Integrand size = 31, antiderivative size = 143 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b^2}{2 (b d-a e)^3 (a+b x)^2}+\frac {3 b^2 e}{(b d-a e)^4 (a+b x)}+\frac {e^2}{2 (b d-a e)^3 (d+e x)^2}+\frac {3 b e^2}{(b d-a e)^4 (d+e x)}+\frac {6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac {6 b^2 e^2 \log (d+e x)}{(b d-a e)^5} \]
-1/2*b^2/(-a*e+b*d)^3/(b*x+a)^2+3*b^2*e/(-a*e+b*d)^4/(b*x+a)+1/2*e^2/(-a*e +b*d)^3/(e*x+d)^2+3*b*e^2/(-a*e+b*d)^4/(e*x+d)+6*b^2*e^2*ln(b*x+a)/(-a*e+b *d)^5-6*b^2*e^2*ln(e*x+d)/(-a*e+b*d)^5
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-\frac {b^2 (b d-a e)^2}{(a+b x)^2}+\frac {6 b^2 e (b d-a e)}{a+b x}+\frac {e^2 (b d-a e)^2}{(d+e x)^2}+\frac {6 b e^2 (b d-a e)}{d+e x}+12 b^2 e^2 \log (a+b x)-12 b^2 e^2 \log (d+e x)}{2 (b d-a e)^5} \]
(-((b^2*(b*d - a*e)^2)/(a + b*x)^2) + (6*b^2*e*(b*d - a*e))/(a + b*x) + (e ^2*(b*d - a*e)^2)/(d + e*x)^2 + (6*b*e^2*(b*d - a*e))/(d + e*x) + 12*b^2*e ^2*Log[a + b*x] - 12*b^2*e^2*Log[d + e*x])/(2*(b*d - a*e)^5)
Time = 0.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 54, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {1}{b^4 (a+b x)^3 (d+e x)^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(a+b x)^3 (d+e x)^3}dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (\frac {6 b^3 e^2}{(a+b x) (b d-a e)^5}-\frac {3 b^3 e}{(a+b x)^2 (b d-a e)^4}+\frac {b^3}{(a+b x)^3 (b d-a e)^3}-\frac {6 b^2 e^3}{(d+e x) (b d-a e)^5}-\frac {3 b e^3}{(d+e x)^2 (b d-a e)^4}-\frac {e^3}{(d+e x)^3 (b d-a e)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac {6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}+\frac {3 b^2 e}{(a+b x) (b d-a e)^4}-\frac {b^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {3 b e^2}{(d+e x) (b d-a e)^4}+\frac {e^2}{2 (d+e x)^2 (b d-a e)^3}\) |
-1/2*b^2/((b*d - a*e)^3*(a + b*x)^2) + (3*b^2*e)/((b*d - a*e)^4*(a + b*x)) + e^2/(2*(b*d - a*e)^3*(d + e*x)^2) + (3*b*e^2)/((b*d - a*e)^4*(d + e*x)) + (6*b^2*e^2*Log[a + b*x])/(b*d - a*e)^5 - (6*b^2*e^2*Log[d + e*x])/(b*d - a*e)^5
3.20.43.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.34 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {b^{2}}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {6 b^{2} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {3 b^{2} e}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {e^{2}}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}+\frac {6 b^{2} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 e^{2} b}{\left (a e -b d \right )^{4} \left (e x +d \right )}\) | \(140\) |
| risch | \(\frac {\frac {6 b^{3} e^{3} x^{3}}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}+\frac {9 b^{2} e^{2} \left (a e +b d \right ) x^{2}}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}+\frac {2 \left (e^{2} a^{2}+7 a b d e +b^{2} d^{2}\right ) b e x}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}-\frac {a^{3} e^{3}-7 a^{2} b d \,e^{2}-7 a \,b^{2} d^{2} e +b^{3} d^{3}}{2 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right )^{2}}-\frac {6 b^{2} e^{2} \ln \left (b x +a \right )}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}+\frac {6 b^{2} e^{2} \ln \left (-e x -d \right )}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}\) | \(477\) |
| parallelrisch | \(-\frac {-24 x \,a^{2} b^{4} d \,e^{5}+24 x a \,b^{5} d^{2} e^{4}+24 \ln \left (b x +a \right ) x^{3} a \,b^{5} e^{6}+24 \ln \left (b x +a \right ) x^{3} b^{6} d \,e^{5}-24 \ln \left (e x +d \right ) x^{3} a \,b^{5} e^{6}-24 \ln \left (e x +d \right ) x^{3} b^{6} d \,e^{5}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} e^{6}+12 \ln \left (b x +a \right ) x^{2} b^{6} d^{2} e^{4}-12 \ln \left (e x +d \right ) x^{2} a^{2} b^{4} e^{6}-12 \ln \left (e x +d \right ) x^{2} b^{6} d^{2} e^{4}+12 \ln \left (b x +a \right ) a^{2} b^{4} d^{2} e^{4}-12 \ln \left (e x +d \right ) a^{2} b^{4} d^{2} e^{4}+12 \ln \left (b x +a \right ) x^{4} b^{6} e^{6}-12 \ln \left (e x +d \right ) x^{4} b^{6} e^{6}-12 x^{3} a \,b^{5} e^{6}+12 x^{3} b^{6} d \,e^{5}-18 x^{2} a^{2} b^{4} e^{6}+18 x^{2} b^{6} d^{2} e^{4}-4 x \,a^{3} b^{3} e^{6}+4 x \,b^{6} d^{3} e^{3}-8 a^{3} b^{3} d \,e^{5}+8 a \,b^{5} d^{3} e^{3}+a^{4} b^{2} e^{6}-b^{6} d^{4} e^{2}+48 \ln \left (b x +a \right ) x^{2} a \,b^{5} d \,e^{5}-48 \ln \left (e x +d \right ) x^{2} a \,b^{5} d \,e^{5}+24 \ln \left (b x +a \right ) x \,a^{2} b^{4} d \,e^{5}+24 \ln \left (b x +a \right ) x a \,b^{5} d^{2} e^{4}-24 \ln \left (e x +d \right ) x \,a^{2} b^{4} d \,e^{5}-24 \ln \left (e x +d \right ) x a \,b^{5} d^{2} e^{4}}{2 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right ) \left (b x +a \right )^{2} \left (e x +d \right )^{2} b^{2} e^{2}}\) | \(577\) |
| norman | \(\frac {\frac {6 b^{4} e^{3} x^{4}}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}+\frac {\left (15 a \,b^{5} e^{5}+9 b^{6} d \,e^{4}\right ) x^{3}}{e^{2} b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {\left (11 e^{5} a^{2} b^{5}+23 d \,e^{4} a \,b^{6}+2 d^{2} e^{3} b^{7}\right ) x^{2}}{e^{2} b^{3} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {a \left (-a^{3} b^{3} e^{5}+7 a^{2} b^{4} d \,e^{4}+7 a \,b^{5} d^{2} e^{3}-b^{6} d^{3} e^{2}\right )}{2 e^{2} b^{3} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {\left (3 e^{5} a^{3} b^{4}+35 d \,e^{4} a^{2} b^{5}+11 d^{2} e^{3} a \,b^{6}-d^{3} e^{2} b^{7}\right ) x}{2 e^{2} b^{3} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}}{\left (b x +a \right )^{3} \left (e x +d \right )^{2}}-\frac {6 b^{2} e^{2} \ln \left (b x +a \right )}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}+\frac {6 b^{2} e^{2} \ln \left (e x +d \right )}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}\) | \(628\) |
1/2/(a*e-b*d)^3*b^2/(b*x+a)^2-6*b^2/(a*e-b*d)^5*e^2*ln(b*x+a)+3*b^2/(a*e-b *d)^4*e/(b*x+a)-1/2*e^2/(a*e-b*d)^3/(e*x+d)^2+6*b^2/(a*e-b*d)^5*e^2*ln(e*x +d)+3*e^2/(a*e-b*d)^4*b/(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (139) = 278\).
Time = 0.31 (sec) , antiderivative size = 760, normalized size of antiderivative = 5.31 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \, {\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} + {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \, {\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} + {\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \, {\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \]
-1/2*(b^4*d^4 - 8*a*b^3*d^3*e + 8*a^3*b*d*e^3 - a^4*e^4 - 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 - 18*(b^4*d^2*e^2 - a^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e + 6*a*b ^3*d^2*e^2 - 6*a^2*b^2*d*e^3 - a^3*b*e^4)*x - 12*(b^4*e^4*x^4 + a^2*b^2*d^ 2*e^2 + 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2 *b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*log(b*x + a) + 12*(b^ 4*e^4*x^4 + a^2*b^2*d^2*e^2 + 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x) *log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^ 5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4 *e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2 *e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3 *d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a ^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2*e ^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b ^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (128) = 256\).
Time = 1.33 (sec) , antiderivative size = 881, normalized size of antiderivative = 6.16 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {6 b^{2} e^{2} \log {\left (x + \frac {- \frac {6 a^{6} b^{2} e^{8}}{\left (a e - b d\right )^{5}} + \frac {36 a^{5} b^{3} d e^{7}}{\left (a e - b d\right )^{5}} - \frac {90 a^{4} b^{4} d^{2} e^{6}}{\left (a e - b d\right )^{5}} + \frac {120 a^{3} b^{5} d^{3} e^{5}}{\left (a e - b d\right )^{5}} - \frac {90 a^{2} b^{6} d^{4} e^{4}}{\left (a e - b d\right )^{5}} + \frac {36 a b^{7} d^{5} e^{3}}{\left (a e - b d\right )^{5}} + 6 a b^{2} e^{3} - \frac {6 b^{8} d^{6} e^{2}}{\left (a e - b d\right )^{5}} + 6 b^{3} d e^{2}}{12 b^{3} e^{3}} \right )}}{\left (a e - b d\right )^{5}} - \frac {6 b^{2} e^{2} \log {\left (x + \frac {\frac {6 a^{6} b^{2} e^{8}}{\left (a e - b d\right )^{5}} - \frac {36 a^{5} b^{3} d e^{7}}{\left (a e - b d\right )^{5}} + \frac {90 a^{4} b^{4} d^{2} e^{6}}{\left (a e - b d\right )^{5}} - \frac {120 a^{3} b^{5} d^{3} e^{5}}{\left (a e - b d\right )^{5}} + \frac {90 a^{2} b^{6} d^{4} e^{4}}{\left (a e - b d\right )^{5}} - \frac {36 a b^{7} d^{5} e^{3}}{\left (a e - b d\right )^{5}} + 6 a b^{2} e^{3} + \frac {6 b^{8} d^{6} e^{2}}{\left (a e - b d\right )^{5}} + 6 b^{3} d e^{2}}{12 b^{3} e^{3}} \right )}}{\left (a e - b d\right )^{5}} + \frac {- a^{3} e^{3} + 7 a^{2} b d e^{2} + 7 a b^{2} d^{2} e - b^{3} d^{3} + 12 b^{3} e^{3} x^{3} + x^{2} \cdot \left (18 a b^{2} e^{3} + 18 b^{3} d e^{2}\right ) + x \left (4 a^{2} b e^{3} + 28 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{2 a^{6} d^{2} e^{4} - 8 a^{5} b d^{3} e^{3} + 12 a^{4} b^{2} d^{4} e^{2} - 8 a^{3} b^{3} d^{5} e + 2 a^{2} b^{4} d^{6} + x^{4} \cdot \left (2 a^{4} b^{2} e^{6} - 8 a^{3} b^{3} d e^{5} + 12 a^{2} b^{4} d^{2} e^{4} - 8 a b^{5} d^{3} e^{3} + 2 b^{6} d^{4} e^{2}\right ) + x^{3} \cdot \left (4 a^{5} b e^{6} - 12 a^{4} b^{2} d e^{5} + 8 a^{3} b^{3} d^{2} e^{4} + 8 a^{2} b^{4} d^{3} e^{3} - 12 a b^{5} d^{4} e^{2} + 4 b^{6} d^{5} e\right ) + x^{2} \cdot \left (2 a^{6} e^{6} - 18 a^{4} b^{2} d^{2} e^{4} + 32 a^{3} b^{3} d^{3} e^{3} - 18 a^{2} b^{4} d^{4} e^{2} + 2 b^{6} d^{6}\right ) + x \left (4 a^{6} d e^{5} - 12 a^{5} b d^{2} e^{4} + 8 a^{4} b^{2} d^{3} e^{3} + 8 a^{3} b^{3} d^{4} e^{2} - 12 a^{2} b^{4} d^{5} e + 4 a b^{5} d^{6}\right )} \]
6*b**2*e**2*log(x + (-6*a**6*b**2*e**8/(a*e - b*d)**5 + 36*a**5*b**3*d*e** 7/(a*e - b*d)**5 - 90*a**4*b**4*d**2*e**6/(a*e - b*d)**5 + 120*a**3*b**5*d **3*e**5/(a*e - b*d)**5 - 90*a**2*b**6*d**4*e**4/(a*e - b*d)**5 + 36*a*b** 7*d**5*e**3/(a*e - b*d)**5 + 6*a*b**2*e**3 - 6*b**8*d**6*e**2/(a*e - b*d)* *5 + 6*b**3*d*e**2)/(12*b**3*e**3))/(a*e - b*d)**5 - 6*b**2*e**2*log(x + ( 6*a**6*b**2*e**8/(a*e - b*d)**5 - 36*a**5*b**3*d*e**7/(a*e - b*d)**5 + 90* a**4*b**4*d**2*e**6/(a*e - b*d)**5 - 120*a**3*b**5*d**3*e**5/(a*e - b*d)** 5 + 90*a**2*b**6*d**4*e**4/(a*e - b*d)**5 - 36*a*b**7*d**5*e**3/(a*e - b*d )**5 + 6*a*b**2*e**3 + 6*b**8*d**6*e**2/(a*e - b*d)**5 + 6*b**3*d*e**2)/(1 2*b**3*e**3))/(a*e - b*d)**5 + (-a**3*e**3 + 7*a**2*b*d*e**2 + 7*a*b**2*d* *2*e - b**3*d**3 + 12*b**3*e**3*x**3 + x**2*(18*a*b**2*e**3 + 18*b**3*d*e* *2) + x*(4*a**2*b*e**3 + 28*a*b**2*d*e**2 + 4*b**3*d**2*e))/(2*a**6*d**2*e **4 - 8*a**5*b*d**3*e**3 + 12*a**4*b**2*d**4*e**2 - 8*a**3*b**3*d**5*e + 2 *a**2*b**4*d**6 + x**4*(2*a**4*b**2*e**6 - 8*a**3*b**3*d*e**5 + 12*a**2*b* *4*d**2*e**4 - 8*a*b**5*d**3*e**3 + 2*b**6*d**4*e**2) + x**3*(4*a**5*b*e** 6 - 12*a**4*b**2*d*e**5 + 8*a**3*b**3*d**2*e**4 + 8*a**2*b**4*d**3*e**3 - 12*a*b**5*d**4*e**2 + 4*b**6*d**5*e) + x**2*(2*a**6*e**6 - 18*a**4*b**2*d* *2*e**4 + 32*a**3*b**3*d**3*e**3 - 18*a**2*b**4*d**4*e**2 + 2*b**6*d**6) + x*(4*a**6*d*e**5 - 12*a**5*b*d**2*e**4 + 8*a**4*b**2*d**3*e**3 + 8*a**3*b **3*d**4*e**2 - 12*a**2*b**4*d**5*e + 4*a*b**5*d**6))
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (139) = 278\).
Time = 0.21 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.15 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {6 \, b^{2} e^{2} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {6 \, b^{2} e^{2} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {12 \, b^{3} e^{3} x^{3} - b^{3} d^{3} + 7 \, a b^{2} d^{2} e + 7 \, a^{2} b d e^{2} - a^{3} e^{3} + 18 \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e + 7 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} + {\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \, {\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \]
6*b^2*e^2*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10* a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 6*b^2*e^2*log(e*x + d)/(b^5*d ^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e ^4 - a^5*e^5) + 1/2*(12*b^3*e^3*x^3 - b^3*d^3 + 7*a*b^2*d^2*e + 7*a^2*b*d* e^2 - a^3*e^3 + 18*(b^3*d*e^2 + a*b^2*e^3)*x^2 + 4*(b^3*d^2*e + 7*a*b^2*d* e^2 + a^2*b*e^3)*x)/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4 *a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4* d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4* e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b*e^6) *x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e ^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2 *a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (139) = 278\).
Time = 0.26 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.41 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {6 \, b^{3} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} - \frac {6 \, b^{2} e^{3} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} + \frac {12 \, b^{3} e^{3} x^{3} + 18 \, b^{3} d e^{2} x^{2} + 18 \, a b^{2} e^{3} x^{2} + 4 \, b^{3} d^{2} e x + 28 \, a b^{2} d e^{2} x + 4 \, a^{2} b e^{3} x - b^{3} d^{3} + 7 \, a b^{2} d^{2} e + 7 \, a^{2} b d e^{2} - a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (b e x^{2} + b d x + a e x + a d\right )}^{2}} \]
6*b^3*e^2*log(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5) - 6*b^2*e^3*log(abs(e* x + d))/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^2 *e^4 + 5*a^4*b*d*e^5 - a^5*e^6) + 1/2*(12*b^3*e^3*x^3 + 18*b^3*d*e^2*x^2 + 18*a*b^2*e^3*x^2 + 4*b^3*d^2*e*x + 28*a*b^2*d*e^2*x + 4*a^2*b*e^3*x - b^3 *d^3 + 7*a*b^2*d^2*e + 7*a^2*b*d*e^2 - a^3*e^3)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(b*e*x^2 + b*d*x + a*e*x + a*d)^2)
Time = 11.00 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.79 \[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {6\,b^3\,e^3\,x^3}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}-\frac {a^3\,e^3-7\,a^2\,b\,d\,e^2-7\,a\,b^2\,d^2\,e+b^3\,d^3}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {9\,b\,e\,x^2\,\left (d\,b^2\,e+a\,b\,e^2\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {2\,b\,e\,x\,\left (a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{x\,\left (2\,e\,a^2\,d+2\,b\,a\,d^2\right )+x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )+x^3\,\left (2\,d\,b^2\,e+2\,a\,b\,e^2\right )+a^2\,d^2+b^2\,e^2\,x^4}-\frac {12\,b^2\,e^2\,\mathrm {atanh}\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,e\,x\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )}{{\left (a\,e-b\,d\right )}^5} \]
((6*b^3*e^3*x^3)/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) - (a^3*e^3 + b^3*d^3 - 7*a*b^2*d^2*e - 7*a^2*b*d*e^2)/(2*(a ^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + ( 9*b*e*x^2*(a*b*e^2 + b^2*d*e))/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4* a*b^3*d^3*e - 4*a^3*b*d*e^3) + (2*b*e*x*(a^2*e^2 + b^2*d^2 + 7*a*b*d*e))/( a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(x *(2*a*b*d^2 + 2*a^2*d*e) + x^2*(a^2*e^2 + b^2*d^2 + 4*a*b*d*e) + x^3*(2*a* b*e^2 + 2*b^2*d*e) + a^2*d^2 + b^2*e^2*x^4) - (12*b^2*e^2*atanh((a^5*e^5 + b^5*d^5 + 2*a^2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 - 3*a*b^4*d^4*e - 3*a^4*b *d*e^4)/(a*e - b*d)^5 + (2*b*e*x*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(a*e - b*d)^5))/(a*e - b*d)^5