Integrand size = 18, antiderivative size = 129 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=\frac {a^3}{b (b c-a d)^3 (a+b x)}+\frac {c^3}{2 d^2 (b c-a d)^2 (c+d x)^2}-\frac {c^2 (b c-3 a d)}{d^2 (b c-a d)^3 (c+d x)}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4} \]
a^3/b/(-a*d+b*c)^3/(b*x+a)+1/2*c^3/d^2/(-a*d+b*c)^2/(d*x+c)^2-c^2*(-3*a*d+ b*c)/d^2/(-a*d+b*c)^3/(d*x+c)+3*a^2*c*ln(b*x+a)/(-a*d+b*c)^4-3*a^2*c*ln(d* x+c)/(-a*d+b*c)^4
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=\frac {a^3}{b (b c-a d)^3 (a+b x)}+\frac {c^3}{2 d^2 (-b c+a d)^2 (c+d x)^2}+\frac {b c^3-3 a c^2 d}{d^2 (-b c+a d)^3 (c+d x)}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4} \]
a^3/(b*(b*c - a*d)^3*(a + b*x)) + c^3/(2*d^2*(-(b*c) + a*d)^2*(c + d*x)^2) + (b*c^3 - 3*a*c^2*d)/(d^2*(-(b*c) + a*d)^3*(c + d*x)) + (3*a^2*c*Log[a + b*x])/(b*c - a*d)^4 - (3*a^2*c*Log[c + d*x])/(b*c - a*d)^4
Time = 0.32 (sec) , antiderivative size = 129, 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.
| \(\displaystyle \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^3}{(a+b x)^2 (b c-a d)^3}+\frac {3 a^2 b c}{(a+b x) (b c-a d)^4}-\frac {3 a^2 c d}{(c+d x) (a d-b c)^4}-\frac {c^3}{d (c+d x)^3 (a d-b c)^2}-\frac {c^2 (b c-3 a d)}{d (c+d x)^2 (a d-b c)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac {c^3}{2 d^2 (c+d x)^2 (b c-a d)^2}-\frac {c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3}\) |
a^3/(b*(b*c - a*d)^3*(a + b*x)) + c^3/(2*d^2*(b*c - a*d)^2*(c + d*x)^2) - (c^2*(b*c - 3*a*d))/(d^2*(b*c - a*d)^3*(c + d*x)) + (3*a^2*c*Log[a + b*x]) /(b*c - a*d)^4 - (3*a^2*c*Log[c + d*x])/(b*c - a*d)^4
3.3.94.3.1 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 = 0.51 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(-\frac {c^{2} \left (3 a d -b c \right )}{\left (a d -b c \right )^{3} d^{2} \left (d x +c \right )}+\frac {c^{3}}{2 d^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}-\frac {3 a^{2} c \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {a^{3}}{\left (a d -b c \right )^{3} b \left (b x +a \right )}+\frac {3 a^{2} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\) | \(130\) |
| norman | \(\frac {\frac {\left (-a^{3} d^{3}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{2}}{d b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (-4 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (-2 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right )}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {3 a^{2} c \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {3 a^{2} c \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(374\) |
| risch | \(\frac {-\frac {\left (a^{3} d^{3}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{d b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c \left (4 a^{3} d^{3}+6 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{2 b \,d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (2 a^{2} d^{2}+5 a b c d -b^{2} c^{2}\right ) a \,c^{2}}{2 b \,d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {3 a^{2} c \ln \left (-b x -a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {3 a^{2} c \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(380\) |
| parallelrisch | \(\frac {2 x^{2} a^{3} b c \,d^{4}-6 x^{2} a^{2} b^{2} c^{2} d^{3}+8 x^{2} a \,b^{3} c^{3} d^{2}-2 x \,a^{3} b \,c^{2} d^{3}+3 x \,a^{2} b^{2} c^{3} d^{2}+4 x a \,b^{3} c^{4} d +6 \ln \left (b x +a \right ) a^{3} b \,c^{3} d^{2}-2 x^{2} a^{4} d^{5}-2 x^{2} b^{4} c^{4} d -4 x \,a^{4} c \,d^{4}-x \,b^{4} c^{5}-2 a^{4} c^{2} d^{3}-a \,b^{3} c^{5}-6 \ln \left (d x +c \right ) a^{3} b \,c^{3} d^{2}-3 a^{3} b \,c^{3} d^{2}+6 a^{2} b^{2} c^{4} d +6 \ln \left (b x +a \right ) x^{3} a^{2} b^{2} c \,d^{4}-6 \ln \left (d x +c \right ) x^{3} a^{2} b^{2} c \,d^{4}+6 \ln \left (b x +a \right ) x^{2} a^{3} b c \,d^{4}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} c^{2} d^{3}-6 \ln \left (d x +c \right ) x^{2} a^{3} b c \,d^{4}-12 \ln \left (d x +c \right ) x^{2} a^{2} b^{2} c^{2} d^{3}+12 \ln \left (b x +a \right ) x \,a^{3} b \,c^{2} d^{3}+6 \ln \left (b x +a \right ) x \,a^{2} b^{2} c^{3} d^{2}-12 \ln \left (d x +c \right ) x \,a^{3} b \,c^{2} d^{3}-6 \ln \left (d x +c \right ) x \,a^{2} b^{2} c^{3} d^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right )^{2} \left (b x +a \right ) b \,d^{2}}\) | \(487\) |
-c^2*(3*a*d-b*c)/(a*d-b*c)^3/d^2/(d*x+c)+1/2*c^3/d^2/(a*d-b*c)^2/(d*x+c)^2 -3*a^2*c/(a*d-b*c)^4*ln(d*x+c)-a^3/(a*d-b*c)^3/b/(b*x+a)+3*a^2*c/(a*d-b*c) ^4*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (127) = 254\).
Time = 0.23 (sec) , antiderivative size = 621, normalized size of antiderivative = 4.81 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} + 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4} + a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3} + 4 \, a^{4} c d^{4}\right )} x - 6 \, {\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} + {\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} + {\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{5} c^{6} d^{2} - 4 \, a^{2} b^{4} c^{5} d^{3} + 6 \, a^{3} b^{3} c^{4} d^{4} - 4 \, a^{4} b^{2} c^{3} d^{5} + a^{5} b c^{2} d^{6} + {\left (b^{6} c^{4} d^{4} - 4 \, a b^{5} c^{3} d^{5} + 6 \, a^{2} b^{4} c^{2} d^{6} - 4 \, a^{3} b^{3} c d^{7} + a^{4} b^{2} d^{8}\right )} x^{3} + {\left (2 \, b^{6} c^{5} d^{3} - 7 \, a b^{5} c^{4} d^{4} + 8 \, a^{2} b^{4} c^{3} d^{5} - 2 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7} + a^{5} b d^{8}\right )} x^{2} + {\left (b^{6} c^{6} d^{2} - 2 \, a b^{5} c^{5} d^{3} - 2 \, a^{2} b^{4} c^{4} d^{4} + 8 \, a^{3} b^{3} c^{3} d^{5} - 7 \, a^{4} b^{2} c^{2} d^{6} + 2 \, a^{5} b c d^{7}\right )} x\right )}} \]
-1/2*(a*b^3*c^5 - 6*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 + 2*a^4*c^2*d^3 + 2*(b ^4*c^4*d - 4*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^ 2 + (b^4*c^5 - 4*a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 2*a^3*b*c^2*d^3 + 4*a^4 *c*d^4)*x - 6*(a^2*b^2*c*d^4*x^3 + a^3*b*c^3*d^2 + (2*a^2*b^2*c^2*d^3 + a^ 3*b*c*d^4)*x^2 + (a^2*b^2*c^3*d^2 + 2*a^3*b*c^2*d^3)*x)*log(b*x + a) + 6*( a^2*b^2*c*d^4*x^3 + a^3*b*c^3*d^2 + (2*a^2*b^2*c^2*d^3 + a^3*b*c*d^4)*x^2 + (a^2*b^2*c^3*d^2 + 2*a^3*b*c^2*d^3)*x)*log(d*x + c))/(a*b^5*c^6*d^2 - 4* a^2*b^4*c^5*d^3 + 6*a^3*b^3*c^4*d^4 - 4*a^4*b^2*c^3*d^5 + a^5*b*c^2*d^6 + (b^6*c^4*d^4 - 4*a*b^5*c^3*d^5 + 6*a^2*b^4*c^2*d^6 - 4*a^3*b^3*c*d^7 + a^4 *b^2*d^8)*x^3 + (2*b^6*c^5*d^3 - 7*a*b^5*c^4*d^4 + 8*a^2*b^4*c^3*d^5 - 2*a ^3*b^3*c^2*d^6 - 2*a^4*b^2*c*d^7 + a^5*b*d^8)*x^2 + (b^6*c^6*d^2 - 2*a*b^5 *c^5*d^3 - 2*a^2*b^4*c^4*d^4 + 8*a^3*b^3*c^3*d^5 - 7*a^4*b^2*c^2*d^6 + 2*a ^5*b*c*d^7)*x)
Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (114) = 228\).
Time = 1.00 (sec) , antiderivative size = 717, normalized size of antiderivative = 5.56 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=- \frac {3 a^{2} c \log {\left (x + \frac {- \frac {3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} - \frac {15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d + \frac {3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {3 a^{2} c \log {\left (x + \frac {\frac {3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + \frac {15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d - \frac {3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {- 2 a^{3} c^{2} d^{2} - 5 a^{2} b c^{3} d + a b^{2} c^{4} + x^{2} \left (- 2 a^{3} d^{4} - 6 a b^{2} c^{2} d^{2} + 2 b^{3} c^{3} d\right ) + x \left (- 4 a^{3} c d^{3} - 6 a^{2} b c^{2} d^{2} - 3 a b^{2} c^{3} d + b^{3} c^{4}\right )}{2 a^{4} b c^{2} d^{5} - 6 a^{3} b^{2} c^{3} d^{4} + 6 a^{2} b^{3} c^{4} d^{3} - 2 a b^{4} c^{5} d^{2} + x^{3} \cdot \left (2 a^{3} b^{2} d^{7} - 6 a^{2} b^{3} c d^{6} + 6 a b^{4} c^{2} d^{5} - 2 b^{5} c^{3} d^{4}\right ) + x^{2} \cdot \left (2 a^{4} b d^{7} - 2 a^{3} b^{2} c d^{6} - 6 a^{2} b^{3} c^{2} d^{5} + 10 a b^{4} c^{3} d^{4} - 4 b^{5} c^{4} d^{3}\right ) + x \left (4 a^{4} b c d^{6} - 10 a^{3} b^{2} c^{2} d^{5} + 6 a^{2} b^{3} c^{3} d^{4} + 2 a b^{4} c^{4} d^{3} - 2 b^{5} c^{5} d^{2}\right )} \]
-3*a**2*c*log(x + (-3*a**7*c*d**5/(a*d - b*c)**4 + 15*a**6*b*c**2*d**4/(a* d - b*c)**4 - 30*a**5*b**2*c**3*d**3/(a*d - b*c)**4 + 30*a**4*b**3*c**4*d* *2/(a*d - b*c)**4 - 15*a**3*b**4*c**5*d/(a*d - b*c)**4 + 3*a**3*c*d + 3*a* *2*b**5*c**6/(a*d - b*c)**4 + 3*a**2*b*c**2)/(6*a**2*b*c*d))/(a*d - b*c)** 4 + 3*a**2*c*log(x + (3*a**7*c*d**5/(a*d - b*c)**4 - 15*a**6*b*c**2*d**4/( a*d - b*c)**4 + 30*a**5*b**2*c**3*d**3/(a*d - b*c)**4 - 30*a**4*b**3*c**4* d**2/(a*d - b*c)**4 + 15*a**3*b**4*c**5*d/(a*d - b*c)**4 + 3*a**3*c*d - 3* a**2*b**5*c**6/(a*d - b*c)**4 + 3*a**2*b*c**2)/(6*a**2*b*c*d))/(a*d - b*c) **4 + (-2*a**3*c**2*d**2 - 5*a**2*b*c**3*d + a*b**2*c**4 + x**2*(-2*a**3*d **4 - 6*a*b**2*c**2*d**2 + 2*b**3*c**3*d) + x*(-4*a**3*c*d**3 - 6*a**2*b*c **2*d**2 - 3*a*b**2*c**3*d + b**3*c**4))/(2*a**4*b*c**2*d**5 - 6*a**3*b**2 *c**3*d**4 + 6*a**2*b**3*c**4*d**3 - 2*a*b**4*c**5*d**2 + x**3*(2*a**3*b** 2*d**7 - 6*a**2*b**3*c*d**6 + 6*a*b**4*c**2*d**5 - 2*b**5*c**3*d**4) + x** 2*(2*a**4*b*d**7 - 2*a**3*b**2*c*d**6 - 6*a**2*b**3*c**2*d**5 + 10*a*b**4* c**3*d**4 - 4*b**5*c**4*d**3) + x*(4*a**4*b*c*d**6 - 10*a**3*b**2*c**2*d** 5 + 6*a**2*b**3*c**3*d**4 + 2*a*b**4*c**4*d**3 - 2*b**5*c**5*d**2))
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (127) = 254\).
Time = 0.21 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.59 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 \, a^{2} c \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {3 \, a^{2} c \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {a b^{2} c^{4} - 5 \, a^{2} b c^{3} d - 2 \, a^{3} c^{2} d^{2} + 2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{3} d^{4}\right )} x^{2} + {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} - 4 \, a^{3} c d^{3}\right )} x}{2 \, {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5} + {\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{3} + {\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{2} + {\left (b^{5} c^{5} d^{2} - a b^{4} c^{4} d^{3} - 3 \, a^{2} b^{3} c^{3} d^{4} + 5 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6}\right )} x\right )}} \]
3*a^2*c*log(b*x + 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*a^2*c*log(d*x + 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) - 1/2*(a*b^2*c^4 - 5*a^2*b*c^3*d - 2*a^3*c^2*d^2 + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 - a^3*d^4)*x^2 + (b^3*c^4 - 3*a*b^2*c^3*d - 6*a^2*b*c^2*d^2 - 4*a^3*c*d^3)*x)/(a*b^4*c^5*d^2 - 3*a^2 *b^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 - a^4*b*c^2*d^5 + (b^5*c^3*d^4 - 3*a*b^4* c^2*d^5 + 3*a^2*b^3*c*d^6 - a^3*b^2*d^7)*x^3 + (2*b^5*c^4*d^3 - 5*a*b^4*c^ 3*d^4 + 3*a^2*b^3*c^2*d^5 + a^3*b^2*c*d^6 - a^4*b*d^7)*x^2 + (b^5*c^5*d^2 - a*b^4*c^4*d^3 - 3*a^2*b^3*c^3*d^4 + 5*a^3*b^2*c^2*d^5 - 2*a^4*b*c*d^6)*x )
Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.78 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {3 \, a^{2} b c \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {a^{3} b^{2}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} + \frac {b^{2} c^{3} - 6 \, a b c^{2} d - \frac {6 \, {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \]
-3*a^2*b*c*log(abs(b*c/(b*x + a) - a*d/(b*x + 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) + a^3*b^2/((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 + a)) + 1/2*(b^2* c^3 - 6*a*b*c^2*d - 6*(a*b^3*c^3 - a^2*b^2*c^2*d)/((b*x + a)*b))/((b*c - a *d)^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2)
Time = 0.53 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.07 \[ \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx=\frac {6\,a^2\,c\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {x^2\,\left (a^3\,d^3+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^2\,d^2+5\,a\,b\,c\,d-b^2\,c^2\right )}{2\,b\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3} \]
(6*a^2*c*atanh((a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3)/(a*d - b*c)^4 + (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a* d - b*c)^4))/(a*d - b*c)^4 - ((x^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d))/(b *d*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (c*x*(4*a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2))/(2*b*d^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (a*c^2*(2*a^2*d^2 - b^2*c^2 + 5*a*b*c*d) )/(2*b*d^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(x*(b*c^2 + 2*a*c*d) + a*c^2 + x^2*(a*d^2 + 2*b*c*d) + b*d^2*x^3)