 54, 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.

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(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]
 

rule 2009

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

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.14

method	result	size
default	\(-\frac {b^{3} \left (-\frac {a d -b c}{b \left (b \,x^{2}+a \right )}+\frac {3 d \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 \left (a d -b c \right )^{4}}+\frac {d^{2} \left (\frac {2 b \left (a d -b c \right )}{d \left (x^{2} d +c \right )}+\frac {3 b^{2} \ln \left (x^{2} d +c \right )}{d}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 d \left (x^{2} d +c \right )^{2}}\right )}{2 \left (a d -b c \right )^{4}}\)	\(144\)
risch	\(\frac {\frac {3 b^{2} d^{2} x^{4}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 \left (a d +3 b c \right ) d b \,x^{2}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {a^{2} d^{2}-5 a b c d -2 b^{2} c^{2}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )^{2}}+\frac {3 d \,b^{2} \ln \left (x^{2} d +c \right )}{2 \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}-\frac {3 d \,b^{2} \ln \left (-b \,x^{2}-a \right )}{2 \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(319\)
norman	\(\frac {\frac {-a^{2} b \,d^{4}+5 a \,b^{2} c \,d^{3}+2 b^{3} c^{2} d^{2}}{4 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 b^{2} d^{2} x^{4}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (3 a \,b^{2} d^{4}+9 b^{3} c \,d^{3}\right ) x^{2}}{4 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )^{2}}-\frac {3 d \,b^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {3 d \,b^{2} \ln \left (x^{2} d +c \right )}{2 \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(346\)
parallelrisch	\(-\frac {6 \ln \left (b \,x^{2}+a \right ) x^{6} b^{4} d^{5}-6 \ln \left (x^{2} d +c \right ) x^{6} b^{4} d^{5}-6 x^{4} a \,b^{3} d^{5}+6 x^{4} b^{4} c \,d^{4}-3 x^{2} a^{2} b^{2} d^{5}+9 x^{2} b^{4} c^{2} d^{3}-6 a^{2} b^{2} c \,d^{4}+3 a \,b^{3} c^{2} d^{3}-6 x^{2} a \,b^{3} c \,d^{4}+6 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{3} d^{5}+12 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c \,d^{4}-6 \ln \left (x^{2} d +c \right ) x^{4} a \,b^{3} d^{5}-12 \ln \left (x^{2} d +c \right ) x^{4} b^{4} c \,d^{4}+6 \ln \left (b \,x^{2}+a \right ) x^{2} b^{4} c^{2} d^{3}-6 \ln \left (x^{2} d +c \right ) x^{2} b^{4} c^{2} d^{3}+6 \ln \left (b \,x^{2}+a \right ) a \,b^{3} c^{2} d^{3}-6 \ln \left (x^{2} d +c \right ) a \,b^{3} c^{2} d^{3}-12 \ln \left (x^{2} d +c \right ) x^{2} a \,b^{3} c \,d^{4}+12 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c \,d^{4}+2 b^{4} c^{3} d^{2}+a^{3} b \,d^{5}}{4 \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \left (x^{2} d +c \right )^{2} \left (b \,x^{2}+a \right ) b \,d^{2}}\)	\(431\)

Input:

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

Output:

-1/2*b^3/(a*d-b*c)^4*(-(a*d-b*c)/b/(b*x^2+a)+3*d/b*ln(b*x^2+a))+1/2*d^2/(a 
*d-b*c)^4*(2*b*(a*d-b*c)/d/(d*x^2+c)+3*b^2/d*ln(d*x^2+c)-1/2*(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. 507 vs. \(2 (118) = 236\).

Time = 0.09 (sec) , antiderivative size = 507, normalized size of antiderivative = 4.02 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 3 \, {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{3} x^{6} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{4} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (b^{3} d^{3} x^{6} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{4} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (109) = 218\).

Time = 6.03 (sec) , antiderivative size = 643, normalized size of antiderivative = 5.10 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {3 b^{2} d \log {\left (x^{2} + \frac {- \frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} + \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{2 \left (a d - b c\right )^{4}} - \frac {3 b^{2} d \log {\left (x^{2} + \frac {\frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} - \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{2 \left (a d - b c\right )^{4}} + \frac {- a^{2} d^{2} + 5 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{4} + x^{2} \cdot \left (3 a b d^{2} + 9 b^{2} c d\right )}{4 a^{4} c^{2} d^{3} - 12 a^{3} b c^{3} d^{2} + 12 a^{2} b^{2} c^{4} d - 4 a b^{3} c^{5} + x^{6} \cdot \left (4 a^{3} b d^{5} - 12 a^{2} b^{2} c d^{4} + 12 a b^{3} c^{2} d^{3} - 4 b^{4} c^{3} d^{2}\right ) + x^{4} \cdot \left (4 a^{4} d^{5} - 4 a^{3} b c d^{4} - 12 a^{2} b^{2} c^{2} d^{3} + 20 a b^{3} c^{3} d^{2} - 8 b^{4} c^{4} d\right ) + x^{2} \cdot \left (8 a^{4} c d^{4} - 20 a^{3} b c^{2} d^{3} + 12 a^{2} b^{2} c^{3} d^{2} + 4 a b^{3} c^{4} d - 4 b^{4} c^{5}\right )} \] Input:

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

Output:

3*b**2*d*log(x**2 + (-3*a**5*b**2*d**6/(a*d - b*c)**4 + 15*a**4*b**3*c*d** 
5/(a*d - b*c)**4 - 30*a**3*b**4*c**2*d**4/(a*d - b*c)**4 + 30*a**2*b**5*c* 
*3*d**3/(a*d - b*c)**4 - 15*a*b**6*c**4*d**2/(a*d - b*c)**4 + 3*a*b**2*d** 
2 + 3*b**7*c**5*d/(a*d - b*c)**4 + 3*b**3*c*d)/(6*b**3*d**2))/(2*(a*d - b* 
c)**4) - 3*b**2*d*log(x**2 + (3*a**5*b**2*d**6/(a*d - b*c)**4 - 15*a**4*b* 
*3*c*d**5/(a*d - b*c)**4 + 30*a**3*b**4*c**2*d**4/(a*d - b*c)**4 - 30*a**2 
*b**5*c**3*d**3/(a*d - b*c)**4 + 15*a*b**6*c**4*d**2/(a*d - b*c)**4 + 3*a* 
b**2*d**2 - 3*b**7*c**5*d/(a*d - b*c)**4 + 3*b**3*c*d)/(6*b**3*d**2))/(2*( 
a*d - b*c)**4) + (-a**2*d**2 + 5*a*b*c*d + 2*b**2*c**2 + 6*b**2*d**2*x**4 
+ x**2*(3*a*b*d**2 + 9*b**2*c*d))/(4*a**4*c**2*d**3 - 12*a**3*b*c**3*d**2 
+ 12*a**2*b**2*c**4*d - 4*a*b**3*c**5 + x**6*(4*a**3*b*d**5 - 12*a**2*b**2 
*c*d**4 + 12*a*b**3*c**2*d**3 - 4*b**4*c**3*d**2) + x**4*(4*a**4*d**5 - 4* 
a**3*b*c*d**4 - 12*a**2*b**2*c**2*d**3 + 20*a*b**3*c**3*d**2 - 8*b**4*c**4 
*d) + x**2*(8*a**4*c*d**4 - 20*a**3*b*c**2*d**3 + 12*a**2*b**2*c**3*d**2 + 
 4*a*b**3*c**4*d - 4*b**4*c**5))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (118) = 236\).

Time = 0.05 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.13 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {3 \, b^{2} d \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac {3 \, b^{2} d \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {6 \, b^{2} d^{2} x^{4} + 2 \, b^{2} c^{2} + 5 \, a b c d - a^{2} d^{2} + 3 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}}{4 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} \] Input:

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

Output:

-3/2*b^2*d*log(b*x^2 + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4 
*a^3*b*c*d^3 + a^4*d^4) + 3/2*b^2*d*log(d*x^2 + c)/(b^4*c^4 - 4*a*b^3*c^3* 
d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 1/4*(6*b^2*d^2*x^4 + 2* 
b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x^2)/(a*b^3*c^5 - 
3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c 
^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 
 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d 
 - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.82 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {3 \, b^{3} d \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{2 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {b^{5}}{2 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} + \frac {5 \, b^{2} d^{3} + \frac {6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}{{\left (b x^{2} + a\right )} b}}{4 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}^{2}} \] Input:

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

Output:

3/2*b^3*d*log(abs(b*c/(b*x^2 + a) - a*d/(b*x^2 + a) + d))/(b^5*c^4 - 4*a*b 
^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 1/2*b^5/((b^ 
6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*(b*x^2 + a)) + 1/4* 
(5*b^2*d^3 + 6*(b^4*c*d^2 - a*b^3*d^3)/((b*x^2 + a)*b))/((b*c - a*d)^4*(b* 
c/(b*x^2 + a) - a*d/(b*x^2 + a) + d)^2)
 

Mupad [B] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 707, normalized size of antiderivative = 5.61 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {a^3\,d^3+2\,b^3\,c^3-3\,a^2\,b\,d^3\,x^2-6\,a\,b^2\,d^3\,x^4+9\,b^3\,c^2\,d\,x^2+6\,b^3\,c\,d^2\,x^4+3\,a\,b^2\,c^2\,d-6\,a^2\,b\,c\,d^2+b^3\,d^3\,x^6\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,12{}\mathrm {i}+a\,b^2\,d^3\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,12{}\mathrm {i}+b^3\,c^2\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,12{}\mathrm {i}+b^3\,c\,d^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,24{}\mathrm {i}-6\,a\,b^2\,c\,d^2\,x^2+a\,b^2\,c^2\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,12{}\mathrm {i}+a\,b^2\,c\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,24{}\mathrm {i}}{4\,a^5\,c^2\,d^4+8\,a^5\,c\,d^5\,x^2+4\,a^5\,d^6\,x^4-16\,a^4\,b\,c^3\,d^3-28\,a^4\,b\,c^2\,d^4\,x^2-8\,a^4\,b\,c\,d^5\,x^4+4\,a^4\,b\,d^6\,x^6+24\,a^3\,b^2\,c^4\,d^2+32\,a^3\,b^2\,c^3\,d^3\,x^2-8\,a^3\,b^2\,c^2\,d^4\,x^4-16\,a^3\,b^2\,c\,d^5\,x^6-16\,a^2\,b^3\,c^5\,d-8\,a^2\,b^3\,c^4\,d^2\,x^2+32\,a^2\,b^3\,c^3\,d^3\,x^4+24\,a^2\,b^3\,c^2\,d^4\,x^6+4\,a\,b^4\,c^6-8\,a\,b^4\,c^5\,d\,x^2-28\,a\,b^4\,c^4\,d^2\,x^4-16\,a\,b^4\,c^3\,d^3\,x^6+4\,b^5\,c^6\,x^2+8\,b^5\,c^5\,d\,x^4+4\,b^5\,c^4\,d^2\,x^6} \] Input:

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

Output:

-(a^3*d^3 + 2*b^3*c^3 - 3*a^2*b*d^3*x^2 - 6*a*b^2*d^3*x^4 + 9*b^3*c^2*d*x^ 
2 + 6*b^3*c*d^2*x^4 + 3*a*b^2*c^2*d - 6*a^2*b*c*d^2 + b^3*d^3*x^6*atan((a* 
d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*12i + a*b^2*d^3*x^4*at 
an((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*12i + b^3*c^2*d* 
x^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*12i + b^3* 
c*d^2*x^4*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*24i 
- 6*a*b^2*c*d^2*x^2 + a*b^2*c^2*d*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + 
a*d*x^2 + b*c*x^2))*12i + a*b^2*c*d^2*x^2*atan((a*d*x^2*1i - b*c*x^2*1i)/( 
2*a*c + a*d*x^2 + b*c*x^2))*24i)/(4*a*b^4*c^6 + 4*a^5*c^2*d^4 + 4*b^5*c^6* 
x^2 + 4*a^5*d^6*x^4 - 16*a^2*b^3*c^5*d - 16*a^4*b*c^3*d^3 + 4*a^4*b*d^6*x^ 
6 + 8*a^5*c*d^5*x^2 + 8*b^5*c^5*d*x^4 + 24*a^3*b^2*c^4*d^2 + 4*b^5*c^4*d^2 
*x^6 - 8*a^2*b^3*c^4*d^2*x^2 + 32*a^3*b^2*c^3*d^3*x^2 + 32*a^2*b^3*c^3*d^3 
*x^4 - 8*a^3*b^2*c^2*d^4*x^4 + 24*a^2*b^3*c^2*d^4*x^6 - 8*a*b^4*c^5*d*x^2 
- 8*a^4*b*c*d^5*x^4 - 28*a^4*b*c^2*d^4*x^2 - 28*a*b^4*c^4*d^2*x^4 - 16*a*b 
^4*c^3*d^3*x^6 - 16*a^3*b^2*c*d^5*x^6)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 893, normalized size of antiderivative = 7.09 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*log(a + b*x**2)*a**2*b**2*c**2*d**2 - 12*log(a + b*x**2)*a**2*b**2*c 
*d**3*x**2 - 6*log(a + b*x**2)*a**2*b**2*d**4*x**4 - 12*log(a + b*x**2)*a* 
b**3*c**3*d - 30*log(a + b*x**2)*a*b**3*c**2*d**2*x**2 - 24*log(a + b*x**2 
)*a*b**3*c*d**3*x**4 - 6*log(a + b*x**2)*a*b**3*d**4*x**6 - 12*log(a + b*x 
**2)*b**4*c**3*d*x**2 - 24*log(a + b*x**2)*b**4*c**2*d**2*x**4 - 12*log(a 
+ b*x**2)*b**4*c*d**3*x**6 + 6*log(c + d*x**2)*a**2*b**2*c**2*d**2 + 12*lo 
g(c + d*x**2)*a**2*b**2*c*d**3*x**2 + 6*log(c + d*x**2)*a**2*b**2*d**4*x** 
4 + 12*log(c + d*x**2)*a*b**3*c**3*d + 30*log(c + d*x**2)*a*b**3*c**2*d**2 
*x**2 + 24*log(c + d*x**2)*a*b**3*c*d**3*x**4 + 6*log(c + d*x**2)*a*b**3*d 
**4*x**6 + 12*log(c + d*x**2)*b**4*c**3*d*x**2 + 24*log(c + d*x**2)*b**4*c 
**2*d**2*x**4 + 12*log(c + d*x**2)*b**4*c*d**3*x**6 - a**4*d**4 + 4*a**3*b 
*c*d**3 + 3*a**3*b*d**4*x**2 + 3*a**2*b**2*c**2*d**2 - 2*a*b**3*c**3*d + 9 
*a*b**3*c**2*d**2*x**2 - 6*a*b**3*d**4*x**6 - 4*b**4*c**4 - 12*b**4*c**3*d 
*x**2 + 6*b**4*c*d**3*x**6)/(4*(a**6*c**2*d**5 + 2*a**6*c*d**6*x**2 + a**6 
*d**7*x**4 - 2*a**5*b*c**3*d**4 - 3*a**5*b*c**2*d**5*x**2 + a**5*b*d**7*x* 
*6 - 2*a**4*b**2*c**4*d**3 - 6*a**4*b**2*c**3*d**4*x**2 - 6*a**4*b**2*c**2 
*d**5*x**4 - 2*a**4*b**2*c*d**6*x**6 + 8*a**3*b**3*c**5*d**2 + 14*a**3*b** 
3*c**4*d**3*x**2 + 4*a**3*b**3*c**3*d**4*x**4 - 2*a**3*b**3*c**2*d**5*x**6 
 - 7*a**2*b**4*c**6*d - 6*a**2*b**4*c**5*d**2*x**2 + 9*a**2*b**4*c**4*d**3 
*x**4 + 8*a**2*b**4*c**3*d**4*x**6 + 2*a*b**5*c**7 - 3*a*b**5*c**6*d*x*...