Integrand size = 18, antiderivative size = 180 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {a (2 b c+a d)}{(b c-a d)^4 (a+b x)}+\frac {c^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {c (b c+2 a d)}{(b c-a d)^4 (c+d x)}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (c+d x)}{(b c-a d)^5} \]
-1/2*a^2/(-a*d+b*c)^3/(b*x+a)^2+a*(a*d+2*b*c)/(-a*d+b*c)^4/(b*x+a)+1/2*c^2 /(-a*d+b*c)^3/(d*x+c)^2+c*(2*a*d+b*c)/(-a*d+b*c)^4/(d*x+c)+(a^2*d^2+4*a*b* c*d+b^2*c^2)*ln(b*x+a)/(-a*d+b*c)^5-(a^2*d^2+4*a*b*c*d+b^2*c^2)*ln(d*x+c)/ (-a*d+b*c)^5
Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {-\frac {a^2 (b c-a d)^2}{(a+b x)^2}+\frac {2 a (b c-a d) (2 b c+a d)}{a+b x}+\frac {c^2 (b c-a d)^2}{(c+d x)^2}+\frac {2 c (b c-a d) (b c+2 a d)}{c+d x}+2 \left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (a+b x)-2 \left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (c+d x)}{2 (b c-a d)^5} \]
(-((a^2*(b*c - a*d)^2)/(a + b*x)^2) + (2*a*(b*c - a*d)*(2*b*c + a*d))/(a + b*x) + (c^2*(b*c - a*d)^2)/(c + d*x)^2 + (2*c*(b*c - a*d)*(b*c + 2*a*d))/ (c + d*x) + 2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*Log[a + b*x] - 2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*Log[c + d*x])/(2*(b*c - a*d)^5)
Time = 0.38 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
| \(\displaystyle \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {b \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{(a+b x) (b c-a d)^5}-\frac {d \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{(c+d x) (b c-a d)^5}+\frac {a^2 b}{(a+b x)^3 (b c-a d)^3}-\frac {c^2 d}{(c+d x)^3 (b c-a d)^3}-\frac {a b (a d+2 b c)}{(a+b x)^2 (b c-a d)^4}-\frac {c d (2 a d+b c)}{(c+d x)^2 (b c-a d)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac {a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac {a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac {c (2 a d+b c)}{(c+d x) (b c-a d)^4}\) |
-1/2*a^2/((b*c - a*d)^3*(a + b*x)^2) + (a*(2*b*c + a*d))/((b*c - a*d)^4*(a + b*x)) + c^2/(2*(b*c - a*d)^3*(c + d*x)^2) + (c*(b*c + 2*a*d))/((b*c - a *d)^4*(c + d*x)) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*Log[a + b*x])/(b*c - a *d)^5 - ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*Log[c + d*x])/(b*c - a*d)^5
3.4.14.3.1 Defintions of rubi rules used
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]))
Time = 0.54 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {c^{2}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {c \left (2 a d +b c \right )}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {a^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {a \left (a d +2 b c \right )}{\left (a d -b c \right )^{4} \left (b x +a \right )}\) | \(177\) |
| norman | \(\frac {\frac {\left (a^{2} b^{2} d^{4}+4 a \,b^{3} c \,d^{3}+b^{4} c^{2} d^{2}\right ) x^{3}}{d b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {c^{2} a^{2} \left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right )}{b^{2} d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (5 a^{2} b^{2} d^{4}+8 a \,b^{3} c \,d^{3}+5 b^{4} c^{2} d^{2}\right ) c a x}{\left (a d -b c \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} d^{2}}+\frac {\left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right ) \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{2}}{2 \left (a d -b c \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} d^{2}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(562\) |
| risch | \(\frac {\frac {b d \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 \left (a d +b c \right ) \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a c \left (5 a^{2} d^{2}+8 a b c d +5 b^{2} c^{2}\right ) x}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 c^{2} a^{2} \left (a d +b c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {\ln \left (-d x -c \right ) a^{2} d^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {4 \ln \left (-d x -c \right ) a b c d}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {\ln \left (-d x -c \right ) b^{2} c^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {\ln \left (b x +a \right ) a^{2} d^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {4 \ln \left (b x +a \right ) a b c d}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {\ln \left (b x +a \right ) b^{2} c^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(807\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1044\) |
-1/2*c^2/(a*d-b*c)^3/(d*x+c)^2+(a^2*d^2+4*a*b*c*d+b^2*c^2)/(a*d-b*c)^5*ln( d*x+c)+c*(2*a*d+b*c)/(a*d-b*c)^4/(d*x+c)+1/2*a^2/(a*d-b*c)^3/(b*x+a)^2-(a^ 2*d^2+4*a*b*c*d+b^2*c^2)/(a*d-b*c)^5*ln(b*x+a)+a*(a*d+2*b*c)/(a*d-b*c)^4/( b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (176) = 352\).
Time = 0.24 (sec) , antiderivative size = 990, normalized size of antiderivative = 5.50 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, a^{2} b^{2} c^{4} - 6 \, a^{4} c^{2} d^{2} + 2 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{4} + 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x + 2 \, {\left (a^{2} b^{2} c^{4} + 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} + 8 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 5 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a^{2} b^{2} c^{4} + 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} + 8 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 5 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \]
1/2*(6*a^2*b^2*c^4 - 6*a^4*c^2*d^2 + 2*(b^4*c^3*d + 3*a*b^3*c^2*d^2 - 3*a^ 2*b^2*c*d^3 - a^3*b*d^4)*x^3 + 3*(b^4*c^4 + 4*a*b^3*c^3*d - 4*a^3*b*c*d^3 - a^4*d^4)*x^2 + 2*(5*a*b^3*c^4 + 3*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 - 5*a^ 4*c*d^3)*x + 2*(a^2*b^2*c^4 + 4*a^3*b*c^3*d + a^4*c^2*d^2 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^4 + 2*(b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2 *b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^4*c^4 + 8*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^ 2 + 8*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3*c^4 + 5*a^2*b^2*c^3*d + 5*a^3* b*c^2*d^2 + a^4*c*d^3)*x)*log(b*x + a) - 2*(a^2*b^2*c^4 + 4*a^3*b*c^3*d + a^4*c^2*d^2 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^4 + 2*(b^4*c^3 *d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^4*c^4 + 8*a*b ^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 8*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3*c^ 4 + 5*a^2*b^2*c^3*d + 5*a^3*b*c^2*d^2 + a^4*c*d^3)*x)*log(d*x + c))/(a^2*b ^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6 *b*c^3*d^4 - a^7*c^2*d^5 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3 *d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 + 2*(b^7*c^ 6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2* c*d^6 - a^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a ^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^ 7*d^7)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^ 2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (162) = 324\).
Time = 1.72 (sec) , antiderivative size = 1299, normalized size of antiderivative = 7.22 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]
(6*a**3*c**2*d + 6*a**2*b*c**3 + x**3*(2*a**2*b*d**3 + 8*a*b**2*c*d**2 + 2 *b**3*c**2*d) + x**2*(3*a**3*d**3 + 15*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 3*b**3*c**3) + x*(10*a**3*c*d**2 + 16*a**2*b*c**2*d + 10*a*b**2*c**3))/(2* a**6*c**2*d**4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4*d**2 - 8*a**3*b**3 *c**5*d + 2*a**2*b**4*c**6 + x**4*(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b**4*c**2*d**4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**3*(4 *a**5*b*d**6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2*d**4 + 8*a**2*b**4*c **3*d**3 - 12*a*b**5*c**4*d**2 + 4*b**6*c**5*d) + x**2*(2*a**6*d**6 - 18*a **4*b**2*c**2*d**4 + 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2 + 2*b **6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4 + 8*a**4*b**2*c**3*d**3 + 8*a**3*b**3*c**4*d**2 - 12*a**2*b**4*c**5*d + 4*a*b**5*c**6)) + (a**2*d **2 + 4*a*b*c*d + b**2*c**2)*log(x + (-a**6*d**6*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 6*a**5*b*c*d**5*(a**2*d**2 + 4*a*b*c*d + b**2* c**2)/(a*d - b*c)**5 - 15*a**4*b**2*c**2*d**4*(a**2*d**2 + 4*a*b*c*d + b** 2*c**2)/(a*d - b*c)**5 + 20*a**3*b**3*c**3*d**3*(a**2*d**2 + 4*a*b*c*d + b **2*c**2)/(a*d - b*c)**5 + a**3*d**3 - 15*a**2*b**4*c**4*d**2*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 5*a**2*b*c*d**2 + 6*a*b**5*c**5*d *(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 5*a*b**2*c**2*d - b* *6*c**6*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + b**3*c**3)/(2 *a**2*b*d**3 + 8*a*b**2*c*d**2 + 2*b**3*c**2*d))/(a*d - b*c)**5 - (a**2...
Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (176) = 352\).
Time = 0.23 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.59 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d + 2 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \, {\left (b^{3} c^{3} + 5 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c^{3} + 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} x}{2 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10 *a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - (b^2*c^ 2 + 4*a*b*c*d + a^2*d^2)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^ 3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) + 1/2*(6*a^2*b*c ^3 + 6*a^3*c^2*d + 2*(b^3*c^2*d + 4*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 3*(b^3* c^3 + 5*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3)*x^2 + 2*(5*a*b^2*c^3 + 8*a^ 2*b*c^2*d + 5*a^3*c*d^2)*x)/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4 *d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6* a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a* b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^ 5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b ^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4 *d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (176) = 352\).
Time = 0.28 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.29 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {{\left (b^{3} c^{2} + 4 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {{\left (b^{2} c^{2} d + 4 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} + \frac {2 \, b^{3} c^{2} d x^{3} + 8 \, a b^{2} c d^{2} x^{3} + 2 \, a^{2} b d^{3} x^{3} + 3 \, b^{3} c^{3} x^{2} + 15 \, a b^{2} c^{2} d x^{2} + 15 \, a^{2} b c d^{2} x^{2} + 3 \, a^{3} d^{3} x^{2} + 10 \, a b^{2} c^{3} x + 16 \, a^{2} b c^{2} d x + 10 \, a^{3} c d^{2} x + 6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]
(b^3*c^2 + 4*a*b^2*c*d + a^2*b*d^2)*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c ^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d ^5) - (b^2*c^2*d + 4*a*b*c*d^2 + a^2*d^3)*log(abs(d*x + c))/(b^5*c^5*d - 5 *a*b^4*c^4*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6) + 1/2*(2*b^3*c^2*d*x^3 + 8*a*b^2*c*d^2*x^3 + 2*a^2*b*d^3*x^3 + 3 *b^3*c^3*x^2 + 15*a*b^2*c^2*d*x^2 + 15*a^2*b*c*d^2*x^2 + 3*a^3*d^3*x^2 + 1 0*a*b^2*c^3*x + 16*a^2*b*c^2*d*x + 10*a^3*c*d^2*x + 6*a^2*b*c^3 + 6*a^3*c^ 2*d)/((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)*(b*d*x^2 + b*c*x + a*d*x + a*c)^2)
Time = 0.76 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.16 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {3\,\left (d\,a^3\,c^2+b\,a^2\,c^3\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {3\,x^2\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b\,d\,x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {a\,c\,x\,\left (5\,a^2\,d^2+8\,a\,b\,c\,d+5\,b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^5} \]
((3*(a^2*b*c^3 + a^3*c^2*d))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a* b^3*c^3*d - 4*a^3*b*c*d^3) + (3*x^2*(a*d + b*c)*(a^2*d^2 + b^2*c^2 + 4*a*b *c*d))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b *c*d^3)) + (b*d*x^3*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3) + (a*c*x*(5*a^2*d^2 + 5 *b^2*c^2 + 8*a*b*c*d))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^ 3*d - 4*a^3*b*c*d^3))/(x*(2*a*b*c^2 + 2*a^2*c*d) + x^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*a*b*d^2 + 2*b^2*c*d) + a^2*c^2 + b^2*d^2*x^4) - (2*a tanh((a^5*d^5 + b^5*c^5 + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a*b^4* c^4*d - 3*a^4*b*c*d^4)/(a*d - b*c)^5 + (2*b*d*x*(a^4*d^4 + b^4*c^4 + 6*a^2 *b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(a*d - b*c)^5)*(a^2*d^2 + b ^2*c^2 + 4*a*b*c*d))/(a*d - b*c)^5