Integrand size = 31, antiderivative size = 199 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {A b-a B}{3 (b d-a e)^2 (a+b x)^3}-\frac {b B d-2 A b e+a B e}{2 (b d-a e)^3 (a+b x)^2}+\frac {e (2 b B d-3 A b e+a B e)}{(b d-a e)^4 (a+b x)}+\frac {e^2 (B d-A e)}{(b d-a e)^4 (d+e x)}+\frac {e^2 (3 b B d-4 A b e+a B e) \log (a+b x)}{(b d-a e)^5}-\frac {e^2 (3 b B d-4 A b e+a B e) \log (d+e x)}{(b d-a e)^5} \]
1/3*(-A*b+B*a)/(-a*e+b*d)^2/(b*x+a)^3+1/2*(2*A*b*e-B*a*e-B*b*d)/(-a*e+b*d) ^3/(b*x+a)^2+e*(-3*A*b*e+B*a*e+2*B*b*d)/(-a*e+b*d)^4/(b*x+a)+e^2*(-A*e+B*d )/(-a*e+b*d)^4/(e*x+d)+e^2*(-4*A*b*e+B*a*e+3*B*b*d)*ln(b*x+a)/(-a*e+b*d)^5 -e^2*(-4*A*b*e+B*a*e+3*B*b*d)*ln(e*x+d)/(-a*e+b*d)^5
Time = 0.08 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {2 (-A b+a B) (b d-a e)^3}{(a+b x)^3}-\frac {3 (b d-a e)^2 (b B d-2 A b e+a B e)}{(a+b x)^2}+\frac {6 e (-b d+a e) (-2 b B d+3 A b e-a B e)}{a+b x}+\frac {6 e^2 (-b d+a e) (-B d+A e)}{d+e x}+6 e^2 (3 b B d-4 A b e+a B e) \log (a+b x)-6 e^2 (3 b B d-4 A b e+a B e) \log (d+e x)}{6 (b d-a e)^5} \]
((2*(-(A*b) + a*B)*(b*d - a*e)^3)/(a + b*x)^3 - (3*(b*d - a*e)^2*(b*B*d - 2*A*b*e + a*B*e))/(a + b*x)^2 + (6*e*(-(b*d) + a*e)*(-2*b*B*d + 3*A*b*e - a*B*e))/(a + b*x) + (6*e^2*(-(b*d) + a*e)*(-(B*d) + A*e))/(d + e*x) + 6*e^ 2*(3*b*B*d - 4*A*b*e + a*B*e)*Log[a + b*x] - 6*e^2*(3*b*B*d - 4*A*b*e + a* B*e)*Log[d + e*x])/(6*(b*d - a*e)^5)
Time = 0.48 (sec) , antiderivative size = 199, 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, 86, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {A+B x}{b^4 (a+b x)^4 (d+e x)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {A+B x}{(a+b x)^4 (d+e x)^2}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^3 (-a B e+4 A b e-3 b B d)}{(d+e x) (b d-a e)^5}+\frac {e^3 (A e-B d)}{(d+e x)^2 (b d-a e)^4}-\frac {b e^2 (-a B e+4 A b e-3 b B d)}{(a+b x) (b d-a e)^5}+\frac {b e (-a B e+3 A b e-2 b B d)}{(a+b x)^2 (b d-a e)^4}+\frac {b (a B e-2 A b e+b B d)}{(a+b x)^3 (b d-a e)^3}+\frac {b (A b-a B)}{(a+b x)^4 (b d-a e)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 (B d-A e)}{(d+e x) (b d-a e)^4}+\frac {e^2 \log (a+b x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}-\frac {e^2 \log (d+e x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}+\frac {e (a B e-3 A b e+2 b B d)}{(a+b x) (b d-a e)^4}-\frac {a B e-2 A b e+b B d}{2 (a+b x)^2 (b d-a e)^3}-\frac {A b-a B}{3 (a+b x)^3 (b d-a e)^2}\) |
-1/3*(A*b - a*B)/((b*d - a*e)^2*(a + b*x)^3) - (b*B*d - 2*A*b*e + a*B*e)/( 2*(b*d - a*e)^3*(a + b*x)^2) + (e*(2*b*B*d - 3*A*b*e + a*B*e))/((b*d - a*e )^4*(a + b*x)) + (e^2*(B*d - A*e))/((b*d - a*e)^4*(d + e*x)) + (e^2*(3*b*B *d - 4*A*b*e + a*B*e)*Log[a + b*x])/(b*d - a*e)^5 - (e^2*(3*b*B*d - 4*A*b* e + a*B*e)*Log[d + e*x])/(b*d - a*e)^5
3.18.7.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.31 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(-\frac {2 A b e -B a e -B b d}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {A b -B a}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}}-\frac {e \left (3 A b e -B a e -2 B b d \right )}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {\left (A e -B d \right ) e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}\) | \(203\) |
| norman | \(\frac {-\frac {6 A \,a^{3} b^{3} e^{4}+26 A \,a^{2} b^{4} d \,e^{3}-10 A a \,b^{5} d^{2} e^{2}+2 A \,b^{6} d^{3} e -17 B \,a^{3} b^{3} d \,e^{3}-8 B \,a^{2} b^{4} d^{2} e^{2}+B a \,b^{5} d^{3} e}{6 e \,b^{3} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (4 A \,b^{4} e^{4}-B a \,b^{3} e^{4}-3 b^{4} B \,e^{3} d \right ) x^{3}}{e b \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (20 A a \,b^{4} e^{4}+4 A \,b^{5} d \,e^{3}-5 B \,a^{2} b^{3} e^{4}-16 B a \,b^{4} d \,e^{3}-3 B \,b^{5} d^{2} e^{2}\right ) x^{2}}{2 e \,b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (44 A \,a^{2} b^{4} e^{4}+32 A a \,b^{5} d \,e^{3}-4 A \,b^{6} d^{2} e^{2}-11 B \,a^{3} b^{3} e^{4}-41 B \,a^{2} b^{4} d \,e^{3}-23 B a \,b^{5} d^{2} e^{2}+3 B \,b^{6} d^{3} e \right ) x}{6 e \,b^{3} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right ) \left (b x +a \right )^{3}}+\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(703\) |
| risch | \(\frac {-\frac {b^{2} e^{2} \left (4 A b e -B a e -3 B b d \right ) 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 {b \left (5 a e +b d \right ) e \left (4 A b e -B a e -3 B b d \right ) x^{2}}{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 {\left (44 A \,a^{2} b \,e^{3}+32 A a \,b^{2} d \,e^{2}-4 A \,b^{3} d^{2} e -11 B \,e^{3} a^{3}-41 B \,a^{2} b d \,e^{2}-23 B a \,b^{2} d^{2} e +3 B \,b^{3} d^{3}\right ) x}{6 \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 {6 A \,a^{3} e^{3}+26 A \,a^{2} b d \,e^{2}-10 A a \,b^{2} d^{2} e +2 A \,b^{3} d^{3}-17 B \,a^{3} d \,e^{2}-8 B \,a^{2} b \,d^{2} e +B a \,b^{2} d^{3}}{6 \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 (b x +a \right ) \left (e x +d \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}+\frac {4 e^{3} \ln \left (-b x -a \right ) A b}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {e^{3} \ln \left (-b x -a \right ) B a}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {3 e^{2} \ln \left (-b x -a \right ) B b d}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {4 e^{3} \ln \left (e x +d \right ) A b}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {e^{3} \ln \left (e x +d \right ) B a}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {3 e^{2} \ln \left (e x +d \right ) B b d}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(930\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1364\) |
-1/2*(2*A*b*e-B*a*e-B*b*d)/(a*e-b*d)^3/(b*x+a)^2-1/3*(A*b-B*a)/(a*e-b*d)^2 /(b*x+a)^3-e*(3*A*b*e-B*a*e-2*B*b*d)/(a*e-b*d)^4/(b*x+a)+e^2*(4*A*b*e-B*a* e-3*B*b*d)/(a*e-b*d)^5*ln(b*x+a)-(A*e-B*d)*e^2/(a*e-b*d)^4/(e*x+d)-e^2*(4* A*b*e-B*a*e-3*B*b*d)/(a*e-b*d)^5*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1228 vs. \(2 (195) = 390\).
Time = 0.35 (sec) , antiderivative size = 1228, normalized size of antiderivative = 6.17 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]
1/6*(6*A*a^4*e^4 - (B*a*b^3 + 2*A*b^4)*d^4 + 3*(3*B*a^2*b^2 + 4*A*a*b^3)*d ^3*e + 9*(B*a^3*b - 4*A*a^2*b^2)*d^2*e^2 - (17*B*a^4 - 20*A*a^3*b)*d*e^3 + 6*(3*B*b^4*d^2*e^2 - 2*(B*a*b^3 + 2*A*b^4)*d*e^3 - (B*a^2*b^2 - 4*A*a*b^3 )*e^4)*x^3 + 3*(3*B*b^4*d^3*e + (13*B*a*b^3 - 4*A*b^4)*d^2*e^2 - (11*B*a^2 *b^2 + 16*A*a*b^3)*d*e^3 - 5*(B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 - (3*B*b^4*d ^4 - 2*(13*B*a*b^3 + 2*A*b^4)*d^3*e - 18*(B*a^2*b^2 - 2*A*a*b^3)*d^2*e^2 + 6*(5*B*a^3*b + 2*A*a^2*b^2)*d*e^3 + 11*(B*a^4 - 4*A*a^3*b)*e^4)*x + 6*(3* B*a^3*b*d^2*e^2 + (B*a^4 - 4*A*a^3*b)*d*e^3 + (3*B*b^4*d*e^3 + (B*a*b^3 - 4*A*b^4)*e^4)*x^4 + (3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 2*A*b^4)*d*e^3 + 3*( B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*B*a*b^3*d^2*e^2 + 4*(B*a^2*b^2 - A* a*b^3)*d*e^3 + (B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 + (9*B*a^2*b^2*d^2*e^2 + 6 *(B*a^3*b - 2*A*a^2*b^2)*d*e^3 + (B*a^4 - 4*A*a^3*b)*e^4)*x)*log(b*x + a) - 6*(3*B*a^3*b*d^2*e^2 + (B*a^4 - 4*A*a^3*b)*d*e^3 + (3*B*b^4*d*e^3 + (B*a *b^3 - 4*A*b^4)*e^4)*x^4 + (3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 2*A*b^4)*d*e^ 3 + 3*(B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*B*a*b^3*d^2*e^2 + 4*(B*a^2*b ^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 + (9*B*a^2*b^2*d^2* e^2 + 6*(B*a^3*b - 2*A*a^2*b^2)*d*e^3 + (B*a^4 - 4*A*a^3*b)*e^4)*x)*log(e* x + d))/(a^3*b^5*d^6 - 5*a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d ^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a ^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^...
Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (190) = 380\).
Time = 2.71 (sec) , antiderivative size = 1445, normalized size of antiderivative = 7.26 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]
e**2*(-4*A*b*e + B*a*e + 3*B*b*d)*log(x + (-4*A*a*b*e**4 - 4*A*b**2*d*e**3 + B*a**2*e**4 + 4*B*a*b*d*e**3 + 3*B*b**2*d**2*e**2 - a**6*e**8*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 6*a**5*b*d*e**7*(-4*A*b*e + B*a*e + 3 *B*b*d)/(a*e - b*d)**5 - 15*a**4*b**2*d**2*e**6*(-4*A*b*e + B*a*e + 3*B*b* d)/(a*e - b*d)**5 + 20*a**3*b**3*d**3*e**5*(-4*A*b*e + B*a*e + 3*B*b*d)/(a *e - b*d)**5 - 15*a**2*b**4*d**4*e**4*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 6*a*b**5*d**5*e**3*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - b**6*d**6*e**2*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5)/(-8*A*b**2*e* *4 + 2*B*a*b*e**4 + 6*B*b**2*d*e**3))/(a*e - b*d)**5 - e**2*(-4*A*b*e + B* a*e + 3*B*b*d)*log(x + (-4*A*a*b*e**4 - 4*A*b**2*d*e**3 + B*a**2*e**4 + 4* B*a*b*d*e**3 + 3*B*b**2*d**2*e**2 + a**6*e**8*(-4*A*b*e + B*a*e + 3*B*b*d) /(a*e - b*d)**5 - 6*a**5*b*d*e**7*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d) **5 + 15*a**4*b**2*d**2*e**6*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 20*a**3*b**3*d**3*e**5*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 15*a **2*b**4*d**4*e**4*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 6*a*b**5* d**5*e**3*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + b**6*d**6*e**2*(-4 *A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5)/(-8*A*b**2*e**4 + 2*B*a*b*e**4 + 6*B*b**2*d*e**3))/(a*e - b*d)**5 + (-6*A*a**3*e**3 - 26*A*a**2*b*d*e**2 + 10*A*a*b**2*d**2*e - 2*A*b**3*d**3 + 17*B*a**3*d*e**2 + 8*B*a**2*b*d**2*e - B*a*b**2*d**3 + x**3*(-24*A*b**3*e**3 + 6*B*a*b**2*e**3 + 18*B*b**3*...
Leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (195) = 390\).
Time = 0.23 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.80 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {{\left (3 \, B b d e^{2} + {\left (B a - 4 \, A b\right )} e^{3}\right )} \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 {{\left (3 \, B b d e^{2} + {\left (B a - 4 \, A b\right )} e^{3}\right )} \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 {6 \, A a^{3} e^{3} + {\left (B a b^{2} + 2 \, A b^{3}\right )} d^{3} - 2 \, {\left (4 \, B a^{2} b + 5 \, A a b^{2}\right )} d^{2} e - {\left (17 \, B a^{3} - 26 \, A a^{2} b\right )} d e^{2} - 6 \, {\left (3 \, B b^{3} d e^{2} + {\left (B a b^{2} - 4 \, A b^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, B b^{3} d^{2} e + 4 \, {\left (4 \, B a b^{2} - A b^{3}\right )} d e^{2} + 5 \, {\left (B a^{2} b - 4 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (3 \, B b^{3} d^{3} - {\left (23 \, B a b^{2} + 4 \, A b^{3}\right )} d^{2} e - {\left (41 \, B a^{2} b - 32 \, A a b^{2}\right )} d e^{2} - 11 \, {\left (B a^{3} - 4 \, A a^{2} b\right )} e^{3}\right )} x}{6 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}} \]
(3*B*b*d*e^2 + (B*a - 4*A*b)*e^3)*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) - (3*B* b*d*e^2 + (B*a - 4*A*b)*e^3)*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) - 1/6*(6*A*a ^3*e^3 + (B*a*b^2 + 2*A*b^3)*d^3 - 2*(4*B*a^2*b + 5*A*a*b^2)*d^2*e - (17*B *a^3 - 26*A*a^2*b)*d*e^2 - 6*(3*B*b^3*d*e^2 + (B*a*b^2 - 4*A*b^3)*e^3)*x^3 - 3*(3*B*b^3*d^2*e + 4*(4*B*a*b^2 - A*b^3)*d*e^2 + 5*(B*a^2*b - 4*A*a*b^2 )*e^3)*x^2 + (3*B*b^3*d^3 - (23*B*a*b^2 + 4*A*b^3)*d^2*e - (41*B*a^2*b - 3 2*A*a*b^2)*d*e^2 - 11*(B*a^3 - 4*A*a^2*b)*e^3)*x)/(a^3*b^4*d^5 - 4*a^4*b^3 *d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4* a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + ( b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 - 11*a^4*b^ 3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4* d^3*e^2 + 2*a^4*b^3*d^2*e^3 - 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^ 5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e^3 - a^6*b* d*e^4 + a^7*e^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (195) = 390\).
Time = 0.29 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.17 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {{\left (3 \, B b d e^{3} + B a e^{4} - 4 \, A b e^{4}\right )} \log \left ({\left | b - \frac {b d}{e x + d} + \frac {a e}{e x + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} + \frac {\frac {B d e^{6}}{e x + d} - \frac {A e^{7}}{e x + d}}{b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}} + \frac {15 \, B b^{4} d e^{2} + 11 \, B a b^{3} e^{3} - 26 \, A b^{4} e^{3} - \frac {3 \, {\left (11 \, B b^{4} d^{2} e^{3} - 2 \, B a b^{3} d e^{4} - 20 \, A b^{4} d e^{4} - 9 \, B a^{2} b^{2} e^{5} + 20 \, A a b^{3} e^{5}\right )}}{{\left (e x + d\right )} e} + \frac {18 \, {\left (B b^{4} d^{3} e^{4} - B a b^{3} d^{2} e^{5} - 2 \, A b^{4} d^{2} e^{5} - B a^{2} b^{2} d e^{6} + 4 \, A a b^{3} d e^{6} + B a^{3} b e^{7} - 2 \, A a^{2} b^{2} e^{7}\right )}}{{\left (e x + d\right )}^{2} e^{2}}}{6 \, {\left (b d - a e\right )}^{5} {\left (b - \frac {b d}{e x + d} + \frac {a e}{e x + d}\right )}^{3}} \]
(3*B*b*d*e^3 + B*a*e^4 - 4*A*b*e^4)*log(abs(b - b*d/(e*x + d) + a*e/(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) + (B*d*e^6/(e*x + d) - A*e^7/(e*x + d))/(b^4 *d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8) + 1/6*(15*B*b^4*d*e^2 + 11*B*a*b^3*e^3 - 26*A*b^4*e^3 - 3*(11*B*b^4*d^2*e^ 3 - 2*B*a*b^3*d*e^4 - 20*A*b^4*d*e^4 - 9*B*a^2*b^2*e^5 + 20*A*a*b^3*e^5)/( (e*x + d)*e) + 18*(B*b^4*d^3*e^4 - B*a*b^3*d^2*e^5 - 2*A*b^4*d^2*e^5 - B*a ^2*b^2*d*e^6 + 4*A*a*b^3*d*e^6 + B*a^3*b*e^7 - 2*A*a^2*b^2*e^7)/((e*x + d) ^2*e^2))/((b*d - a*e)^5*(b - b*d/(e*x + d) + a*e/(e*x + d))^3)
Time = 11.29 (sec) , antiderivative size = 711, normalized size of antiderivative = 3.57 \[ \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {x\,\left (11\,a^2\,e^2+8\,a\,b\,d\,e-b^2\,d^2\right )\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{6\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}-\frac {-17\,B\,a^3\,d\,e^2+6\,A\,a^3\,e^3-8\,B\,a^2\,b\,d^2\,e+26\,A\,a^2\,b\,d\,e^2+B\,a\,b^2\,d^3-10\,A\,a\,b^2\,d^2\,e+2\,A\,b^3\,d^3}{6\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {b^2\,e^2\,x^3\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {e\,x^2\,\left (d\,b^2+5\,a\,e\,b\right )\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}}{x^3\,\left (d\,b^3+3\,a\,e\,b^2\right )+x^2\,\left (3\,e\,a^2\,b+3\,d\,a\,b^2\right )+a^3\,d+x\,\left (e\,a^3+3\,b\,d\,a^2\right )+b^3\,e\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (e^3\,\left (4\,A\,b-B\,a\right )-3\,B\,b\,d\,e^2\right )\,\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+2\,b\,e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5\,\left (B\,a\,e^3-4\,A\,b\,e^3+3\,B\,b\,d\,e^2\right )}\right )\,\left (e^3\,\left (4\,A\,b-B\,a\right )-3\,B\,b\,d\,e^2\right )}{{\left (a\,e-b\,d\right )}^5} \]
((x*(11*a^2*e^2 - b^2*d^2 + 8*a*b*d*e)*(B*a*e - 4*A*b*e + 3*B*b*d))/(6*(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)) - (6 *A*a^3*e^3 + 2*A*b^3*d^3 + B*a*b^2*d^3 - 17*B*a^3*d*e^2 - 10*A*a*b^2*d^2*e + 26*A*a^2*b*d*e^2 - 8*B*a^2*b*d^2*e)/(6*(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)) + (b^2*e^2*x^3*(B*a*e - 4*A*b*e + 3*B*b*d))/(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) + (e*x^2*(b^2*d + 5*a*b*e)*(B*a*e - 4*A*b*e + 3*B*b*d))/(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)))/(x^3*( b^3*d + 3*a*b^2*e) + x^2*(3*a*b^2*d + 3*a^2*b*e) + a^3*d + x*(a^3*e + 3*a^ 2*b*d) + b^3*e*x^4) - (2*atanh(((e^3*(4*A*b - B*a) - 3*B*b*d*e^2)*((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^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^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*(B*a*e^3 - 4*A*b*e^3 + 3*B*b*d*e^2)))*(e ^3*(4*A*b - B*a) - 3*B*b*d*e^2))/(a*e - b*d)^5