Integrand size = 19, antiderivative size = 193 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}+\frac {c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
-1/2/b^3/d/x^2+(b*e+3*c*d)/b^4/d^2/x+1/2*c^3/b^3/(-b*e+c*d)/(c*x+b)^2+c^3* (-4*b*e+3*c*d)/b^4/(-b*e+c*d)^2/(c*x+b)+(b^2*e^2+3*b*c*d*e+6*c^2*d^2)*ln(x )/b^5/d^3-c^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^2)*ln(c*x+b)/b^5/(-b*e+c*d)^3 +e^5*ln(e*x+d)/d^3/(-b*e+c*d)^3
Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}-\frac {c^3}{2 b^3 (-c d+b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}+\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (-c d+b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
-1/2*1/(b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) - c^3/(2*b^3*(-(c*d) + b*e) *(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/(b^4*(c*d - b*e)^2*(b + c*x)) + ((6* c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) + (c^3*(6*c^2*d^2 - 15*b* c*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(-(c*d) + b*e)^3) + (e^5*Log[d + e* x])/(d^3*(c*d - b*e)^3)
Time = 0.48 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1141, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1141 |
| \(\displaystyle c^3 \int \left (\frac {e^6}{c^3 d^3 (c d-b e)^3 (d+e x)}+\frac {6 c^2 d^2+3 b c e d+b^2 e^2}{b^5 c^3 d^3 x}-\frac {c \left (6 c^2 d^2-15 b c e d+10 b^2 e^2\right )}{b^5 (c d-b e)^3 (b+c x)}-\frac {3 c d+b e}{b^4 c^3 d^2 x^2}-\frac {c (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)^2}+\frac {1}{b^3 c^3 d x^3}-\frac {c}{b^3 (c d-b e) (b+c x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^3 \left (\frac {b e+3 c d}{b^4 c^3 d^2 x}+\frac {3 c d-4 b e}{b^4 (b+c x) (c d-b e)^2}-\frac {1}{2 b^3 c^3 d x^2}+\frac {1}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {\left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 c^3 d^3}+\frac {e^5 \log (d+e x)}{c^3 d^3 (c d-b e)^3}\right )\) |
c^3*(-1/2*1/(b^3*c^3*d*x^2) + (3*c*d + b*e)/(b^4*c^3*d^2*x) + 1/(2*b^3*(c* d - b*e)*(b + c*x)^2) + (3*c*d - 4*b*e)/(b^4*(c*d - b*e)^2*(b + c*x)) + (( 6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*c^3*d^3) - ((6*c^2*d^2 - 15* b*c*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^3) + (e^5*Log[d + e*x ])/(c^3*d^3*(c*d - b*e)^3))
3.3.83.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Time = 1.95 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {1}{2 b^{3} d \,x^{2}}-\frac {-b e -3 c d}{b^{4} d^{2} x}+\frac {\left (b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}-\frac {c^{3}}{2 \left (b e -c d \right ) b^{3} \left (c x +b \right )^{2}}-\frac {c^{3} \left (4 b e -3 c d \right )}{\left (b e -c d \right )^{2} b^{4} \left (c x +b \right )}+\frac {c^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{5}}-\frac {e^{5} \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}\) | \(193\) |
| norman | \(\frac {\frac {\left (b e +2 c d \right ) x}{b^{2} d^{2}}+\frac {\left (-3 b^{3} c \,e^{3}-2 b^{2} c^{2} d \,e^{2}+18 b \,c^{3} d^{2} e -12 c^{4} d^{3}\right ) c \,x^{3}}{d^{2} b^{4} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {1}{2 b d}+\frac {c^{2} \left (-4 b^{3} c \,e^{3}-3 b^{2} c^{2} d \,e^{2}+27 b \,c^{3} d^{2} e -18 c^{4} d^{3}\right ) x^{4}}{2 d^{2} b^{5} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}+\frac {c^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {e^{5} \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(346\) |
| risch | \(\frac {\frac {c^{2} \left (b^{3} e^{3}+b^{2} d \,e^{2} c -9 b \,c^{2} d^{2} e +6 c^{3} d^{3}\right ) x^{3}}{b^{4} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {c \left (4 b^{3} e^{3}+3 b^{2} d \,e^{2} c -27 b \,c^{2} d^{2} e +18 c^{3} d^{3}\right ) x^{2}}{2 b^{3} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (b e +2 c d \right ) x}{b^{2} d^{2}}-\frac {1}{2 b d}}{x^{2} \left (c x +b \right )^{2}}-\frac {e^{5} \ln \left (-e x -d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {\ln \left (-x \right ) e^{2}}{b^{3} d^{3}}+\frac {3 \ln \left (-x \right ) c e}{b^{4} d^{2}}+\frac {6 \ln \left (-x \right ) c^{2}}{b^{5} d}+\frac {10 c^{3} \ln \left (c x +b \right ) e^{2}}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {15 c^{4} \ln \left (c x +b \right ) d e}{b^{4} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {6 c^{5} \ln \left (c x +b \right ) d^{2}}{b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(443\) |
| parallelrisch | \(\frac {-45 x^{4} b \,c^{6} d^{4} e +2 x^{3} b^{4} c^{3} d^{2} e^{3}-2 x \,b^{6} c \,d^{2} e^{3}-6 x \,b^{5} c^{2} d^{3} e^{2}+2 \ln \left (x \right ) x^{4} b^{5} c^{2} e^{5}-2 \ln \left (e x +d \right ) x^{4} b^{5} c^{2} e^{5}+4 \ln \left (x \right ) x^{3} b^{6} c \,e^{5}-4 \ln \left (e x +d \right ) x^{3} b^{6} c \,e^{5}-20 \ln \left (x \right ) x^{4} b^{2} c^{5} d^{3} e^{2}+30 \ln \left (x \right ) x^{4} b \,c^{6} d^{4} e +20 \ln \left (c x +b \right ) x^{4} b^{2} c^{5} d^{3} e^{2}-30 \ln \left (c x +b \right ) x^{4} b \,c^{6} d^{4} e -40 \ln \left (x \right ) x^{3} b^{3} c^{4} d^{3} e^{2}+60 \ln \left (x \right ) x^{3} b^{2} c^{5} d^{4} e +40 \ln \left (c x +b \right ) x^{3} b^{3} c^{4} d^{3} e^{2}+b^{4} c^{3} d^{5}-b^{7} d^{2} e^{3}+3 b^{6} c \,d^{3} e^{2}-3 b^{5} c^{2} d^{4} e -60 \ln \left (c x +b \right ) x^{3} b^{2} c^{5} d^{4} e -20 \ln \left (x \right ) x^{2} b^{4} c^{3} d^{3} e^{2}+30 \ln \left (x \right ) x^{2} b^{3} c^{4} d^{4} e +20 \ln \left (c x +b \right ) x^{2} b^{4} c^{3} d^{3} e^{2}-30 \ln \left (c x +b \right ) x^{2} b^{3} c^{4} d^{4} e -6 x^{3} b^{5} c^{2} d \,e^{4}+40 x^{3} b^{3} c^{4} d^{3} e^{2}-60 x^{3} b^{2} c^{5} d^{4} e -12 \ln \left (x \right ) x^{4} c^{7} d^{5}+12 \ln \left (c x +b \right ) x^{4} c^{7} d^{5}+24 x^{3} b \,c^{6} d^{5}-4 x \,b^{3} c^{4} d^{5}+18 x^{4} c^{7} d^{5}+2 x \,b^{7} d \,e^{4}+2 \ln \left (x \right ) x^{2} b^{7} e^{5}-2 \ln \left (e x +d \right ) x^{2} b^{7} e^{5}+10 x \,b^{4} c^{3} d^{4} e -12 \ln \left (x \right ) x^{2} b^{2} c^{5} d^{5}+12 \ln \left (c x +b \right ) x^{2} b^{2} c^{5} d^{5}-24 \ln \left (x \right ) x^{3} b \,c^{6} d^{5}+24 \ln \left (c x +b \right ) x^{3} b \,c^{6} d^{5}-4 x^{4} b^{4} c^{3} d \,e^{4}+x^{4} b^{3} c^{4} d^{2} e^{3}+30 x^{4} b^{2} c^{5} d^{3} e^{2}}{2 \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \left (c x +b \right )^{2} x^{2} b^{5} d^{3}}\) | \(746\) |
-1/2/b^3/d/x^2-(-b*e-3*c*d)/b^4/d^2/x+(b^2*e^2+3*b*c*d*e+6*c^2*d^2)*ln(x)/ b^5/d^3-1/2*c^3/(b*e-c*d)/b^3/(c*x+b)^2-c^3*(4*b*e-3*c*d)/(b*e-c*d)^2/b^4/ (c*x+b)+c^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^2)/(b*e-c*d)^3/b^5*ln(c*x+b)-e^ 5/d^3/(b*e-c*d)^3*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (189) = 378\).
Time = 18.68 (sec) , antiderivative size = 716, normalized size of antiderivative = 3.71 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} c^{3} d^{5} - 3 \, b^{5} c^{2} d^{4} e + 3 \, b^{6} c d^{3} e^{2} - b^{7} d^{2} e^{3} - 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{5} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} + b^{5} c^{2} d^{2} e^{3} - 4 \, b^{6} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e + 3 \, b^{5} c^{2} d^{3} e^{2} + b^{6} c d^{2} e^{3} - b^{7} d e^{4}\right )} x + 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left (b^{5} c^{2} e^{5} x^{4} + 2 \, b^{6} c e^{5} x^{3} + b^{7} e^{5} x^{2}\right )} \log \left (e x + d\right ) - 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (b^{5} c^{5} d^{6} - 3 \, b^{6} c^{4} d^{5} e + 3 \, b^{7} c^{3} d^{4} e^{2} - b^{8} c^{2} d^{3} e^{3}\right )} x^{4} + 2 \, {\left (b^{6} c^{4} d^{6} - 3 \, b^{7} c^{3} d^{5} e + 3 \, b^{8} c^{2} d^{4} e^{2} - b^{9} c d^{3} e^{3}\right )} x^{3} + {\left (b^{7} c^{3} d^{6} - 3 \, b^{8} c^{2} d^{5} e + 3 \, b^{9} c d^{4} e^{2} - b^{10} d^{3} e^{3}\right )} x^{2}\right )}} \]
-1/2*(b^4*c^3*d^5 - 3*b^5*c^2*d^4*e + 3*b^6*c*d^3*e^2 - b^7*d^2*e^3 - 2*(6 *b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^5*c^2*d*e^4)*x^3 - (18*b^2*c^5*d^5 - 45*b^3*c^4*d^4*e + 30*b^4*c^3*d^3*e^2 + b^5*c^2*d^2*e^3 - 4*b^6*c*d*e^4)*x^2 - 2*(2*b^3*c^4*d^5 - 5*b^4*c^3*d^4*e + 3*b^5*c^2*d^3* e^2 + b^6*c*d^2*e^3 - b^7*d*e^4)*x + 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b ^2*c^5*d^3*e^2)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e ^2)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b^4*c^3*d^3*e^2)*x^2)*log (c*x + b) - 2*(b^5*c^2*e^5*x^4 + 2*b^6*c*e^5*x^3 + b^7*e^5*x^2)*log(e*x + d) - 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2 - b^5*c^2*e^5)*x^ 4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^6*c*e^5)*x^ 3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b^4*c^3*d^3*e^2 - b^7*e^5)*x^2) *log(x))/((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e^2 - b^8*c^2*d^3 *e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2 - b^9*c*d ^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*d^3* e^3)*x^2)
Timed out. \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (189) = 378\).
Time = 0.22 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.27 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac {b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \, {\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} - {\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{3}} \]
e^5*log(e*x + d)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3) - (6*c^5*d^2 - 15*b*c^4*d*e + 10*b^2*c^3*e^2)*log(c*x + b)/(b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3) - 1/2*(b^3*c^2*d^3 - 2*b^4*c*d ^2*e + b^5*d*e^2 - 2*(6*c^5*d^3 - 9*b*c^4*d^2*e + b^2*c^3*d*e^2 + b^3*c^2* e^3)*x^3 - (18*b*c^4*d^3 - 27*b^2*c^3*d^2*e + 3*b^3*c^2*d*e^2 + 4*b^4*c*e^ 3)*x^2 - 2*(2*b^2*c^3*d^3 - 3*b^3*c^2*d^2*e + b^5*e^3)*x)/((b^4*c^4*d^4 - 2*b^5*c^3*d^3*e + b^6*c^2*d^2*e^2)*x^4 + 2*(b^5*c^3*d^4 - 2*b^6*c^2*d^3*e + b^7*c*d^2*e^2)*x^3 + (b^6*c^2*d^4 - 2*b^7*c*d^3*e + b^8*d^2*e^2)*x^2) + (6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (189) = 378\).
Time = 0.27 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.19 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {e^{6} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} - \frac {{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \]
e^6*log(abs(e*x + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3 *d^3*e^4) - (6*c^6*d^2 - 15*b*c^5*d*e + 10*b^2*c^4*e^2)*log(abs(c*x + b))/ (b^5*c^4*d^3 - 3*b^6*c^3*d^2*e + 3*b^7*c^2*d*e^2 - b^8*c*e^3) + (6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*log(abs(x))/(b^5*d^3) - 1/2*(b^3*c^3*d^5 - 3*b^4*c ^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*d^2*e^3 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - b^4*c^2*d*e^4)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4* e + 30*b^3*c^3*d^3*e^2 + b^4*c^2*d^2*e^3 - 4*b^5*c*d*e^4)*x^2 - 2*(2*b^2*c ^4*d^5 - 5*b^3*c^3*d^4*e + 3*b^4*c^2*d^3*e^2 + b^5*c*d^2*e^3 - b^6*d*e^4)* x)/((c*d - b*e)^3*(c*x + b)^2*b^4*d^3*x^2)
Time = 10.16 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {\frac {x\,\left (b\,e+2\,c\,d\right )}{b^2\,d^2}-\frac {1}{2\,b\,d}+\frac {x^2\,\left (4\,b^3\,c\,e^3+3\,b^2\,c^2\,d\,e^2-27\,b\,c^3\,d^2\,e+18\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^3\,\left (b^3\,c^2\,e^3+b^2\,c^3\,d\,e^2-9\,b\,c^4\,d^2\,e+6\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,\left (10\,b^2\,c^3\,e^2-15\,b\,c^4\,d\,e+6\,c^5\,d^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}-\frac {e^5\,\ln \left (d+e\,x\right )}{d^3\,{\left (b\,e-c\,d\right )}^3}+\frac {\ln \left (x\right )\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,d^3} \]
((x*(b*e + 2*c*d))/(b^2*d^2) - 1/(2*b*d) + (x^2*(18*c^4*d^3 + 4*b^3*c*e^3 + 3*b^2*c^2*d*e^2 - 27*b*c^3*d^2*e))/(2*b^3*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c *d*e)) + (x^3*(6*c^5*d^3 + b^3*c^2*e^3 + b^2*c^3*d*e^2 - 9*b*c^4*d^2*e))/( b^4*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)))/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) + (log(b + c*x)*(6*c^5*d^2 + 10*b^2*c^3*e^2 - 15*b*c^4*d*e))/(b^8*e^3 - b^ 5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2) - (e^5*log(d + e*x))/(d^3*(b* e - c*d)^3) + (log(x)*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(b^5*d^3)