Integrand size = 18, antiderivative size = 164 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^4}{b^2 (b c-a d)^3 (a+b x)}-\frac {c^4}{2 d^3 (b c-a d)^2 (c+d x)^2}+\frac {2 c^3 (b c-2 a d)}{d^3 (b c-a d)^3 (c+d x)}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^4} \]
-a^4/b^2/(-a*d+b*c)^3/(b*x+a)-1/2*c^4/d^3/(-a*d+b*c)^2/(d*x+c)^2+2*c^3*(-2 *a*d+b*c)/d^3/(-a*d+b*c)^3/(d*x+c)-a^3*(-a*d+4*b*c)*ln(b*x+a)/b^2/(-a*d+b* c)^4+c^2*(6*a^2*d^2-4*a*b*c*d+b^2*c^2)*ln(d*x+c)/d^3/(-a*d+b*c)^4
Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^4}{b^2 (b c-a d)^3 (a+b x)}-\frac {c^4}{2 d^3 (b c-a d)^2 (c+d x)^2}-\frac {2 c^3 (b c-2 a d)}{d^3 (-b c+a d)^3 (c+d x)}+\frac {a^3 (-4 b c+a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^4} \]
-(a^4/(b^2*(b*c - a*d)^3*(a + b*x))) - c^4/(2*d^3*(b*c - a*d)^2*(c + d*x)^ 2) - (2*c^3*(b*c - 2*a*d))/(d^3*(-(b*c) + a*d)^3*(c + d*x)) + (a^3*(-4*b*c + a*d)*Log[a + b*x])/(b^2*(b*c - a*d)^4) + (c^2*(b^2*c^2 - 4*a*b*c*d + 6* a^2*d^2)*Log[c + d*x])/(d^3*(b*c - a*d)^4)
Time = 0.38 (sec) , antiderivative size = 164, 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.
| \(\displaystyle \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^4}{b (a+b x)^2 (b c-a d)^3}+\frac {a^3 (a d-4 b c)}{b (a+b x) (b c-a d)^4}+\frac {c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right )}{d^2 (c+d x) (a d-b c)^4}+\frac {c^4}{d^2 (c+d x)^3 (a d-b c)^2}+\frac {2 c^3 (b c-2 a d)}{d^2 (c+d x)^2 (a d-b c)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4}{b^2 (a+b x) (b c-a d)^3}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac {c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}+\frac {2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3}\) |
-(a^4/(b^2*(b*c - a*d)^3*(a + b*x))) - c^4/(2*d^3*(b*c - a*d)^2*(c + d*x)^ 2) + (2*c^3*(b*c - 2*a*d))/(d^3*(b*c - a*d)^3*(c + d*x)) - (a^3*(4*b*c - a *d)*Log[a + b*x])/(b^2*(b*c - a*d)^4) + (c^2*(b^2*c^2 - 4*a*b*c*d + 6*a^2* d^2)*Log[c + d*x])/(d^3*(b*c - a*d)^4)
3.3.93.3.1 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 = 161, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {c^{4}}{2 d^{3} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {c^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{3}}+\frac {2 c^{3} \left (2 a d -b c \right )}{d^{3} \left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {a^{3} \left (a d -4 b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{2}}+\frac {a^{4}}{b^{2} \left (a d -b c \right )^{3} \left (b x +a \right )}\) | \(161\) |
| norman | \(\frac {\frac {\left (a^{4} d^{4}+4 a \,b^{3} c^{3} d -2 b^{4} c^{4}\right ) x^{2}}{d^{2} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (4 a^{4} d^{4}+8 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (2 a^{3} d^{3}+7 a \,b^{2} c^{2} d -3 b^{3} c^{3}\right )}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {a^{3} \left (a d -4 b c \right ) \ln \left (b x +a \right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2}}+\frac {c^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(416\) |
| risch | \(\frac {\frac {\left (a^{4} d^{4}+4 a \,b^{3} c^{3} d -2 b^{4} c^{4}\right ) x^{2}}{d^{2} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (4 a^{4} d^{4}+8 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (2 a^{3} d^{3}+7 a \,b^{2} c^{2} d -3 b^{3} c^{3}\right )}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {a^{4} \ln \left (-b x -a \right ) d}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2}}-\frac {4 a^{3} \ln \left (-b x -a \right ) c}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b}+\frac {6 c^{2} \ln \left (d x +c \right ) a^{2}}{d \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {4 c^{3} \ln \left (d x +c \right ) a b}{d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {c^{4} \ln \left (d x +c \right ) b^{2}}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(597\) |
| parallelrisch | \(\frac {2 \ln \left (b x +a \right ) x^{2} a^{5} d^{6}+2 \ln \left (d x +c \right ) x \,b^{5} c^{6}+2 \ln \left (b x +a \right ) a^{5} c^{2} d^{4}+2 \ln \left (d x +c \right ) a \,b^{4} c^{6}+4 x^{2} b^{5} c^{5} d +4 x \,a^{5} c \,d^{5}-2 a^{4} b \,c^{3} d^{3}+7 a^{3} b^{2} c^{4} d^{2}-10 a^{2} b^{3} c^{5} d +3 x \,b^{5} c^{6}+2 a^{5} c^{2} d^{4}+3 a \,b^{4} c^{6}+2 x^{2} a^{5} d^{6}-2 x^{2} a^{4} b c \,d^{5}+8 x^{2} a^{2} b^{3} c^{3} d^{3}-12 x^{2} a \,b^{4} c^{4} d^{2}-4 x \,a^{4} b \,c^{2} d^{4}+8 x \,a^{3} b^{2} c^{3} d^{3}-5 x \,a^{2} b^{3} c^{4} d^{2}-6 x a \,b^{4} c^{5} d +2 \ln \left (b x +a \right ) x^{3} a^{4} b \,d^{6}+2 \ln \left (d x +c \right ) x^{3} b^{5} c^{4} d^{2}+4 \ln \left (d x +c \right ) x^{2} b^{5} c^{5} d +4 \ln \left (b x +a \right ) x \,a^{5} c \,d^{5}-8 \ln \left (b x +a \right ) a^{4} b \,c^{3} d^{3}+12 \ln \left (d x +c \right ) a^{3} b^{2} c^{4} d^{2}-8 \ln \left (d x +c \right ) a^{2} b^{3} c^{5} d -8 \ln \left (b x +a \right ) x^{3} a^{3} b^{2} c \,d^{5}+12 \ln \left (d x +c \right ) x^{3} a^{2} b^{3} c^{2} d^{4}-8 \ln \left (d x +c \right ) x^{3} a \,b^{4} c^{3} d^{3}-4 \ln \left (b x +a \right ) x^{2} a^{4} b c \,d^{5}-16 \ln \left (b x +a \right ) x^{2} a^{3} b^{2} c^{2} d^{4}+12 \ln \left (d x +c \right ) x^{2} a^{3} b^{2} c^{2} d^{4}+16 \ln \left (d x +c \right ) x^{2} a^{2} b^{3} c^{3} d^{3}-14 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{4} d^{2}-14 \ln \left (b x +a \right ) x \,a^{4} b \,c^{2} d^{4}-8 \ln \left (b x +a \right ) x \,a^{3} b^{2} c^{3} d^{3}+24 \ln \left (d x +c \right ) x \,a^{3} b^{2} c^{3} d^{3}-4 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{4} d^{2}-4 \ln \left (d x +c \right ) x a \,b^{4} c^{5} d}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right )^{2} \left (b x +a \right ) b^{2} d^{3}}\) | \(739\) |
-1/2*c^4/d^3/(a*d-b*c)^2/(d*x+c)^2+c^2*(6*a^2*d^2-4*a*b*c*d+b^2*c^2)/(a*d- b*c)^4/d^3*ln(d*x+c)+2*c^3*(2*a*d-b*c)/d^3/(a*d-b*c)^3/(d*x+c)+a^3*(a*d-4* b*c)/(a*d-b*c)^4/b^2*ln(b*x+a)+a^4/b^2/(a*d-b*c)^3/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (162) = 324\).
Time = 0.26 (sec) , antiderivative size = 797, normalized size of antiderivative = 4.86 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 \, a b^{4} c^{6} - 10 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 2 \, a^{4} b c^{3} d^{3} + 2 \, a^{5} c^{2} d^{4} + 2 \, {\left (2 \, b^{5} c^{5} d - 6 \, a b^{4} c^{4} d^{2} + 4 \, a^{2} b^{3} c^{3} d^{3} - a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (3 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 5 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 4 \, a^{4} b c^{2} d^{4} + 4 \, a^{5} c d^{5}\right )} x - 2 \, {\left (4 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4} + {\left (4 \, a^{3} b^{2} c d^{5} - a^{4} b d^{6}\right )} x^{3} + {\left (8 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - a^{5} d^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{2} c^{3} d^{3} + 7 \, a^{4} b c^{2} d^{4} - 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} + 6 \, a^{3} b^{2} c^{2} d^{4}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 12 \, a^{3} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{6} c^{6} d^{3} - 4 \, a^{2} b^{5} c^{5} d^{4} + 6 \, a^{3} b^{4} c^{4} d^{5} - 4 \, a^{4} b^{3} c^{3} d^{6} + a^{5} b^{2} c^{2} d^{7} + {\left (b^{7} c^{4} d^{5} - 4 \, a b^{6} c^{3} d^{6} + 6 \, a^{2} b^{5} c^{2} d^{7} - 4 \, a^{3} b^{4} c d^{8} + a^{4} b^{3} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{5} d^{4} - 7 \, a b^{6} c^{4} d^{5} + 8 \, a^{2} b^{5} c^{3} d^{6} - 2 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8} + a^{5} b^{2} d^{9}\right )} x^{2} + {\left (b^{7} c^{6} d^{3} - 2 \, a b^{6} c^{5} d^{4} - 2 \, a^{2} b^{5} c^{4} d^{5} + 8 \, a^{3} b^{4} c^{3} d^{6} - 7 \, a^{4} b^{3} c^{2} d^{7} + 2 \, a^{5} b^{2} c d^{8}\right )} x\right )}} \]
1/2*(3*a*b^4*c^6 - 10*a^2*b^3*c^5*d + 7*a^3*b^2*c^4*d^2 - 2*a^4*b*c^3*d^3 + 2*a^5*c^2*d^4 + 2*(2*b^5*c^5*d - 6*a*b^4*c^4*d^2 + 4*a^2*b^3*c^3*d^3 - a ^4*b*c*d^5 + a^5*d^6)*x^2 + (3*b^5*c^6 - 6*a*b^4*c^5*d - 5*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 4*a^4*b*c^2*d^4 + 4*a^5*c*d^5)*x - 2*(4*a^4*b*c^3*d ^3 - a^5*c^2*d^4 + (4*a^3*b^2*c*d^5 - a^4*b*d^6)*x^3 + (8*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - a^5*d^6)*x^2 + (4*a^3*b^2*c^3*d^3 + 7*a^4*b*c^2*d^4 - 2* a^5*c*d^5)*x)*log(b*x + a) + 2*(a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^ 4*d^2 + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4)*x^3 + (2*b^5*c ^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 + 6*a^3*b^2*c^2*d^4)*x^2 + (b^5 *c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 12*a^3*b^2*c^3*d^3)*x)*log(d*x + c))/(a*b^6*c^6*d^3 - 4*a^2*b^5*c^5*d^4 + 6*a^3*b^4*c^4*d^5 - 4*a^4*b^3*c ^3*d^6 + a^5*b^2*c^2*d^7 + (b^7*c^4*d^5 - 4*a*b^6*c^3*d^6 + 6*a^2*b^5*c^2* d^7 - 4*a^3*b^4*c*d^8 + a^4*b^3*d^9)*x^3 + (2*b^7*c^5*d^4 - 7*a*b^6*c^4*d^ 5 + 8*a^2*b^5*c^3*d^6 - 2*a^3*b^4*c^2*d^7 - 2*a^4*b^3*c*d^8 + a^5*b^2*d^9) *x^2 + (b^7*c^6*d^3 - 2*a*b^6*c^5*d^4 - 2*a^2*b^5*c^4*d^5 + 8*a^3*b^4*c^3* d^6 - 7*a^4*b^3*c^2*d^7 + 2*a^5*b^2*c*d^8)*x)
Timed out. \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (162) = 324\).
Time = 0.23 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.16 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {{\left (4 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}} + \frac {{\left (b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}} + \frac {3 \, a b^{3} c^{5} - 7 \, a^{2} b^{2} c^{4} d - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} - a^{4} d^{5}\right )} x^{2} + {\left (3 \, b^{4} c^{5} - 3 \, a b^{3} c^{4} d - 8 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{4} c d^{4}\right )} x}{2 \, {\left (a b^{5} c^{5} d^{3} - 3 \, a^{2} b^{4} c^{4} d^{4} + 3 \, a^{3} b^{3} c^{3} d^{5} - a^{4} b^{2} c^{2} d^{6} + {\left (b^{6} c^{3} d^{5} - 3 \, a b^{5} c^{2} d^{6} + 3 \, a^{2} b^{4} c d^{7} - a^{3} b^{3} d^{8}\right )} x^{3} + {\left (2 \, b^{6} c^{4} d^{4} - 5 \, a b^{5} c^{3} d^{5} + 3 \, a^{2} b^{4} c^{2} d^{6} + a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )} x^{2} + {\left (b^{6} c^{5} d^{3} - a b^{5} c^{4} d^{4} - 3 \, a^{2} b^{4} c^{3} d^{5} + 5 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7}\right )} x\right )}} \]
-(4*a^3*b*c - a^4*d)*log(b*x + a)/(b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2 *d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4) + (b^2*c^4 - 4*a*b*c^3*d + 6*a^2*c^2 *d^2)*log(d*x + c)/(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4* a^3*b*c*d^6 + a^4*d^7) + 1/2*(3*a*b^3*c^5 - 7*a^2*b^2*c^4*d - 2*a^4*c^2*d^ 3 + 2*(2*b^4*c^4*d - 4*a*b^3*c^3*d^2 - a^4*d^5)*x^2 + (3*b^4*c^5 - 3*a*b^3 *c^4*d - 8*a^2*b^2*c^3*d^2 - 4*a^4*c*d^4)*x)/(a*b^5*c^5*d^3 - 3*a^2*b^4*c^ 4*d^4 + 3*a^3*b^3*c^3*d^5 - a^4*b^2*c^2*d^6 + (b^6*c^3*d^5 - 3*a*b^5*c^2*d ^6 + 3*a^2*b^4*c*d^7 - a^3*b^3*d^8)*x^3 + (2*b^6*c^4*d^4 - 5*a*b^5*c^3*d^5 + 3*a^2*b^4*c^2*d^6 + a^3*b^3*c*d^7 - a^4*b^2*d^8)*x^2 + (b^6*c^5*d^3 - a *b^5*c^4*d^4 - 3*a^2*b^4*c^3*d^5 + 5*a^3*b^3*c^2*d^6 - 2*a^4*b^2*c*d^7)*x)
Time = 0.30 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.89 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^{4} b^{3}}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} {\left (b x + a\right )}} + \frac {{\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 6 \, a^{2} b c^{2} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}} - \frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2} d^{3}} - \frac {3 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + \frac {2 \, {\left (b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 4 \, a^{2} b^{2} c^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{3}} \]
-a^4*b^3/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*(b*x + a)) + (b^3*c^4 - 4*a*b^2*c^3*d + 6*a^2*b*c^2*d^2)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4*d^3 - 4*a*b^4*c^3*d^4 + 6*a^2*b^3*c^2*d^5 - 4*a^3*b^2*c*d^6 + a^4*b*d^7) - log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^2 *d^3) - 1/2*(3*b^2*c^4*d^2 - 8*a*b*c^3*d^3 + 2*(b^4*c^5*d - 5*a*b^3*c^4*d^ 2 + 4*a^2*b^2*c^3*d^3)/((b*x + a)*b))/((b*c - a*d)^4*(b*c/(b*x + a) - a*d/ (b*x + a) + d)^2*d^3)
Time = 0.80 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.85 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {x^2\,\left (a^4\,d^4+4\,a\,b^3\,c^3\,d-2\,b^4\,c^4\right )}{b^2\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^3\,d^3+7\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{2\,b^2\,d^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^4\,d^4+8\,a^2\,b^2\,c^2\,d^2+3\,a\,b^3\,c^3\,d-3\,b^4\,c^4\right )}{2\,b^2\,d^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}+\frac {a^4\,d\,\ln \left (a+b\,x\right )}{a^4\,b^2\,d^4-4\,a^3\,b^3\,c\,d^3+6\,a^2\,b^4\,c^2\,d^2-4\,a\,b^5\,c^3\,d+b^6\,c^4}+\frac {c^2\,\ln \left (c+d\,x\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{d^3\,{\left (a\,d-b\,c\right )}^4}-\frac {4\,a^3\,b\,c\,\ln \left (a+b\,x\right )}{a^4\,b^2\,d^4-4\,a^3\,b^3\,c\,d^3+6\,a^2\,b^4\,c^2\,d^2-4\,a\,b^5\,c^3\,d+b^6\,c^4} \]
((x^2*(a^4*d^4 - 2*b^4*c^4 + 4*a*b^3*c^3*d))/(b^2*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^3*d^3 - 3*b^3*c^3 + 7*a*b^2 *c^2*d))/(2*b^2*d^3*(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^4*d^4 - 3*b^4*c^4 + 8*a^2*b^2*c^2*d^2 + 3*a*b^3*c^3*d))/(2*b^2* d^3*(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) + (a^4*d*log(a + b*x))/( b^6*c^4 + a^4*b^2*d^4 - 4*a^3*b^3*c*d^3 + 6*a^2*b^4*c^2*d^2 - 4*a*b^5*c^3* d) + (c^2*log(c + d*x)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(d^3*(a*d - b*c) ^4) - (4*a^3*b*c*log(a + b*x))/(b^6*c^4 + a^4*b^2*d^4 - 4*a^3*b^3*c*d^3 + 6*a^2*b^4*c^2*d^2 - 4*a*b^5*c^3*d)