Integrand size = 29, antiderivative size = 130 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {1}{4 (b c-a d) (a+b x)^4}+\frac {d}{3 (b c-a d)^2 (a+b x)^3}-\frac {d^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5} \]
-1/4/(-a*d+b*c)/(b*x+a)^4+1/3*d/(-a*d+b*c)^2/(b*x+a)^3-1/2*d^2/(-a*d+b*c)^ 3/(b*x+a)^2+d^3/(-a*d+b*c)^4/(b*x+a)+d^4*ln(b*x+a)/(-a*d+b*c)^5-d^4*ln(d*x +c)/(-a*d+b*c)^5
Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {1}{4 (-b c+a d) (a+b x)^4}+\frac {d}{3 (b c-a d)^2 (a+b x)^3}-\frac {d^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5} \]
1/(4*(-(b*c) + a*d)*(a + b*x)^4) + d/(3*(b*c - a*d)^2*(a + b*x)^3) - d^2/( 2*(b*c - a*d)^3*(a + b*x)^2) + d^3/((b*c - a*d)^4*(a + b*x)) + (d^4*Log[a + b*x])/(b*c - a*d)^5 - (d^4*Log[c + d*x])/(b*c - a*d)^5
Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1121, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^4 \left (x (a d+b c)+a c+b d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (-\frac {d^5}{(c+d x) (b c-a d)^5}+\frac {b d^4}{(a+b x) (b c-a d)^5}-\frac {b d^3}{(a+b x)^2 (b c-a d)^4}+\frac {b d^2}{(a+b x)^3 (b c-a d)^3}-\frac {b d}{(a+b x)^4 (b c-a d)^2}+\frac {b}{(a+b x)^5 (b c-a d)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}+\frac {d^3}{(a+b x) (b c-a d)^4}-\frac {d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {d}{3 (a+b x)^3 (b c-a d)^2}-\frac {1}{4 (a+b x)^4 (b c-a d)}\) |
-1/4*1/((b*c - a*d)*(a + b*x)^4) + d/(3*(b*c - a*d)^2*(a + b*x)^3) - d^2/( 2*(b*c - a*d)^3*(a + b*x)^2) + d^3/((b*c - a*d)^4*(a + b*x)) + (d^4*Log[a + b*x])/(b*c - a*d)^5 - (d^4*Log[c + d*x])/(b*c - a*d)^5
3.19.9.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 2.57 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {1}{4 \left (a d -b c \right ) \left (b x +a \right )^{4}}+\frac {d}{3 \left (a d -b c \right )^{2} \left (b x +a \right )^{3}}+\frac {d^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {d^{3}}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {d^{4} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}\) | \(125\) |
| parallelrisch | \(-\frac {-3 b^{8} c^{4}-25 a^{4} b^{4} d^{4}+48 a^{3} b^{5} c \,d^{3}-36 a^{2} b^{6} c^{2} d^{2}+16 a \,b^{7} c^{3} d -12 x^{3} a \,b^{7} d^{4}+12 x^{3} b^{8} c \,d^{3}-42 x^{2} a^{2} b^{6} d^{4}-6 x^{2} b^{8} c^{2} d^{2}-52 x \,a^{3} b^{5} d^{4}+4 x \,b^{8} c^{3} d +12 \ln \left (b x +a \right ) x^{4} b^{8} d^{4}-12 \ln \left (d x +c \right ) x^{4} b^{8} d^{4}+12 \ln \left (b x +a \right ) a^{4} b^{4} d^{4}-12 \ln \left (d x +c \right ) a^{4} b^{4} d^{4}+48 x^{2} a \,b^{7} c \,d^{3}+72 x \,a^{2} b^{6} c \,d^{3}-24 x a \,b^{7} c^{2} d^{2}+48 \ln \left (b x +a \right ) x^{3} a \,b^{7} d^{4}-48 \ln \left (d x +c \right ) x^{3} a \,b^{7} d^{4}+72 \ln \left (b x +a \right ) x^{2} a^{2} b^{6} d^{4}-72 \ln \left (d x +c \right ) x^{2} a^{2} b^{6} d^{4}+48 \ln \left (b x +a \right ) x \,a^{3} b^{5} d^{4}-48 \ln \left (d x +c \right ) x \,a^{3} b^{5} d^{4}}{12 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \left (b x +a \right )^{4} b^{4}}\) | \(428\) |
| risch | \(\frac {\frac {b^{3} d^{3} x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {\left (7 a d -b c \right ) b^{2} d^{2} x^{2}}{2 a^{4} d^{4}-8 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-8 a \,b^{3} c^{3} d +2 b^{4} c^{4}}+\frac {b d \left (13 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) x}{3 a^{4} d^{4}-12 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-12 a \,b^{3} c^{3} d +3 b^{4} c^{4}}+\frac {25 a^{3} d^{3}-23 a^{2} b c \,d^{2}+13 a \,b^{2} c^{2} d -3 b^{3} c^{3}}{12 a^{4} d^{4}-48 a^{3} b c \,d^{3}+72 a^{2} b^{2} c^{2} d^{2}-48 a \,b^{3} c^{3} d +12 b^{4} c^{4}}}{\left (b x +a \right )^{4}}+\frac {d^{4} \ln \left (-d x -c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {d^{4} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(467\) |
| norman | \(\frac {\frac {b^{3} d^{3} x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {25 a^{3} b^{4} d^{3}-23 a^{2} b^{5} c \,d^{2}+13 a \,c^{2} d \,b^{6}-3 c^{3} b^{7}}{12 b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (7 a \,b^{4} d^{3}-b^{5} d^{2} c \right ) x^{2}}{2 b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (13 d^{3} a^{2} b^{4}-5 c \,d^{2} a \,b^{5}+c^{2} d \,b^{6}\right ) x}{3 b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{4}}+\frac {d^{4} \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {d^{4} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(488\) |
d^4/(a*d-b*c)^5*ln(d*x+c)+1/4/(a*d-b*c)/(b*x+a)^4+1/3*d/(a*d-b*c)^2/(b*x+a )^3+1/2*d^2/(a*d-b*c)^3/(b*x+a)^2+d^3/(a*d-b*c)^4/(b*x+a)-d^4/(a*d-b*c)^5* ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (124) = 248\).
Time = 0.30 (sec) , antiderivative size = 657, normalized size of antiderivative = 5.05 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x - 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \, {\left (a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5} + {\left (b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}\right )} x^{4} + 4 \, {\left (a b^{8} c^{5} - 5 \, a^{2} b^{7} c^{4} d + 10 \, a^{3} b^{6} c^{3} d^{2} - 10 \, a^{4} b^{5} c^{2} d^{3} + 5 \, a^{5} b^{4} c d^{4} - a^{6} b^{3} d^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} c^{5} - 5 \, a^{3} b^{6} c^{4} d + 10 \, a^{4} b^{5} c^{3} d^{2} - 10 \, a^{5} b^{4} c^{2} d^{3} + 5 \, a^{6} b^{3} c d^{4} - a^{7} b^{2} d^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} c^{5} - 5 \, a^{4} b^{5} c^{4} d + 10 \, a^{5} b^{4} c^{3} d^{2} - 10 \, a^{6} b^{3} c^{2} d^{3} + 5 \, a^{7} b^{2} c d^{4} - a^{8} b d^{5}\right )} x\right )}} \]
-1/12*(3*b^4*c^4 - 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 - 48*a^3*b*c*d^3 + 25*a^4*d^4 - 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 8*a*b^3*c*d ^3 + 7*a^2*b^2*d^4)*x^2 - 4*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 18*a^2*b^2*c*d^ 3 - 13*a^3*b*d^4)*x - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^ 2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(b^4*d^4*x^4 + 4*a*b^3*d^4* x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c))/(a^4*b^5* c^5 - 5*a^5*b^4*c^4*d + 10*a^6*b^3*c^3*d^2 - 10*a^7*b^2*c^2*d^3 + 5*a^8*b* c*d^4 - a^9*d^5 + (b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b ^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*x^4 + 4*(a*b^8*c^5 - 5*a^2*b^7 *c^4*d + 10*a^3*b^6*c^3*d^2 - 10*a^4*b^5*c^2*d^3 + 5*a^5*b^4*c*d^4 - a^6*b ^3*d^5)*x^3 + 6*(a^2*b^7*c^5 - 5*a^3*b^6*c^4*d + 10*a^4*b^5*c^3*d^2 - 10*a ^5*b^4*c^2*d^3 + 5*a^6*b^3*c*d^4 - a^7*b^2*d^5)*x^2 + 4*(a^3*b^6*c^5 - 5*a ^4*b^5*c^4*d + 10*a^5*b^4*c^3*d^2 - 10*a^6*b^3*c^2*d^3 + 5*a^7*b^2*c*d^4 - a^8*b*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (109) = 218\).
Time = 1.54 (sec) , antiderivative size = 802, normalized size of antiderivative = 6.17 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {d^{4} \log {\left (x + \frac {- \frac {a^{6} d^{10}}{\left (a d - b c\right )^{5}} + \frac {6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} - \frac {15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} + \frac {20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} - \frac {15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} + \frac {6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} - \frac {b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} - \frac {d^{4} \log {\left (x + \frac {\frac {a^{6} d^{10}}{\left (a d - b c\right )^{5}} - \frac {6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} + \frac {15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} - \frac {20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} + \frac {15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} - \frac {6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} + \frac {b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} + \frac {25 a^{3} d^{3} - 23 a^{2} b c d^{2} + 13 a b^{2} c^{2} d - 3 b^{3} c^{3} + 12 b^{3} d^{3} x^{3} + x^{2} \cdot \left (42 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (52 a^{2} b d^{3} - 20 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{12 a^{8} d^{4} - 48 a^{7} b c d^{3} + 72 a^{6} b^{2} c^{2} d^{2} - 48 a^{5} b^{3} c^{3} d + 12 a^{4} b^{4} c^{4} + x^{4} \cdot \left (12 a^{4} b^{4} d^{4} - 48 a^{3} b^{5} c d^{3} + 72 a^{2} b^{6} c^{2} d^{2} - 48 a b^{7} c^{3} d + 12 b^{8} c^{4}\right ) + x^{3} \cdot \left (48 a^{5} b^{3} d^{4} - 192 a^{4} b^{4} c d^{3} + 288 a^{3} b^{5} c^{2} d^{2} - 192 a^{2} b^{6} c^{3} d + 48 a b^{7} c^{4}\right ) + x^{2} \cdot \left (72 a^{6} b^{2} d^{4} - 288 a^{5} b^{3} c d^{3} + 432 a^{4} b^{4} c^{2} d^{2} - 288 a^{3} b^{5} c^{3} d + 72 a^{2} b^{6} c^{4}\right ) + x \left (48 a^{7} b d^{4} - 192 a^{6} b^{2} c d^{3} + 288 a^{5} b^{3} c^{2} d^{2} - 192 a^{4} b^{4} c^{3} d + 48 a^{3} b^{5} c^{4}\right )} \]
d**4*log(x + (-a**6*d**10/(a*d - b*c)**5 + 6*a**5*b*c*d**9/(a*d - b*c)**5 - 15*a**4*b**2*c**2*d**8/(a*d - b*c)**5 + 20*a**3*b**3*c**3*d**7/(a*d - b* c)**5 - 15*a**2*b**4*c**4*d**6/(a*d - b*c)**5 + 6*a*b**5*c**5*d**5/(a*d - b*c)**5 + a*d**5 - b**6*c**6*d**4/(a*d - b*c)**5 + b*c*d**4)/(2*b*d**5))/( a*d - b*c)**5 - d**4*log(x + (a**6*d**10/(a*d - b*c)**5 - 6*a**5*b*c*d**9/ (a*d - b*c)**5 + 15*a**4*b**2*c**2*d**8/(a*d - b*c)**5 - 20*a**3*b**3*c**3 *d**7/(a*d - b*c)**5 + 15*a**2*b**4*c**4*d**6/(a*d - b*c)**5 - 6*a*b**5*c* *5*d**5/(a*d - b*c)**5 + a*d**5 + b**6*c**6*d**4/(a*d - b*c)**5 + b*c*d**4 )/(2*b*d**5))/(a*d - b*c)**5 + (25*a**3*d**3 - 23*a**2*b*c*d**2 + 13*a*b** 2*c**2*d - 3*b**3*c**3 + 12*b**3*d**3*x**3 + x**2*(42*a*b**2*d**3 - 6*b**3 *c*d**2) + x*(52*a**2*b*d**3 - 20*a*b**2*c*d**2 + 4*b**3*c**2*d))/(12*a**8 *d**4 - 48*a**7*b*c*d**3 + 72*a**6*b**2*c**2*d**2 - 48*a**5*b**3*c**3*d + 12*a**4*b**4*c**4 + x**4*(12*a**4*b**4*d**4 - 48*a**3*b**5*c*d**3 + 72*a** 2*b**6*c**2*d**2 - 48*a*b**7*c**3*d + 12*b**8*c**4) + x**3*(48*a**5*b**3*d **4 - 192*a**4*b**4*c*d**3 + 288*a**3*b**5*c**2*d**2 - 192*a**2*b**6*c**3* d + 48*a*b**7*c**4) + x**2*(72*a**6*b**2*d**4 - 288*a**5*b**3*c*d**3 + 432 *a**4*b**4*c**2*d**2 - 288*a**3*b**5*c**3*d + 72*a**2*b**6*c**4) + x*(48*a **7*b*d**4 - 192*a**6*b**2*c*d**3 + 288*a**5*b**3*c**2*d**2 - 192*a**4*b** 4*c**3*d + 48*a**3*b**5*c**4))
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (124) = 248\).
Time = 0.22 (sec) , antiderivative size = 558, normalized size of antiderivative = 4.29 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {d^{4} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {d^{4} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{12 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]
d^4*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) - d^4*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 ) + 1/12*(12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 2 5*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4 *a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4* a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^4 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3* b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4 - 4*a^3* b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^ 3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b* d^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (124) = 248\).
Time = 0.28 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.60 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {b d^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {d^{5} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x}{12 \, {\left (b c - a d\right )}^{5} {\left (b x + a\right )}^{4}} \]
b*d^4*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) - d^5*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) - 1/12*(3*b^4*c^4 - 16*a*b^3*c^3*d + 36*a^2*b^2*c^2* d^2 - 48*a^3*b*c*d^3 + 25*a^4*d^4 - 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(b^ 4*c^2*d^2 - 8*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*x^2 - 4*(b^4*c^3*d - 6*a*b^3*c^ 2*d^2 + 18*a^2*b^2*c*d^3 - 13*a^3*b*d^4)*x)/((b*c - a*d)^5*(b*x + a)^4)
Time = 10.19 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.88 \[ \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {\frac {25\,a^3\,d^3-23\,a^2\,b\,c\,d^2+13\,a\,b^2\,c^2\,d-3\,b^3\,c^3}{12\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {d^2\,x^2\,\left (b^3\,c-7\,a\,b^2\,d\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {d\,x\,\left (13\,a^2\,b\,d^2-5\,a\,b^2\,c\,d+b^3\,c^2\right )}{3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b^3\,d^3\,x^3}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4}-\frac {2\,d^4\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \]
((25*a^3*d^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2)/(12*(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)) - (d^2*x^2* (b^3*c - 7*a*b^2*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)) + (d*x*(b^3*c^2 + 13*a^2*b*d^2 - 5*a*b^2*c*d))/(3*( 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^3*d^3*x^3)/(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^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x) - ( 2*d^4*atanh((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*d - b*c)^5