Integrand size = 24, antiderivative size = 245 \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\frac {B d-A e}{3 d (c d-b e) (d+e x)^3}+\frac {B c d^2-A e (2 c d-b e)}{2 d^2 (c d-b e)^2 (d+e x)^2}+\frac {B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 (d+e x)}+\frac {A \log (x)}{b d^4}+\frac {c^3 (b B-A c) \log (b+c x)}{b (c d-b e)^4}-\frac {\left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \]
1/3*(-A*e+B*d)/d/(-b*e+c*d)/(e*x+d)^3+1/2*(B*c*d^2-A*e*(-b*e+2*c*d))/d^2/( -b*e+c*d)^2/(e*x+d)^2+(B*c^2*d^3-A*e*(b^2*e^2-3*b*c*d*e+3*c^2*d^2))/d^3/(- b*e+c*d)^3/(e*x+d)+A*ln(x)/b/d^4+c^3*(-A*c+B*b)*ln(c*x+b)/b/(-b*e+c*d)^4-( B*c^3*d^4-A*e*(-b^3*e^3+4*b^2*c*d*e^2-6*b*c^2*d^2*e+4*c^3*d^3))*ln(e*x+d)/ d^4/(-b*e+c*d)^4
Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\frac {B d-A e}{3 d (c d-b e) (d+e x)^3}+\frac {B c d^2+A e (-2 c d+b e)}{2 d^2 (c d-b e)^2 (d+e x)^2}+\frac {B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 (d+e x)}+\frac {A \log (x)}{b d^4}+\frac {c^3 (b B-A c) \log (b+c x)}{b (c d-b e)^4}-\frac {\left (B c^3 d^4+A e \left (-4 c^3 d^3+6 b c^2 d^2 e-4 b^2 c d e^2+b^3 e^3\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \]
(B*d - A*e)/(3*d*(c*d - b*e)*(d + e*x)^3) + (B*c*d^2 + A*e*(-2*c*d + b*e)) /(2*d^2*(c*d - b*e)^2*(d + e*x)^2) + (B*c^2*d^3 - A*e*(3*c^2*d^2 - 3*b*c*d *e + b^2*e^2))/(d^3*(c*d - b*e)^3*(d + e*x)) + (A*Log[x])/(b*d^4) + (c^3*( b*B - A*c)*Log[b + c*x])/(b*(c*d - b*e)^4) - ((B*c^3*d^4 + A*e*(-4*c^3*d^3 + 6*b*c^2*d^2*e - 4*b^2*c*d*e^2 + b^3*e^3))*Log[d + e*x])/(d^4*(c*d - b*e )^4)
Time = 0.52 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1200, 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.
| \(\displaystyle \int \frac {A+B x}{\left (b x+c x^2\right ) (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {e \left (A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )-B c^2 d^3\right )}{d^3 (d+e x)^2 (c d-b e)^3}+\frac {e \left (A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )-B c^3 d^4\right )}{d^4 (d+e x) (c d-b e)^4}+\frac {c^4 (b B-A c)}{b (b+c x) (b e-c d)^4}+\frac {e \left (A e (2 c d-b e)-B c d^2\right )}{d^2 (d+e x)^3 (c d-b e)^2}-\frac {e (B d-A e)}{d (d+e x)^4 (c d-b e)}+\frac {A}{b d^4 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 (d+e x) (c d-b e)^3}-\frac {\log (d+e x) \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d^4 (c d-b e)^4}+\frac {c^3 (b B-A c) \log (b+c x)}{b (c d-b e)^4}+\frac {B c d^2-A e (2 c d-b e)}{2 d^2 (d+e x)^2 (c d-b e)^2}+\frac {B d-A e}{3 d (d+e x)^3 (c d-b e)}+\frac {A \log (x)}{b d^4}\) |
(B*d - A*e)/(3*d*(c*d - b*e)*(d + e*x)^3) + (B*c*d^2 - A*e*(2*c*d - b*e))/ (2*d^2*(c*d - b*e)^2*(d + e*x)^2) + (B*c^2*d^3 - A*e*(3*c^2*d^2 - 3*b*c*d* e + b^2*e^2))/(d^3*(c*d - b*e)^3*(d + e*x)) + (A*Log[x])/(b*d^4) + (c^3*(b *B - A*c)*Log[b + c*x])/(b*(c*d - b*e)^4) - ((B*c^3*d^4 - A*e*(4*c^3*d^3 - 6*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3))*Log[d + e*x])/(d^4*(c*d - b*e)^ 4)
3.12.45.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Time = 0.34 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(\frac {A \ln \left (x \right )}{b \,d^{4}}-\frac {\left (A c -B b \right ) c^{3} \ln \left (c x +b \right )}{b \left (b e -c d \right )^{4}}+\frac {A b \,e^{2}-2 A c d e +B c \,d^{2}}{2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{2}}+\frac {A \,b^{2} e^{3}-3 A b c d \,e^{2}+3 A \,c^{2} d^{2} e -B \,c^{2} d^{3}}{d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )}-\frac {\left (A \,b^{3} e^{4}-4 A \,b^{2} c d \,e^{3}+6 A b \,c^{2} d^{2} e^{2}-4 A \,c^{3} d^{3} e +B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{d^{4} \left (b e -c d \right )^{4}}+\frac {A e -B d}{3 d \left (b e -c d \right ) \left (e x +d \right )^{3}}\) | \(245\) |
| norman | \(\frac {-\frac {e^{3} \left (11 A \,b^{2} e^{3}-31 A b c d \,e^{2}+26 A \,c^{2} d^{2} e -2 B \,b^{2} d \,e^{2}+7 B b c \,d^{2} e -11 B \,c^{2} d^{3}\right ) x^{3}}{6 d^{4} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {\left (9 A \,b^{2} e^{3}-25 A b c d \,e^{2}+20 A \,c^{2} d^{2} e -2 B \,b^{2} d \,e^{2}+7 B b c \,d^{2} e -9 B \,c^{2} d^{3}\right ) e^{2} x^{2}}{2 d^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {\left (3 A \,b^{2} e^{3}-8 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+3 B b c \,d^{2} e -3 B \,c^{2} d^{3}\right ) e x}{d^{2} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}}{\left (e x +d \right )^{3}}+\frac {A \ln \left (x \right )}{b \,d^{4}}-\frac {\left (A \,b^{3} e^{4}-4 A \,b^{2} c d \,e^{3}+6 A b \,c^{2} d^{2} e^{2}-4 A \,c^{3} d^{3} e +B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{d^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {c^{3} \left (A c -B b \right ) \ln \left (c x +b \right )}{\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) b}\) | \(521\) |
| risch | \(\frac {\frac {e^{2} \left (A \,b^{2} e^{3}-3 A b c d \,e^{2}+3 A \,c^{2} d^{2} e -B \,c^{2} d^{3}\right ) x^{2}}{d^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {e \left (5 A \,b^{2} e^{3}-15 A b c d \,e^{2}+14 A \,c^{2} d^{2} e +B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) x}{2 d^{2} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {11 A \,b^{2} e^{3}-31 A b c d \,e^{2}+26 A \,c^{2} d^{2} e -2 B \,b^{2} d \,e^{2}+7 B b c \,d^{2} e -11 B \,c^{2} d^{3}}{6 d \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}}{\left (e x +d \right )^{3}}-\frac {c^{4} \ln \left (c x +b \right ) A}{\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) b}+\frac {c^{3} \ln \left (c x +b \right ) B}{b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}}+\frac {A \ln \left (-x \right )}{d^{4} b}-\frac {\ln \left (-e x -d \right ) A \,b^{3} e^{4}}{d^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}+\frac {4 \ln \left (-e x -d \right ) A \,b^{2} c \,e^{3}}{d^{3} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {6 \ln \left (-e x -d \right ) A b \,c^{2} e^{2}}{d^{2} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}+\frac {4 \ln \left (-e x -d \right ) A \,c^{3} e}{d \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {\ln \left (-e x -d \right ) B \,c^{3}}{b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}}\) | \(781\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1309\) |
A*ln(x)/b/d^4-(A*c-B*b)*c^3/b/(b*e-c*d)^4*ln(c*x+b)+1/2*(A*b*e^2-2*A*c*d*e +B*c*d^2)/d^2/(b*e-c*d)^2/(e*x+d)^2+(A*b^2*e^3-3*A*b*c*d*e^2+3*A*c^2*d^2*e -B*c^2*d^3)/d^3/(b*e-c*d)^3/(e*x+d)-(A*b^3*e^4-4*A*b^2*c*d*e^3+6*A*b*c^2*d ^2*e^2-4*A*c^3*d^3*e+B*c^3*d^4)/d^4/(b*e-c*d)^4*ln(e*x+d)+1/3*(A*e-B*d)/d/ (b*e-c*d)/(e*x+d)^3
Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (241) = 482\).
Time = 84.70 (sec) , antiderivative size = 1133, normalized size of antiderivative = 4.62 \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\frac {11 \, B b c^{3} d^{7} + 11 \, A b^{4} d^{3} e^{4} - 2 \, {\left (9 \, B b^{2} c^{2} + 13 \, A b c^{3}\right )} d^{6} e + 3 \, {\left (3 \, B b^{3} c + 19 \, A b^{2} c^{2}\right )} d^{5} e^{2} - 2 \, {\left (B b^{4} + 21 \, A b^{3} c\right )} d^{4} e^{3} + 6 \, {\left (B b c^{3} d^{5} e^{2} + 6 \, A b^{2} c^{2} d^{3} e^{4} - 4 \, A b^{3} c d^{2} e^{5} + A b^{4} d e^{6} - {\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} d^{4} e^{3}\right )} x^{2} + 3 \, {\left (5 \, B b c^{3} d^{6} e - 20 \, A b^{3} c d^{3} e^{4} + 5 \, A b^{4} d^{2} e^{5} - 2 \, {\left (3 \, B b^{2} c^{2} + 7 \, A b c^{3}\right )} d^{5} e^{2} + {\left (B b^{3} c + 29 \, A b^{2} c^{2}\right )} d^{4} e^{3}\right )} x + 6 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{4} e^{3} x^{3} + 3 \, {\left (B b c^{3} - A c^{4}\right )} d^{5} e^{2} x^{2} + 3 \, {\left (B b c^{3} - A c^{4}\right )} d^{6} e x + {\left (B b c^{3} - A c^{4}\right )} d^{7}\right )} \log \left (c x + b\right ) - 6 \, {\left (B b c^{3} d^{7} - 4 \, A b c^{3} d^{6} e + 6 \, A b^{2} c^{2} d^{5} e^{2} - 4 \, A b^{3} c d^{4} e^{3} + A b^{4} d^{3} e^{4} + {\left (B b c^{3} d^{4} e^{3} - 4 \, A b c^{3} d^{3} e^{4} + 6 \, A b^{2} c^{2} d^{2} e^{5} - 4 \, A b^{3} c d e^{6} + A b^{4} e^{7}\right )} x^{3} + 3 \, {\left (B b c^{3} d^{5} e^{2} - 4 \, A b c^{3} d^{4} e^{3} + 6 \, A b^{2} c^{2} d^{3} e^{4} - 4 \, A b^{3} c d^{2} e^{5} + A b^{4} d e^{6}\right )} x^{2} + 3 \, {\left (B b c^{3} d^{6} e - 4 \, A b c^{3} d^{5} e^{2} + 6 \, A b^{2} c^{2} d^{4} e^{3} - 4 \, A b^{3} c d^{3} e^{4} + A b^{4} d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right ) + 6 \, {\left (A c^{4} d^{7} - 4 \, A b c^{3} d^{6} e + 6 \, A b^{2} c^{2} d^{5} e^{2} - 4 \, A b^{3} c d^{4} e^{3} + A b^{4} d^{3} e^{4} + {\left (A c^{4} d^{4} e^{3} - 4 \, A b c^{3} d^{3} e^{4} + 6 \, A b^{2} c^{2} d^{2} e^{5} - 4 \, A b^{3} c d e^{6} + A b^{4} e^{7}\right )} x^{3} + 3 \, {\left (A c^{4} d^{5} e^{2} - 4 \, A b c^{3} d^{4} e^{3} + 6 \, A b^{2} c^{2} d^{3} e^{4} - 4 \, A b^{3} c d^{2} e^{5} + A b^{4} d e^{6}\right )} x^{2} + 3 \, {\left (A c^{4} d^{6} e - 4 \, A b c^{3} d^{5} e^{2} + 6 \, A b^{2} c^{2} d^{4} e^{3} - 4 \, A b^{3} c d^{3} e^{4} + A b^{4} d^{2} e^{5}\right )} x\right )} \log \left (x\right )}{6 \, {\left (b c^{4} d^{11} - 4 \, b^{2} c^{3} d^{10} e + 6 \, b^{3} c^{2} d^{9} e^{2} - 4 \, b^{4} c d^{8} e^{3} + b^{5} d^{7} e^{4} + {\left (b c^{4} d^{8} e^{3} - 4 \, b^{2} c^{3} d^{7} e^{4} + 6 \, b^{3} c^{2} d^{6} e^{5} - 4 \, b^{4} c d^{5} e^{6} + b^{5} d^{4} e^{7}\right )} x^{3} + 3 \, {\left (b c^{4} d^{9} e^{2} - 4 \, b^{2} c^{3} d^{8} e^{3} + 6 \, b^{3} c^{2} d^{7} e^{4} - 4 \, b^{4} c d^{6} e^{5} + b^{5} d^{5} e^{6}\right )} x^{2} + 3 \, {\left (b c^{4} d^{10} e - 4 \, b^{2} c^{3} d^{9} e^{2} + 6 \, b^{3} c^{2} d^{8} e^{3} - 4 \, b^{4} c d^{7} e^{4} + b^{5} d^{6} e^{5}\right )} x\right )}} \]
1/6*(11*B*b*c^3*d^7 + 11*A*b^4*d^3*e^4 - 2*(9*B*b^2*c^2 + 13*A*b*c^3)*d^6* e + 3*(3*B*b^3*c + 19*A*b^2*c^2)*d^5*e^2 - 2*(B*b^4 + 21*A*b^3*c)*d^4*e^3 + 6*(B*b*c^3*d^5*e^2 + 6*A*b^2*c^2*d^3*e^4 - 4*A*b^3*c*d^2*e^5 + A*b^4*d*e ^6 - (B*b^2*c^2 + 3*A*b*c^3)*d^4*e^3)*x^2 + 3*(5*B*b*c^3*d^6*e - 20*A*b^3* c*d^3*e^4 + 5*A*b^4*d^2*e^5 - 2*(3*B*b^2*c^2 + 7*A*b*c^3)*d^5*e^2 + (B*b^3 *c + 29*A*b^2*c^2)*d^4*e^3)*x + 6*((B*b*c^3 - A*c^4)*d^4*e^3*x^3 + 3*(B*b* c^3 - A*c^4)*d^5*e^2*x^2 + 3*(B*b*c^3 - A*c^4)*d^6*e*x + (B*b*c^3 - A*c^4) *d^7)*log(c*x + b) - 6*(B*b*c^3*d^7 - 4*A*b*c^3*d^6*e + 6*A*b^2*c^2*d^5*e^ 2 - 4*A*b^3*c*d^4*e^3 + A*b^4*d^3*e^4 + (B*b*c^3*d^4*e^3 - 4*A*b*c^3*d^3*e ^4 + 6*A*b^2*c^2*d^2*e^5 - 4*A*b^3*c*d*e^6 + A*b^4*e^7)*x^3 + 3*(B*b*c^3*d ^5*e^2 - 4*A*b*c^3*d^4*e^3 + 6*A*b^2*c^2*d^3*e^4 - 4*A*b^3*c*d^2*e^5 + A*b ^4*d*e^6)*x^2 + 3*(B*b*c^3*d^6*e - 4*A*b*c^3*d^5*e^2 + 6*A*b^2*c^2*d^4*e^3 - 4*A*b^3*c*d^3*e^4 + A*b^4*d^2*e^5)*x)*log(e*x + d) + 6*(A*c^4*d^7 - 4*A *b*c^3*d^6*e + 6*A*b^2*c^2*d^5*e^2 - 4*A*b^3*c*d^4*e^3 + A*b^4*d^3*e^4 + ( A*c^4*d^4*e^3 - 4*A*b*c^3*d^3*e^4 + 6*A*b^2*c^2*d^2*e^5 - 4*A*b^3*c*d*e^6 + A*b^4*e^7)*x^3 + 3*(A*c^4*d^5*e^2 - 4*A*b*c^3*d^4*e^3 + 6*A*b^2*c^2*d^3* e^4 - 4*A*b^3*c*d^2*e^5 + A*b^4*d*e^6)*x^2 + 3*(A*c^4*d^6*e - 4*A*b*c^3*d^ 5*e^2 + 6*A*b^2*c^2*d^4*e^3 - 4*A*b^3*c*d^3*e^4 + A*b^4*d^2*e^5)*x)*log(x) )/(b*c^4*d^11 - 4*b^2*c^3*d^10*e + 6*b^3*c^2*d^9*e^2 - 4*b^4*c*d^8*e^3 + b ^5*d^7*e^4 + (b*c^4*d^8*e^3 - 4*b^2*c^3*d^7*e^4 + 6*b^3*c^2*d^6*e^5 - 4...
Timed out. \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (241) = 482\).
Time = 0.23 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.26 \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\frac {{\left (B b c^{3} - A c^{4}\right )} \log \left (c x + b\right )}{b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}} - \frac {{\left (B c^{3} d^{4} - 4 \, A c^{3} d^{3} e + 6 \, A b c^{2} d^{2} e^{2} - 4 \, A b^{2} c d e^{3} + A b^{3} e^{4}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} + \frac {11 \, B c^{2} d^{5} - 11 \, A b^{2} d^{2} e^{3} - {\left (7 \, B b c + 26 \, A c^{2}\right )} d^{4} e + {\left (2 \, B b^{2} + 31 \, A b c\right )} d^{3} e^{2} + 6 \, {\left (B c^{2} d^{3} e^{2} - 3 \, A c^{2} d^{2} e^{3} + 3 \, A b c d e^{4} - A b^{2} e^{5}\right )} x^{2} + 3 \, {\left (5 \, B c^{2} d^{4} e + 15 \, A b c d^{2} e^{3} - 5 \, A b^{2} d e^{4} - {\left (B b c + 14 \, A c^{2}\right )} d^{3} e^{2}\right )} x}{6 \, {\left (c^{3} d^{9} - 3 \, b c^{2} d^{8} e + 3 \, b^{2} c d^{7} e^{2} - b^{3} d^{6} e^{3} + {\left (c^{3} d^{6} e^{3} - 3 \, b c^{2} d^{5} e^{4} + 3 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, b c^{2} d^{6} e^{3} + 3 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, b c^{2} d^{7} e^{2} + 3 \, b^{2} c d^{6} e^{3} - b^{3} d^{5} e^{4}\right )} x\right )}} + \frac {A \log \left (x\right )}{b d^{4}} \]
(B*b*c^3 - A*c^4)*log(c*x + b)/(b*c^4*d^4 - 4*b^2*c^3*d^3*e + 6*b^3*c^2*d^ 2*e^2 - 4*b^4*c*d*e^3 + b^5*e^4) - (B*c^3*d^4 - 4*A*c^3*d^3*e + 6*A*b*c^2* d^2*e^2 - 4*A*b^2*c*d*e^3 + A*b^3*e^4)*log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7 *e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4) + 1/6*(11*B*c^2*d^ 5 - 11*A*b^2*d^2*e^3 - (7*B*b*c + 26*A*c^2)*d^4*e + (2*B*b^2 + 31*A*b*c)*d ^3*e^2 + 6*(B*c^2*d^3*e^2 - 3*A*c^2*d^2*e^3 + 3*A*b*c*d*e^4 - A*b^2*e^5)*x ^2 + 3*(5*B*c^2*d^4*e + 15*A*b*c*d^2*e^3 - 5*A*b^2*d*e^4 - (B*b*c + 14*A*c ^2)*d^3*e^2)*x)/(c^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*( c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^ 3*d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c*d^6*e^3 - b^3*d^5*e^4)*x) + A*log(x)/( b*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (241) = 482\).
Time = 0.28 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.02 \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\frac {{\left (B b c^{4} - A c^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{5} d^{4} - 4 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - 4 \, b^{4} c^{2} d e^{3} + b^{5} c e^{4}} - \frac {{\left (B c^{3} d^{4} e - 4 \, A c^{3} d^{3} e^{2} + 6 \, A b c^{2} d^{2} e^{3} - 4 \, A b^{2} c d e^{4} + A b^{3} e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, b c^{3} d^{7} e^{2} + 6 \, b^{2} c^{2} d^{6} e^{3} - 4 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5}} + \frac {A \log \left ({\left | x \right |}\right )}{b d^{4}} + \frac {11 \, B c^{3} d^{7} - 18 \, B b c^{2} d^{6} e - 26 \, A c^{3} d^{6} e + 9 \, B b^{2} c d^{5} e^{2} + 57 \, A b c^{2} d^{5} e^{2} - 2 \, B b^{3} d^{4} e^{3} - 42 \, A b^{2} c d^{4} e^{3} + 11 \, A b^{3} d^{3} e^{4} + 6 \, {\left (B c^{3} d^{5} e^{2} - B b c^{2} d^{4} e^{3} - 3 \, A c^{3} d^{4} e^{3} + 6 \, A b c^{2} d^{3} e^{4} - 4 \, A b^{2} c d^{2} e^{5} + A b^{3} d e^{6}\right )} x^{2} + 3 \, {\left (5 \, B c^{3} d^{6} e - 6 \, B b c^{2} d^{5} e^{2} - 14 \, A c^{3} d^{5} e^{2} + B b^{2} c d^{4} e^{3} + 29 \, A b c^{2} d^{4} e^{3} - 20 \, A b^{2} c d^{3} e^{4} + 5 \, A b^{3} d^{2} e^{5}\right )} x}{6 \, {\left (c d - b e\right )}^{4} {\left (e x + d\right )}^{3} d^{4}} \]
(B*b*c^4 - A*c^5)*log(abs(c*x + b))/(b*c^5*d^4 - 4*b^2*c^4*d^3*e + 6*b^3*c ^3*d^2*e^2 - 4*b^4*c^2*d*e^3 + b^5*c*e^4) - (B*c^3*d^4*e - 4*A*c^3*d^3*e^2 + 6*A*b*c^2*d^2*e^3 - 4*A*b^2*c*d*e^4 + A*b^3*e^5)*log(abs(e*x + d))/(c^4 *d^8*e - 4*b*c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 - 4*b^3*c*d^5*e^4 + b^4*d^4*e ^5) + A*log(abs(x))/(b*d^4) + 1/6*(11*B*c^3*d^7 - 18*B*b*c^2*d^6*e - 26*A* c^3*d^6*e + 9*B*b^2*c*d^5*e^2 + 57*A*b*c^2*d^5*e^2 - 2*B*b^3*d^4*e^3 - 42* A*b^2*c*d^4*e^3 + 11*A*b^3*d^3*e^4 + 6*(B*c^3*d^5*e^2 - B*b*c^2*d^4*e^3 - 3*A*c^3*d^4*e^3 + 6*A*b*c^2*d^3*e^4 - 4*A*b^2*c*d^2*e^5 + A*b^3*d*e^6)*x^2 + 3*(5*B*c^3*d^6*e - 6*B*b*c^2*d^5*e^2 - 14*A*c^3*d^5*e^2 + B*b^2*c*d^4*e ^3 + 29*A*b*c^2*d^4*e^3 - 20*A*b^2*c*d^3*e^4 + 5*A*b^3*d^2*e^5)*x)/((c*d - b*e)^4*(e*x + d)^3*d^4)
Time = 11.15 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.92 \[ \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx=\frac {\frac {-2\,B\,b^2\,d\,e^2+11\,A\,b^2\,e^3+7\,B\,b\,c\,d^2\,e-31\,A\,b\,c\,d\,e^2-11\,B\,c^2\,d^3+26\,A\,c^2\,d^2\,e}{6\,d\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x^2\,\left (A\,b^2\,e^5-3\,A\,b\,c\,d\,e^4-B\,c^2\,d^3\,e^2+3\,A\,c^2\,d^2\,e^3\right )}{d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x\,\left (5\,A\,b^2\,e^4+B\,b\,c\,d^2\,e^2-15\,A\,b\,c\,d\,e^3-5\,B\,c^2\,d^3\,e+14\,A\,c^2\,d^2\,e^2\right )}{2\,d^2\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3}-\frac {\ln \left (b+c\,x\right )\,\left (A\,c^4-B\,b\,c^3\right )}{b^5\,e^4-4\,b^4\,c\,d\,e^3+6\,b^3\,c^2\,d^2\,e^2-4\,b^2\,c^3\,d^3\,e+b\,c^4\,d^4}+\frac {A\,\ln \left (x\right )}{b\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (A\,b^3\,e^4-4\,A\,b^2\,c\,d\,e^3+6\,A\,b\,c^2\,d^2\,e^2+B\,c^3\,d^4-4\,A\,c^3\,d^3\,e\right )}{d^4\,{\left (b\,e-c\,d\right )}^4} \]
((11*A*b^2*e^3 - 11*B*c^2*d^3 + 26*A*c^2*d^2*e - 2*B*b^2*d*e^2 - 31*A*b*c* d*e^2 + 7*B*b*c*d^2*e)/(6*d*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d *e^2)) + (x^2*(A*b^2*e^5 + 3*A*c^2*d^2*e^3 - B*c^2*d^3*e^2 - 3*A*b*c*d*e^4 ))/(d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)) + (x*(5*A*b^2 *e^4 - 5*B*c^2*d^3*e + 14*A*c^2*d^2*e^2 - 15*A*b*c*d*e^3 + B*b*c*d^2*e^2)) /(2*d^2*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)))/(d^3 + e^3*x ^3 + 3*d*e^2*x^2 + 3*d^2*e*x) - (log(b + c*x)*(A*c^4 - B*b*c^3))/(b^5*e^4 + b*c^4*d^4 - 4*b^2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 - 4*b^4*c*d*e^3) + (A*lo g(x))/(b*d^4) - (log(d + e*x)*(A*b^3*e^4 + B*c^3*d^4 - 4*A*c^3*d^3*e + 6*A *b*c^2*d^2*e^2 - 4*A*b^2*c*d*e^3))/(d^4*(b*e - c*d)^4)