\(\int \frac {x^5}{(a+b x^2)^3 (c+d x^2)} \, dx\) [645]

Optimal result

Integrand size = 22, antiderivative size = 116 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=-\frac {a^2}{4 b^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {a (2 b c-a d)}{2 b^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {a (-b c+a d) \left (-3 a b c+a^2 d-4 b^2 c x^2+2 a b d x^2\right )+2 b^2 c^2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )-2 b^2 c^2 \left (a+b x^2\right )^2 \log \left (c+d x^2\right )}{4 b^2 (b c-a d)^3 \left (a+b x^2\right )^2} \] Input:

Integrate[x^5/((a + b*x^2)^3*(c + d*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^5/((a + b*x^2)^3*(c + d*x^2)),x]
 

Output:

(-1/2*a^2/(b^2*(b*c - a*d)*(a + b*x^2)^2) + (a*(2*b*c - a*d))/(b^2*(b*c - 
a*d)^2*(a + b*x^2)) + (c^2*Log[a + b*x^2])/(b*c - a*d)^3 - (c^2*Log[c + d* 
x^2])/(b*c - a*d)^3)/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 = 0.56 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.07

method	result	size
default	\(-\frac {-\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{2} \left (b \,x^{2}+a \right )^{2}}+\frac {a \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{b^{2} \left (b \,x^{2}+a \right )}+c^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{3}}+\frac {c^{2} \ln \left (x^{2} d +c \right )}{2 \left (a d -b c \right )^{3}}\)	\(124\)
norman	\(\frac {\frac {\left (-a d +3 b c \right ) a^{2}}{4 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a \left (-a d +2 b c \right ) x^{2}}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right )^{2}}-\frac {c^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} \ln \left (x^{2} d +c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\)	\(196\)
risch	\(\frac {-\frac {a \left (a d -2 b c \right ) x^{2}}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a^{2} \left (a d -3 b c \right )}{4 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right )^{2}}+\frac {c^{2} \ln \left (x^{2} d +c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}-\frac {c^{2} \ln \left (-b \,x^{2}-a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(197\)
parallelrisch	\(-\frac {2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{2}-2 \ln \left (x^{2} d +c \right ) x^{4} b^{4} c^{2}+4 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{2}-4 \ln \left (x^{2} d +c \right ) x^{2} a \,b^{3} c^{2}+2 a^{3} b \,d^{2} x^{2}-6 a^{2} b^{2} c d \,x^{2}+4 a \,b^{3} c^{2} x^{2}+2 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2}-2 \ln \left (x^{2} d +c \right ) a^{2} b^{2} c^{2}+a^{4} d^{2}-4 a^{3} c d b +3 a^{2} b^{2} c^{2}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b \,x^{2}+a \right )^{2} b^{2}}\)	\(233\)

Input:

int(x^5/(b*x^2+a)^3/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (108) = 216\).

Time = 0.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.50 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \, {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 2 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2}\right )}} \] Input:

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (97) = 194\).

Time = 1.62 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.60 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {c^{2} \log {\left (x^{2} + \frac {- \frac {a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d - \frac {b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac {c^{2} \log {\left (x^{2} + \frac {\frac {a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d + \frac {b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac {- a^{3} d + 3 a^{2} b c + x^{2} \left (- 2 a^{2} b d + 4 a b^{2} c\right )}{4 a^{4} b^{2} d^{2} - 8 a^{3} b^{3} c d + 4 a^{2} b^{4} c^{2} + x^{4} \cdot \left (4 a^{2} b^{4} d^{2} - 8 a b^{5} c d + 4 b^{6} c^{2}\right ) + x^{2} \cdot \left (8 a^{3} b^{3} d^{2} - 16 a^{2} b^{4} c d + 8 a b^{5} c^{2}\right )} \] Input:

integrate(x**5/(b*x**2+a)**3/(d*x**2+c),x)
 

Output:

c**2*log(x**2 + (-a**4*c**2*d**4/(a*d - b*c)**3 + 4*a**3*b*c**3*d**3/(a*d 
- b*c)**3 - 6*a**2*b**2*c**4*d**2/(a*d - b*c)**3 + 4*a*b**3*c**5*d/(a*d - 
b*c)**3 + a*c**2*d - b**4*c**6/(a*d - b*c)**3 + b*c**3)/(2*b*c**2*d))/(2*( 
a*d - b*c)**3) - c**2*log(x**2 + (a**4*c**2*d**4/(a*d - b*c)**3 - 4*a**3*b 
*c**3*d**3/(a*d - b*c)**3 + 6*a**2*b**2*c**4*d**2/(a*d - b*c)**3 - 4*a*b** 
3*c**5*d/(a*d - b*c)**3 + a*c**2*d + b**4*c**6/(a*d - b*c)**3 + b*c**3)/(2 
*b*c**2*d))/(2*(a*d - b*c)**3) + (-a**3*d + 3*a**2*b*c + x**2*(-2*a**2*b*d 
 + 4*a*b**2*c))/(4*a**4*b**2*d**2 - 8*a**3*b**3*c*d + 4*a**2*b**4*c**2 + x 
**4*(4*a**2*b**4*d**2 - 8*a*b**5*c*d + 4*b**6*c**2) + x**2*(8*a**3*b**3*d* 
*2 - 16*a**2*b**4*c*d + 8*a*b**5*c**2))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (108) = 216\).

Time = 0.06 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.03 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {c^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {c^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {3 \, a^{2} b c - a^{3} d + 2 \, {\left (2 \, a b^{2} c - a^{2} b d\right )} x^{2}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2}\right )}} \] Input:

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (108) = 216\).

Time = 0.12 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.00 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {b c^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {c^{2} d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} - \frac {3 \, b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + 6 \, a^{2} b^{2} c d x^{2} - 2 \, a^{3} b d^{2} x^{2} + 4 \, a^{3} b c d - a^{4} d^{2}}{4 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (b x^{2} + a\right )}^{2}} \] Input:

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.19 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {b^3\,\left (4\,a\,c^2\,x^2+a\,c^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,8{}\mathrm {i}\right )+b\,\left (2\,a^3\,d^2\,x^2-4\,a^3\,c\,d\right )+a^4\,d^2+b^2\,\left (3\,a^2\,c^2-6\,a^2\,c\,d\,x^2+a^2\,c^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}\right )+b^4\,c^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}}{-4\,a^5\,b^2\,d^3+12\,a^4\,b^3\,c\,d^2-8\,a^4\,b^3\,d^3\,x^2-12\,a^3\,b^4\,c^2\,d+24\,a^3\,b^4\,c\,d^2\,x^2-4\,a^3\,b^4\,d^3\,x^4+4\,a^2\,b^5\,c^3-24\,a^2\,b^5\,c^2\,d\,x^2+12\,a^2\,b^5\,c\,d^2\,x^4+8\,a\,b^6\,c^3\,x^2-12\,a\,b^6\,c^2\,d\,x^4+4\,b^7\,c^3\,x^4} \] Input:

int(x^5/((a + b*x^2)^3*(c + d*x^2)),x)
 

Output:

(b^3*(4*a*c^2*x^2 + a*c^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))*8i) + b*(2*a^3*d^2*x^2 - 4*a^3*c*d) + a^4*d^2 + b^2*(3*a^2 
*c^2 + a^2*c^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2)) 
*4i - 6*a^2*c*d*x^2) + b^4*c^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))*4i)/(4*a^2*b^5*c^3 - 4*a^5*b^2*d^3 + 4*b^7*c^3*x^4 - 
12*a^3*b^4*c^2*d + 12*a^4*b^3*c*d^2 + 8*a*b^6*c^3*x^2 - 8*a^4*b^3*d^3*x^2 
- 4*a^3*b^4*d^3*x^4 - 12*a*b^6*c^2*d*x^4 - 24*a^2*b^5*c^2*d*x^2 + 24*a^3*b 
^4*c*d^2*x^2 + 12*a^2*b^5*c*d^2*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.72 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b \,c^{2}-4 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c^{2} x^{2}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} c^{2} x^{4}+2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} b \,c^{2}+4 \,\mathrm {log}\left (d \,x^{2}+c \right ) a \,b^{2} c^{2} x^{2}+2 \,\mathrm {log}\left (d \,x^{2}+c \right ) b^{3} c^{2} x^{4}+a^{3} c d -a^{2} b \,c^{2}+a^{2} b \,d^{2} x^{4}-3 a \,b^{2} c d \,x^{4}+2 b^{3} c^{2} x^{4}}{4 b \left (a^{3} b^{2} d^{3} x^{4}-3 a^{2} b^{3} c \,d^{2} x^{4}+3 a \,b^{4} c^{2} d \,x^{4}-b^{5} c^{3} x^{4}+2 a^{4} b \,d^{3} x^{2}-6 a^{3} b^{2} c \,d^{2} x^{2}+6 a^{2} b^{3} c^{2} d \,x^{2}-2 a \,b^{4} c^{3} x^{2}+a^{5} d^{3}-3 a^{4} b c \,d^{2}+3 a^{3} b^{2} c^{2} d -a^{2} b^{3} c^{3}\right )} \] Input:

int(x^5/(b*x^2+a)^3/(d*x^2+c),x)
 

Output:

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