Integrand size = 18, antiderivative size = 161 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=-\frac {(3 b c+a d) x}{b^2 d^4}+\frac {x^2}{2 b d^3}-\frac {c^5}{2 d^5 (b c-a d) (c+d x)^2}+\frac {c^4 (4 b c-5 a d)}{d^5 (b c-a d)^2 (c+d x)}-\frac {a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac {c^3 \left (6 b^2 c^2-15 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^3} \]
-(a*d+3*b*c)*x/b^2/d^4+1/2*x^2/b/d^3-1/2*c^5/d^5/(-a*d+b*c)/(d*x+c)^2+c^4* (-5*a*d+4*b*c)/d^5/(-a*d+b*c)^2/(d*x+c)-a^5*ln(b*x+a)/b^3/(-a*d+b*c)^3+c^3 *(10*a^2*d^2-15*a*b*c*d+6*b^2*c^2)*ln(d*x+c)/d^5/(-a*d+b*c)^3
Time = 0.13 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=\frac {1}{2} \left (-\frac {2 (3 b c+a d) x}{b^2 d^4}+\frac {x^2}{b d^3}+\frac {c^5}{d^5 (-b c+a d) (c+d x)^2}+\frac {2 c^4 (4 b c-5 a d)}{d^5 (b c-a d)^2 (c+d x)}-\frac {2 a^5 \log (a+b x)}{b^3 (b c-a d)^3}-\frac {2 c^3 \left (6 b^2 c^2-15 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^5 (-b c+a d)^3}\right ) \]
((-2*(3*b*c + a*d)*x)/(b^2*d^4) + x^2/(b*d^3) + c^5/(d^5*(-(b*c) + a*d)*(c + d*x)^2) + (2*c^4*(4*b*c - 5*a*d))/(d^5*(b*c - a*d)^2*(c + d*x)) - (2*a^ 5*Log[a + b*x])/(b^3*(b*c - a*d)^3) - (2*c^3*(6*b^2*c^2 - 15*a*b*c*d + 10* a^2*d^2)*Log[c + d*x])/(d^5*(-(b*c) + a*d)^3))/2
Time = 0.38 (sec) , antiderivative size = 161, 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.
| \(\displaystyle \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^5}{b^2 (a+b x) (b c-a d)^3}-\frac {c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right )}{d^4 (c+d x) (a d-b c)^3}+\frac {-a d-3 b c}{b^2 d^4}-\frac {c^5}{d^4 (c+d x)^3 (a d-b c)}-\frac {c^4 (4 b c-5 a d)}{d^4 (c+d x)^2 (a d-b c)^2}+\frac {x}{b d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac {c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}-\frac {x (a d+3 b c)}{b^2 d^4}-\frac {c^5}{2 d^5 (c+d x)^2 (b c-a d)}+\frac {c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac {x^2}{2 b d^3}\) |
-(((3*b*c + a*d)*x)/(b^2*d^4)) + x^2/(2*b*d^3) - c^5/(2*d^5*(b*c - a*d)*(c + d*x)^2) + (c^4*(4*b*c - 5*a*d))/(d^5*(b*c - a*d)^2*(c + d*x)) - (a^5*Lo g[a + b*x])/(b^3*(b*c - a*d)^3) + (c^3*(6*b^2*c^2 - 15*a*b*c*d + 10*a^2*d^ 2)*Log[c + d*x])/(d^5*(b*c - a*d)^3)
3.3.51.3.1 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 = 0.50 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {-\frac {1}{2} b d \,x^{2}+a d x +3 b c x}{b^{2} d^{4}}-\frac {c^{4} \left (5 a d -4 b c \right )}{d^{5} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {c^{5}}{2 d^{5} \left (a d -b c \right ) \left (d x +c \right )^{2}}-\frac {c^{3} \left (10 a^{2} d^{2}-15 a b c d +6 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{3}}+\frac {a^{5} \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )^{3}}\) | \(156\) |
| norman | \(\frac {\frac {\left (3 a^{3} c \,d^{3}+2 a^{2} d^{2} b \,c^{2}-18 a \,b^{2} c^{3} d +12 c^{4} b^{3}\right ) c x}{d^{4} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {x^{4}}{2 b d}-\frac {\left (a d +2 b c \right ) x^{3}}{b^{2} d^{2}}+\frac {\left (4 a^{3} c \,d^{3}+3 a^{2} d^{2} b \,c^{2}-27 a \,b^{2} c^{3} d +18 c^{4} b^{3}\right ) c^{2}}{2 d^{5} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {a^{5} \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3}}-\frac {c^{3} \left (10 a^{2} d^{2}-15 a b c d +6 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(313\) |
| risch | \(\frac {x^{2}}{2 b \,d^{3}}-\frac {a x}{b^{2} d^{3}}-\frac {3 c x}{b \,d^{4}}+\frac {-\frac {b^{2} c^{4} \left (5 a d -4 b c \right ) x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}-\frac {b^{2} c^{5} \left (9 a d -7 b c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}}{b^{2} d^{4} \left (d x +c \right )^{2}}-\frac {10 c^{3} \ln \left (d x +c \right ) a^{2}}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {15 c^{4} \ln \left (d x +c \right ) a b}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {6 c^{5} \ln \left (d x +c \right ) b^{2}}{d^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {a^{5} \ln \left (-b x -a \right )}{b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(350\) |
| parallelrisch | \(\frac {-20 \ln \left (d x +c \right ) x^{2} a^{2} b^{3} c^{3} d^{4}+30 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{4} d^{3}-40 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{4} d^{3}+60 \ln \left (d x +c \right ) x a \,b^{4} c^{5} d^{2}+x^{4} a^{3} b^{2} d^{7}-x^{4} b^{5} c^{3} d^{4}-2 x^{3} a^{4} b \,d^{7}+4 x^{3} b^{5} c^{4} d^{3}-24 x \,b^{5} c^{6} d +2 \ln \left (b x +a \right ) x^{2} a^{5} d^{7}+2 \ln \left (b x +a \right ) a^{5} c^{2} d^{5}+4 a^{4} b \,c^{3} d^{4}-a^{3} b^{2} c^{4} d^{3}-30 a^{2} b^{3} c^{5} d^{2}+45 a \,b^{4} c^{6} d -3 x^{4} a^{2} b^{3} c \,d^{6}+3 x^{4} a \,b^{4} c^{2} d^{5}+2 x^{3} a^{3} b^{2} c \,d^{6}+6 x^{3} a^{2} b^{3} c^{2} d^{5}-10 x^{3} a \,b^{4} c^{3} d^{4}+6 x \,a^{4} b \,c^{2} d^{5}-2 x \,a^{3} b^{2} c^{3} d^{4}-40 x \,a^{2} b^{3} c^{4} d^{3}+60 x a \,b^{4} c^{5} d^{2}-12 \ln \left (d x +c \right ) x^{2} b^{5} c^{5} d^{2}+4 \ln \left (b x +a \right ) x \,a^{5} c \,d^{6}-24 \ln \left (d x +c \right ) x \,b^{5} c^{6} d -20 \ln \left (d x +c \right ) a^{2} b^{3} c^{5} d^{2}+30 \ln \left (d x +c \right ) a \,b^{4} c^{6} d -12 \ln \left (d x +c \right ) b^{5} c^{7}-18 b^{5} c^{7}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right )^{2} b^{3} d^{5}}\) | \(529\) |
-1/b^2/d^4*(-1/2*b*d*x^2+a*d*x+3*b*c*x)-1/d^5*c^4*(5*a*d-4*b*c)/(a*d-b*c)^ 2/(d*x+c)+1/2/d^5*c^5/(a*d-b*c)/(d*x+c)^2-1/d^5*c^3*(10*a^2*d^2-15*a*b*c*d +6*b^2*c^2)/(a*d-b*c)^3*ln(d*x+c)+1/b^3*a^5/(a*d-b*c)^3*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (157) = 314\).
Time = 0.26 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.60 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=\frac {7 \, b^{5} c^{7} - 16 \, a b^{4} c^{6} d + 9 \, a^{2} b^{3} c^{5} d^{2} + {\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{4} - 2 \, {\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{3} - {\left (11 \, b^{5} c^{5} d^{2} - 29 \, a b^{4} c^{4} d^{3} + 21 \, a^{2} b^{3} c^{3} d^{4} + a^{3} b^{2} c^{2} d^{5} - 4 \, a^{4} b c d^{6}\right )} x^{2} + 2 \, {\left (b^{5} c^{6} d - a b^{4} c^{5} d^{2} - a^{2} b^{3} c^{4} d^{3} + a^{4} b c^{2} d^{5}\right )} x - 2 \, {\left (a^{5} d^{7} x^{2} + 2 \, a^{5} c d^{6} x + a^{5} c^{2} d^{5}\right )} \log \left (b x + a\right ) + 2 \, {\left (6 \, b^{5} c^{7} - 15 \, a b^{4} c^{6} d + 10 \, a^{2} b^{3} c^{5} d^{2} + {\left (6 \, b^{5} c^{5} d^{2} - 15 \, a b^{4} c^{4} d^{3} + 10 \, a^{2} b^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (6 \, b^{5} c^{6} d - 15 \, a b^{4} c^{5} d^{2} + 10 \, a^{2} b^{3} c^{4} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{6} c^{5} d^{5} - 3 \, a b^{5} c^{4} d^{6} + 3 \, a^{2} b^{4} c^{3} d^{7} - a^{3} b^{3} c^{2} d^{8} + {\left (b^{6} c^{3} d^{7} - 3 \, a b^{5} c^{2} d^{8} + 3 \, a^{2} b^{4} c d^{9} - a^{3} b^{3} d^{10}\right )} x^{2} + 2 \, {\left (b^{6} c^{4} d^{6} - 3 \, a b^{5} c^{3} d^{7} + 3 \, a^{2} b^{4} c^{2} d^{8} - a^{3} b^{3} c d^{9}\right )} x\right )}} \]
1/2*(7*b^5*c^7 - 16*a*b^4*c^6*d + 9*a^2*b^3*c^5*d^2 + (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^4 - 2*(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^3 - (11*b^5* c^5*d^2 - 29*a*b^4*c^4*d^3 + 21*a^2*b^3*c^3*d^4 + a^3*b^2*c^2*d^5 - 4*a^4* b*c*d^6)*x^2 + 2*(b^5*c^6*d - a*b^4*c^5*d^2 - a^2*b^3*c^4*d^3 + a^4*b*c^2* d^5)*x - 2*(a^5*d^7*x^2 + 2*a^5*c*d^6*x + a^5*c^2*d^5)*log(b*x + a) + 2*(6 *b^5*c^7 - 15*a*b^4*c^6*d + 10*a^2*b^3*c^5*d^2 + (6*b^5*c^5*d^2 - 15*a*b^4 *c^4*d^3 + 10*a^2*b^3*c^3*d^4)*x^2 + 2*(6*b^5*c^6*d - 15*a*b^4*c^5*d^2 + 1 0*a^2*b^3*c^4*d^3)*x)*log(d*x + c))/(b^6*c^5*d^5 - 3*a*b^5*c^4*d^6 + 3*a^2 *b^4*c^3*d^7 - a^3*b^3*c^2*d^8 + (b^6*c^3*d^7 - 3*a*b^5*c^2*d^8 + 3*a^2*b^ 4*c*d^9 - a^3*b^3*d^10)*x^2 + 2*(b^6*c^4*d^6 - 3*a*b^5*c^3*d^7 + 3*a^2*b^4 *c^2*d^8 - a^3*b^3*c*d^9)*x)
Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (150) = 300\).
Time = 123.67 (sec) , antiderivative size = 748, normalized size of antiderivative = 4.65 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=\frac {a^{5} \log {\left (x + \frac {\frac {a^{9} d^{8}}{b \left (a d - b c\right )^{3}} - \frac {4 a^{8} c d^{7}}{\left (a d - b c\right )^{3}} + \frac {6 a^{7} b c^{2} d^{6}}{\left (a d - b c\right )^{3}} - \frac {4 a^{6} b^{2} c^{3} d^{5}}{\left (a d - b c\right )^{3}} + \frac {a^{5} b^{3} c^{4} d^{4}}{\left (a d - b c\right )^{3}} + a^{5} c d^{4} + 10 a^{3} b^{2} c^{3} d^{2} - 15 a^{2} b^{3} c^{4} d + 6 a b^{4} c^{5}}{a^{5} d^{5} + 10 a^{2} b^{3} c^{3} d^{2} - 15 a b^{4} c^{4} d + 6 b^{5} c^{5}} \right )}}{b^{3} \left (a d - b c\right )^{3}} - \frac {c^{3} \cdot \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right ) \log {\left (x + \frac {a^{5} c d^{4} - \frac {a^{4} b^{2} c^{3} d^{3} \cdot \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{3} c^{4} d^{2} \cdot \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 10 a^{3} b^{2} c^{3} d^{2} - \frac {6 a^{2} b^{4} c^{5} d \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 15 a^{2} b^{3} c^{4} d + \frac {4 a b^{5} c^{6} \cdot \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 6 a b^{4} c^{5} - \frac {b^{6} c^{7} \cdot \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{5} d^{5} + 10 a^{2} b^{3} c^{3} d^{2} - 15 a b^{4} c^{4} d + 6 b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )^{3}} + x \left (- \frac {a}{b^{2} d^{3}} - \frac {3 c}{b d^{4}}\right ) + \frac {- 9 a c^{5} d + 7 b c^{6} + x \left (- 10 a c^{4} d^{2} + 8 b c^{5} d\right )}{2 a^{2} c^{2} d^{7} - 4 a b c^{3} d^{6} + 2 b^{2} c^{4} d^{5} + x^{2} \cdot \left (2 a^{2} d^{9} - 4 a b c d^{8} + 2 b^{2} c^{2} d^{7}\right ) + x \left (4 a^{2} c d^{8} - 8 a b c^{2} d^{7} + 4 b^{2} c^{3} d^{6}\right )} + \frac {x^{2}}{2 b d^{3}} \]
a**5*log(x + (a**9*d**8/(b*(a*d - b*c)**3) - 4*a**8*c*d**7/(a*d - b*c)**3 + 6*a**7*b*c**2*d**6/(a*d - b*c)**3 - 4*a**6*b**2*c**3*d**5/(a*d - b*c)**3 + a**5*b**3*c**4*d**4/(a*d - b*c)**3 + a**5*c*d**4 + 10*a**3*b**2*c**3*d* *2 - 15*a**2*b**3*c**4*d + 6*a*b**4*c**5)/(a**5*d**5 + 10*a**2*b**3*c**3*d **2 - 15*a*b**4*c**4*d + 6*b**5*c**5))/(b**3*(a*d - b*c)**3) - c**3*(10*a* *2*d**2 - 15*a*b*c*d + 6*b**2*c**2)*log(x + (a**5*c*d**4 - a**4*b**2*c**3* d**3*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 + 4*a**3*b** 3*c**4*d**2*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 + 10* a**3*b**2*c**3*d**2 - 6*a**2*b**4*c**5*d*(10*a**2*d**2 - 15*a*b*c*d + 6*b* *2*c**2)/(a*d - b*c)**3 - 15*a**2*b**3*c**4*d + 4*a*b**5*c**6*(10*a**2*d** 2 - 15*a*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 + 6*a*b**4*c**5 - b**6*c**7*( 10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(d*(a*d - b*c)**3))/(a**5*d**5 + 10*a**2*b**3*c**3*d**2 - 15*a*b**4*c**4*d + 6*b**5*c**5))/(d**5*(a*d - b*c )**3) + x*(-a/(b**2*d**3) - 3*c/(b*d**4)) + (-9*a*c**5*d + 7*b*c**6 + x*(- 10*a*c**4*d**2 + 8*b*c**5*d))/(2*a**2*c**2*d**7 - 4*a*b*c**3*d**6 + 2*b**2 *c**4*d**5 + x**2*(2*a**2*d**9 - 4*a*b*c*d**8 + 2*b**2*c**2*d**7) + x*(4*a **2*c*d**8 - 8*a*b*c**2*d**7 + 4*b**2*c**3*d**6)) + x**2/(2*b*d**3)
Time = 0.20 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.80 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=-\frac {a^{5} \log \left (b x + a\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac {{\left (6 \, b^{2} c^{5} - 15 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} + \frac {7 \, b c^{6} - 9 \, a c^{5} d + 2 \, {\left (4 \, b c^{5} d - 5 \, a c^{4} d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d^{5} - 2 \, a b c^{3} d^{6} + a^{2} c^{2} d^{7} + {\left (b^{2} c^{2} d^{7} - 2 \, a b c d^{8} + a^{2} d^{9}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{6} - 2 \, a b c^{2} d^{7} + a^{2} c d^{8}\right )} x\right )}} + \frac {b d x^{2} - 2 \, {\left (3 \, b c + a d\right )} x}{2 \, b^{2} d^{4}} \]
-a^5*log(b*x + a)/(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 ) + (6*b^2*c^5 - 15*a*b*c^4*d + 10*a^2*c^3*d^2)*log(d*x + c)/(b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) + 1/2*(7*b*c^6 - 9*a*c^5*d + 2*(4*b*c^5*d - 5*a*c^4*d^2)*x)/(b^2*c^4*d^5 - 2*a*b*c^3*d^6 + a^2*c^2*d^7 + (b^2*c^2*d^7 - 2*a*b*c*d^8 + a^2*d^9)*x^2 + 2*(b^2*c^3*d^6 - 2*a*b*c^2*d ^7 + a^2*c*d^8)*x) + 1/2*(b*d*x^2 - 2*(3*b*c + a*d)*x)/(b^2*d^4)
Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.56 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=-\frac {a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac {{\left (6 \, b^{2} c^{5} - 15 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} + \frac {b d^{3} x^{2} - 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2} d^{6}} + \frac {7 \, b^{2} c^{7} - 16 \, a b c^{6} d + 9 \, a^{2} c^{5} d^{2} + 2 \, {\left (4 \, b^{2} c^{6} d - 9 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d^{5}} \]
-a^5*log(abs(b*x + a))/(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) + (6*b^2*c^5 - 15*a*b*c^4*d + 10*a^2*c^3*d^2)*log(abs(d*x + c))/(b^ 3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) + 1/2*(b*d^3*x^2 - 6*b*c*d^2*x - 2*a*d^3*x)/(b^2*d^6) + 1/2*(7*b^2*c^7 - 16*a*b*c^6*d + 9*a^2 *c^5*d^2 + 2*(4*b^2*c^6*d - 9*a*b*c^5*d^2 + 5*a^2*c^4*d^3)*x)/((b*c - a*d) ^3*(d*x + c)^2*d^5)
Time = 0.68 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.82 \[ \int \frac {x^5}{(a+b x) (c+d x)^3} \, dx=\frac {\frac {x\,\left (4\,b^3\,c^5-5\,a\,b^2\,c^4\,d\right )}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {7\,b^3\,c^6-9\,a\,b^2\,c^5\,d}{2\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b^2\,c^2\,d^4+2\,b^2\,c\,d^5\,x+b^2\,d^6\,x^2}-\frac {a^5\,\ln \left (a+b\,x\right )}{-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}+\frac {x^2}{2\,b\,d^3}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^2-15\,a\,b\,c^4\,d+6\,b^2\,c^5\right )}{a^3\,d^8-3\,a^2\,b\,c\,d^7+3\,a\,b^2\,c^2\,d^6-b^3\,c^3\,d^5}-\frac {x\,\left (a\,d^3+3\,b\,c\,d^2\right )}{b^2\,d^6} \]
((x*(4*b^3*c^5 - 5*a*b^2*c^4*d))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + (7*b^3* c^6 - 9*a*b^2*c^5*d)/(2*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(b^2*c^2*d^4 + b^2*d^6*x^2 + 2*b^2*c*d^5*x) - (a^5*log(a + b*x))/(b^6*c^3 - a^3*b^3*d^3 + 3*a^2*b^4*c*d^2 - 3*a*b^5*c^2*d) + x^2/(2*b*d^3) - (log(c + d*x)*(6*b^2* c^5 + 10*a^2*c^3*d^2 - 15*a*b*c^4*d))/(a^3*d^8 - b^3*c^3*d^5 + 3*a*b^2*c^2 *d^6 - 3*a^2*b*c*d^7) - (x*(a*d^3 + 3*b*c*d^2))/(b^2*d^6)