\(\int \frac {1}{x^3 (a+b x^2) (c+d x^2)^2} \, dx\) [643]

Optimal result

Integrand size = 22, antiderivative size = 126 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {1}{2 a c^2 x^2}+\frac {d^2}{2 c^2 (b c-a d) \left (c+d x^2\right )}-\frac {(b c+2 a d) \log (x)}{a^2 c^3}+\frac {b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac {d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {1}{a c^2 x^2}+\frac {d^2}{c^2 (b c-a d) \left (c+d x^2\right )}-\frac {2 (b c+2 a d) \log (x)}{a^2 c^3}+\frac {b^3 \log \left (a+b x^2\right )}{a^2 (b c-a d)^2}+\frac {d^2 (-3 b c+2 a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 122, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-(1/(a*c^2*x^2)) + d^2/(c^2*(b*c - a*d)*(c + d*x^2)) - ((b*c + 2*a*d)*Log 
[x^2])/(a^2*c^3) + (b^3*Log[a + b*x^2])/(a^2*(b*c - a*d)^2) - (d^2*(3*b*c 
- 2*a*d)*Log[c + d*x^2])/(c^3*(b*c - a*d)^2))/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 = 0.57 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a d -b c \right )^{2}}-\frac {1}{2 a \,c^{2} x^{2}}+\frac {\left (-2 a d -b c \right ) \ln \left (x \right )}{a^{2} c^{3}}+\frac {d^{3} \left (-\frac {\left (a d -b c \right ) c}{d \left (x^{2} d +c \right )}+\frac {\left (2 a d -3 b c \right ) \ln \left (x^{2} d +c \right )}{d}\right )}{2 c^{3} \left (a d -b c \right )^{2}}\)	\(120\)
norman	\(\frac {-\frac {1}{2 a c}+\frac {\left (-2 a \,d^{3}+b c \,d^{2}\right ) x^{2}}{2 c^{2} d a \left (a d -b c \right )}}{x^{2} \left (x^{2} d +c \right )}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a d +b c \right ) \ln \left (x \right )}{a^{2} c^{3}}+\frac {d^{2} \left (2 a d -3 b c \right ) \ln \left (x^{2} d +c \right )}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(166\)
risch	\(\frac {-\frac {d \left (2 a d -b c \right ) x^{2}}{2 a \,c^{2} \left (a d -b c \right )}-\frac {1}{2 a c}}{x^{2} \left (x^{2} d +c \right )}-\frac {2 \ln \left (x \right ) d}{a \,c^{3}}-\frac {\ln \left (x \right ) b}{a^{2} c^{2}}+\frac {d^{3} \ln \left (-x^{2} d -c \right ) a}{c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {3 d^{2} \ln \left (-x^{2} d -c \right ) b}{2 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(201\)
parallelrisch	\(-\frac {4 \ln \left (x \right ) x^{4} a^{3} d^{5}-6 \ln \left (x \right ) x^{4} a^{2} b c \,d^{4}+2 \ln \left (x \right ) x^{4} b^{3} c^{3} d^{2}-\ln \left (b \,x^{2}+a \right ) x^{4} b^{3} c^{3} d^{2}-2 \ln \left (x^{2} d +c \right ) x^{4} a^{3} d^{5}+3 \ln \left (x^{2} d +c \right ) x^{4} a^{2} b c \,d^{4}+4 \ln \left (x \right ) x^{2} a^{3} c \,d^{4}-6 \ln \left (x \right ) x^{2} a^{2} b \,c^{2} d^{3}+2 \ln \left (x \right ) x^{2} b^{3} c^{4} d -\ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{4} d -2 \ln \left (x^{2} d +c \right ) x^{2} a^{3} c \,d^{4}+3 \ln \left (x^{2} d +c \right ) x^{2} a^{2} b \,c^{2} d^{3}+2 x^{2} a^{3} c \,d^{4}-3 x^{2} a^{2} b \,c^{2} d^{3}+x^{2} a \,b^{2} c^{3} d^{2}+a^{3} c^{2} d^{3}-2 a^{2} b \,c^{3} d^{2}+a \,b^{2} c^{4} d}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (x^{2} d +c \right ) x^{2} a^{2} c^{3} d}\)	\(334\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (118) = 236\).

Time = 1.25 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.40 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} + {\left (a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2} - {\left (b^{3} c^{3} d x^{4} + b^{3} c^{4} x^{2}\right )} \log \left (b x^{2} + a\right ) + {\left ({\left (3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x^{4} + {\left (3 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left ({\left (b^{3} c^{3} d - 3 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{4} + {\left (b^{3} c^{4} - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{2} b^{2} c^{5} d - 2 \, a^{3} b c^{4} d^{2} + a^{4} c^{3} d^{3}\right )} x^{4} + {\left (a^{2} b^{2} c^{6} - 2 \, a^{3} b c^{5} d + a^{4} c^{4} d^{2}\right )} x^{2}\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{3} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac {{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {b c^{2} - a c d + {\left (b c d - 2 \, a d^{2}\right )} x^{2}}{2 \, {\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{4} + {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{2}\right )}} - \frac {{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (118) = 236\).

Time = 0.11 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )}} - \frac {{\left (3 \, b c d^{3} - 2 \, a d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} + \frac {b^{3} c^{2} d x^{4} + b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - 4 \, a^{3} d^{3} x^{2} - 2 \, a b^{2} c^{3} + 4 \, a^{2} b c^{2} d - 2 \, a^{3} c d^{2}}{4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (d x^{4} + c x^{2}\right )}} - \frac {{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^3\,\ln \left (b\,x^2+a\right )}{2\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2\right )}-\frac {\frac {1}{2\,a\,c}+\frac {x^2\,\left (2\,a\,d^2-b\,c\,d\right )}{2\,a\,c^2\,\left (a\,d-b\,c\right )}}{d\,x^4+c\,x^2}+\frac {\ln \left (d\,x^2+c\right )\,\left (2\,a\,d^3-3\,b\,c\,d^2\right )}{2\,a^2\,c^3\,d^2-4\,a\,b\,c^4\,d+2\,b^2\,c^5}-\frac {\ln \left (x\right )\,\left (2\,a\,d+b\,c\right )}{a^2\,c^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} c^{4} x^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} c^{3} d \,x^{4}+2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{3} c \,d^{3} x^{2}+2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{3} d^{4} x^{4}-3 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} b \,c^{2} d^{2} x^{2}-3 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} b c \,d^{3} x^{4}-4 \,\mathrm {log}\left (x \right ) a^{3} c \,d^{3} x^{2}-4 \,\mathrm {log}\left (x \right ) a^{3} d^{4} x^{4}+6 \,\mathrm {log}\left (x \right ) a^{2} b \,c^{2} d^{2} x^{2}+6 \,\mathrm {log}\left (x \right ) a^{2} b c \,d^{3} x^{4}-2 \,\mathrm {log}\left (x \right ) b^{3} c^{4} x^{2}-2 \,\mathrm {log}\left (x \right ) b^{3} c^{3} d \,x^{4}-a^{3} c^{2} d^{2}+2 a^{3} d^{4} x^{4}+2 a^{2} b \,c^{3} d -3 a^{2} b c \,d^{3} x^{4}-a \,b^{2} c^{4}+a \,b^{2} c^{2} d^{2} x^{4}}{2 a^{2} c^{3} x^{2} \left (a^{2} d^{3} x^{2}-2 a b c \,d^{2} x^{2}+b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:

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

Output:

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