Integrand size = 19, antiderivative size = 144 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {1}{b^2 d^2 x}-\frac {c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac {e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac {2 (c d+b e) \log (x)}{b^3 d^3}+\frac {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]
-1/b^2/d^2/x-c^3/b^2/(-b*e+c*d)^2/(c*x+b)-e^3/d^2/(-b*e+c*d)^2/(e*x+d)-2*( b*e+c*d)*ln(x)/b^3/d^3+2*c^3*(-2*b*e+c*d)*ln(c*x+b)/b^3/(-b*e+c*d)^3+2*e^3 *(-b*e+2*c*d)*ln(e*x+d)/d^3/(-b*e+c*d)^3
Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {1}{b^2 d^2 x}-\frac {c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac {e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac {2 (c d+b e) \log (x)}{b^3 d^3}+\frac {2 c^3 (-c d+2 b e) \log (b+c x)}{b^3 (-c d+b e)^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]
-(1/(b^2*d^2*x)) - c^3/(b^2*(c*d - b*e)^2*(b + c*x)) - e^3/(d^2*(c*d - b*e )^2*(d + e*x)) - (2*(c*d + b*e)*Log[x])/(b^3*d^3) + (2*c^3*(-(c*d) + 2*b*e )*Log[b + c*x])/(b^3*(-(c*d) + b*e)^3) + (2*e^3*(2*c*d - b*e)*Log[d + e*x] )/(d^3*(c*d - b*e)^3)
Time = 0.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^2 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1141 |
| \(\displaystyle c^2 \int \left (\frac {2 (2 c d-b e) e^4}{c^2 d^3 (c d-b e)^3 (d+e x)}+\frac {e^4}{c^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {2 (c d+b e)}{b^3 c^2 d^3 x}+\frac {2 c^2 (c d-2 b e)}{b^3 (c d-b e)^3 (b+c x)}+\frac {1}{b^2 c^2 d^2 x^2}+\frac {c^2}{b^2 (c d-b e)^2 (b+c x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^2 \left (-\frac {2 \log (x) (b e+c d)}{b^3 c^2 d^3}+\frac {2 c (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac {1}{b^2 c^2 d^2 x}-\frac {c}{b^2 (b+c x) (c d-b e)^2}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{c^2 d^3 (c d-b e)^3}-\frac {e^3}{c^2 d^2 (d+e x) (c d-b e)^2}\right )\) |
c^2*(-(1/(b^2*c^2*d^2*x)) - c/(b^2*(c*d - b*e)^2*(b + c*x)) - e^3/(c^2*d^2 *(c*d - b*e)^2*(d + e*x)) - (2*(c*d + b*e)*Log[x])/(b^3*c^2*d^3) + (2*c*(c *d - 2*b*e)*Log[b + c*x])/(b^3*(c*d - b*e)^3) + (2*e^3*(2*c*d - b*e)*Log[d + e*x])/(c^2*d^3*(c*d - b*e)^3))
3.3.74.3.1 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 = 146, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(-\frac {1}{b^{2} d^{2} x}+\frac {\left (-2 b e -2 c d \right ) \ln \left (x \right )}{b^{3} d^{3}}-\frac {c^{3}}{\left (b e -c d \right )^{2} b^{2} \left (c x +b \right )}+\frac {2 c^{3} \left (2 b e -c d \right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {e^{3}}{d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )}+\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}\) | \(146\) |
| norman | \(\frac {\frac {\left (2 b^{4} e^{4}-b^{3} c d \,e^{3}-b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x^{2}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (2 b^{3} e^{3}-b^{2} d \,e^{2} c -b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) c e \,x^{3}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {1}{b d}}{\left (c x +b \right ) x \left (e x +d \right )}-\frac {2 \left (b e +c d \right ) \ln \left (x \right )}{b^{3} d^{3}}+\frac {2 c^{3} \left (2 b e -c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(309\) |
| risch | \(\frac {-\frac {2 c e \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) x^{2}}{b^{2} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (2 b^{3} e^{3}-b^{2} d \,e^{2} c -b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) x}{d^{2} b^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {1}{b d}}{\left (c x +b \right ) x \left (e x +d \right )}-\frac {2 \ln \left (-x \right ) e}{b^{2} d^{3}}-\frac {2 \ln \left (-x \right ) c}{b^{3} d^{2}}+\frac {2 e^{4} \ln \left (-e x -d \right ) b}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {4 e^{3} \ln \left (-e x -d \right ) c}{d^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {4 c^{3} \ln \left (c x +b \right ) e}{b^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {2 c^{4} \ln \left (c x +b \right ) d}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(399\) |
| parallelrisch | \(-\frac {-4 \ln \left (c x +b \right ) x^{2} b^{2} c^{3} d^{3} e^{2}+2 x^{2} c^{5} d^{5}+e^{3} d^{2} b^{5}-3 e^{2} d^{3} c \,b^{4}+3 e \,d^{4} c^{2} b^{3}-d^{5} c^{3} b^{2}+2 \ln \left (x \right ) x^{2} b^{5} e^{5}-2 \ln \left (x \right ) x^{2} c^{5} d^{5}+2 \ln \left (c x +b \right ) x^{2} c^{5} d^{5}-2 \ln \left (e x +d \right ) x^{2} b^{5} e^{5}-2 x^{3} b^{4} c \,e^{5}+2 x^{3} c^{5} d^{4} e -4 \ln \left (c x +b \right ) x^{3} b \,c^{4} d^{3} e^{2}+4 \ln \left (e x +d \right ) x^{3} b^{3} c^{2} d \,e^{4}-2 \ln \left (x \right ) x^{2} b^{4} c d \,e^{4}-4 \ln \left (x \right ) x^{2} b^{3} c^{2} d^{2} e^{3}+4 \ln \left (x \right ) x^{2} b^{2} c^{3} d^{3} e^{2}+2 \ln \left (x \right ) x^{2} b \,c^{4} d^{4} e -2 \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{4} e +2 \ln \left (e x +d \right ) x^{2} b^{4} c d \,e^{4}+4 \ln \left (e x +d \right ) x^{2} b^{3} c^{2} d^{2} e^{3}-4 \ln \left (x \right ) x \,b^{4} c \,d^{2} e^{3}+4 \ln \left (x \right ) x \,b^{2} c^{3} d^{4} e -4 \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{4} e +4 \ln \left (e x +d \right ) x \,b^{4} c \,d^{2} e^{3}-4 \ln \left (x \right ) x^{3} b^{3} c^{2} d \,e^{4}+4 \ln \left (x \right ) x^{3} b \,c^{4} d^{3} e^{2}-2 x^{2} b^{5} e^{5}+3 x^{3} b^{3} c^{2} d \,e^{4}-2 \ln \left (x \right ) x b \,c^{4} d^{5}+2 \ln \left (c x +b \right ) x b \,c^{4} d^{5}-2 \ln \left (e x +d \right ) x \,b^{5} d \,e^{4}-x^{2} b^{3} c^{2} d^{2} e^{3}+x^{2} b^{2} c^{3} d^{3} e^{2}+2 \ln \left (x \right ) x^{3} b^{4} c \,e^{5}-2 \ln \left (x \right ) x^{3} c^{5} d^{4} e +2 \ln \left (c x +b \right ) x^{3} c^{5} d^{4} e -2 \ln \left (e x +d \right ) x^{3} b^{4} c \,e^{5}+2 \ln \left (x \right ) x \,b^{5} d \,e^{4}+3 x^{2} b^{4} c d \,e^{4}-3 x^{2} b \,c^{4} d^{4} e -3 x^{3} b \,c^{4} d^{3} e^{2}}{\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \left (e x +d \right ) x \left (c x +b \right ) b^{3} d^{3}}\) | \(721\) |
-1/b^2/d^2/x+(-2*b*e-2*c*d)/b^3/d^3*ln(x)-c^3/(b*e-c*d)^2/b^2/(c*x+b)+2*c^ 3*(2*b*e-c*d)/(b*e-c*d)^3/b^3*ln(c*x+b)-e^3/d^2/(b*e-c*d)^2/(e*x+d)+2*e^3* (b*e-2*c*d)/d^3/(b*e-c*d)^3*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (144) = 288\).
Time = 7.89 (sec) , antiderivative size = 653, normalized size of antiderivative = 4.53 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {b^{2} c^{3} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - b^{5} d^{2} e^{3} + 2 \, {\left (b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4}\right )} x^{2} + {\left (2 \, b c^{4} d^{5} - 3 \, b^{2} c^{3} d^{4} e + 3 \, b^{4} c d^{2} e^{3} - 2 \, b^{5} d e^{4}\right )} x - 2 \, {\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2}\right )} x^{3} + {\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2}\right )} x^{2} + {\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e\right )} x\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} + {\left (2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \, {\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} + {\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} + {\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e + 2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} + {\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} + {\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \]
-(b^2*c^3*d^5 - 3*b^3*c^2*d^4*e + 3*b^4*c*d^3*e^2 - b^5*d^2*e^3 + 2*(b*c^4 *d^4*e - 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - b^4*c*d*e^4)*x^2 + (2*b*c ^4*d^5 - 3*b^2*c^3*d^4*e + 3*b^4*c*d^2*e^3 - 2*b^5*d*e^4)*x - 2*((c^5*d^4* e - 2*b*c^4*d^3*e^2)*x^3 + (c^5*d^5 - b*c^4*d^4*e - 2*b^2*c^3*d^3*e^2)*x^2 + (b*c^4*d^5 - 2*b^2*c^3*d^4*e)*x)*log(c*x + b) - 2*((2*b^3*c^2*d*e^4 - b ^4*c*e^5)*x^3 + (2*b^3*c^2*d^2*e^3 + b^4*c*d*e^4 - b^5*e^5)*x^2 + (2*b^4*c *d^2*e^3 - b^5*d*e^4)*x)*log(e*x + d) + 2*((c^5*d^4*e - 2*b*c^4*d^3*e^2 + 2*b^3*c^2*d*e^4 - b^4*c*e^5)*x^3 + (c^5*d^5 - b*c^4*d^4*e - 2*b^2*c^3*d^3* e^2 + 2*b^3*c^2*d^2*e^3 + b^4*c*d*e^4 - b^5*e^5)*x^2 + (b*c^4*d^5 - 2*b^2* c^3*d^4*e + 2*b^4*c*d^2*e^3 - b^5*d*e^4)*x)*log(x))/((b^3*c^4*d^6*e - 3*b^ 4*c^3*d^5*e^2 + 3*b^5*c^2*d^4*e^3 - b^6*c*d^3*e^4)*x^3 + (b^3*c^4*d^7 - 2* b^4*c^3*d^6*e + 2*b^6*c*d^4*e^3 - b^7*d^3*e^4)*x^2 + (b^4*c^3*d^7 - 3*b^5* c^2*d^6*e + 3*b^6*c*d^5*e^2 - b^7*d^4*e^3)*x)
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (144) = 288\).
Time = 0.23 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.59 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=\frac {2 \, {\left (c^{4} d - 2 \, b c^{3} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} + \frac {2 \, {\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + 2 \, {\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} + {\left (2 \, c^{3} d^{3} - b c^{2} d^{2} e - b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} + {\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} + {\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac {2 \, {\left (c d + b e\right )} \log \left (x\right )}{b^{3} d^{3}} \]
2*(c^4*d - 2*b*c^3*e)*log(c*x + b)/(b^3*c^3*d^3 - 3*b^4*c^2*d^2*e + 3*b^5* c*d*e^2 - b^6*e^3) + 2*(2*c*d*e^3 - b*e^4)*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) - (b*c^2*d^3 - 2*b^2*c*d^2*e + b^3 *d*e^2 + 2*(c^3*d^2*e - b*c^2*d*e^2 + b^2*c*e^3)*x^2 + (2*c^3*d^3 - b*c^2* d^2*e - b^2*c*d*e^2 + 2*b^3*e^3)*x)/((b^2*c^3*d^4*e - 2*b^3*c^2*d^3*e^2 + b^4*c*d^2*e^3)*x^3 + (b^2*c^3*d^5 - b^3*c^2*d^4*e - b^4*c*d^3*e^2 + b^5*d^ 2*e^3)*x^2 + (b^3*c^2*d^5 - 2*b^4*c*d^4*e + b^5*d^3*e^2)*x) - 2*(c*d + b*e )*log(x)/(b^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (144) = 288\).
Time = 0.28 (sec) , antiderivative size = 557, normalized size of antiderivative = 3.87 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )} {\left (e x + d\right )}} - \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left ({\left | -c + \frac {2 \, c d}{e x + d} - \frac {c d^{2}}{{\left (e x + d\right )}^{2}} - \frac {b e}{e x + d} + \frac {b d e}{{\left (e x + d\right )}^{2}} \right |}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} + \frac {{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 2 \, b^{3} c d e^{5} - b^{4} e^{6}\right )} \log \left (\frac {{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{e x + d} + b e^{2} - \frac {2 \, b d e^{2}}{e x + d} - e^{2} {\left | b \right |} \right |}}{{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{e x + d} + b e^{2} - \frac {2 \, b d e^{2}}{e x + d} + e^{2} {\left | b \right |} \right |}}\right )}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} e^{2} {\left | b \right |}} - \frac {\frac {2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}}{c d^{2} - b d e} - \frac {2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + b^{4} e^{6}}{{\left (c d^{2} - b d e\right )} {\left (e x + d\right )} e}}{{\left (c d - b e\right )}^{2} b^{2} {\left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}\right )} d^{2}} \]
-e^7/((c^2*d^4*e^4 - 2*b*c*d^3*e^5 + b^2*d^2*e^6)*(e*x + d)) - (2*c*d*e^3 - b*e^4)*log(abs(-c + 2*c*d/(e*x + d) - c*d^2/(e*x + d)^2 - b*e/(e*x + d) + b*d*e/(e*x + d)^2))/(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) + (2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 2*b^3*c*d*e^5 - b^4*e^6)*log(ab s(-2*c*d*e + 2*c*d^2*e/(e*x + d) + b*e^2 - 2*b*d*e^2/(e*x + d) - e^2*abs(b ))/abs(-2*c*d*e + 2*c*d^2*e/(e*x + d) + b*e^2 - 2*b*d*e^2/(e*x + d) + e^2* abs(b)))/((b^2*c^3*d^6 - 3*b^3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)* e^2*abs(b)) - ((2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*b^2*c^2*d*e^3 - b^3*c*e^ 4)/(c*d^2 - b*d*e) - (2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*b^3*c*d*e^5 + b^4*e^6)/((c*d^2 - b*d*e)*(e*x + d)*e))/((c*d - b*e)^2*b ^2*(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2)*d^2)
Time = 10.02 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {\frac {1}{b\,d}+\frac {2\,x^2\,\left (b^2\,c\,e^3-b\,c^2\,d\,e^2+c^3\,d^2\,e\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x\,\left (b\,e+c\,d\right )\,\left (2\,b^2\,e^2-3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+b\,d\,x}-\frac {\ln \left (b+c\,x\right )\,\left (2\,c^4\,d-4\,b\,c^3\,e\right )}{b^6\,e^3-3\,b^5\,c\,d\,e^2+3\,b^4\,c^2\,d^2\,e-b^3\,c^3\,d^3}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b\,e^4-4\,c\,d\,e^3\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {2\,\ln \left (x\right )\,\left (b\,e+c\,d\right )}{b^3\,d^3} \]
- (1/(b*d) + (2*x^2*(b^2*c*e^3 + c^3*d^2*e - b*c^2*d*e^2))/(b^2*d^2*(b^2*e ^2 + c^2*d^2 - 2*b*c*d*e)) + (x*(b*e + c*d)*(2*b^2*e^2 + 2*c^2*d^2 - 3*b*c *d*e))/(b^2*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)))/(x^2*(b*e + c*d) + b*d*x + c*e*x^3) - (log(b + c*x)*(2*c^4*d - 4*b*c^3*e))/(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2) - (log(d + e*x)*(2*b*e^4 - 4*c*d*e^3))/ (c^3*d^6 - b^3*d^3*e^3 + 3*b^2*c*d^4*e^2 - 3*b*c^2*d^5*e) - (2*log(x)*(b*e + c*d))/(b^3*d^3)