Integrand size = 20, antiderivative size = 248 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=-\frac {b^2 (A b-a B)}{2 (b d-a e)^4 (a+b x)^2}-\frac {b^2 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)}-\frac {e (B d-A e)}{3 (b d-a e)^3 (d+e x)^3}-\frac {e (2 b B d-3 A b e+a B e)}{2 (b d-a e)^4 (d+e x)^2}-\frac {3 b e (b B d-2 A b e+a B e)}{(b d-a e)^5 (d+e x)}-\frac {2 b^2 e (2 b B d-5 A b e+3 a B e) \log (a+b x)}{(b d-a e)^6}+\frac {2 b^2 e (2 b B d-5 A b e+3 a B e) \log (d+e x)}{(b d-a e)^6} \]
-1/2*b^2*(A*b-B*a)/(-a*e+b*d)^4/(b*x+a)^2-b^2*(-4*A*b*e+3*B*a*e+B*b*d)/(-a *e+b*d)^5/(b*x+a)-1/3*e*(-A*e+B*d)/(-a*e+b*d)^3/(e*x+d)^3-1/2*e*(-3*A*b*e+ B*a*e+2*B*b*d)/(-a*e+b*d)^4/(e*x+d)^2-3*b*e*(-2*A*b*e+B*a*e+B*b*d)/(-a*e+b *d)^5/(e*x+d)-2*b^2*e*(-5*A*b*e+3*B*a*e+2*B*b*d)*ln(b*x+a)/(-a*e+b*d)^6+2* b^2*e*(-5*A*b*e+3*B*a*e+2*B*b*d)*ln(e*x+d)/(-a*e+b*d)^6
Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=\frac {-\frac {3 b^2 (A b-a B) (b d-a e)^2}{(a+b x)^2}-\frac {6 b^2 (b d-a e) (b B d-4 A b e+3 a B e)}{a+b x}+\frac {2 e (b d-a e)^3 (-B d+A e)}{(d+e x)^3}+\frac {3 e (b d-a e)^2 (-2 b B d+3 A b e-a B e)}{(d+e x)^2}+\frac {18 b e (-b d+a e) (b B d-2 A b e+a B e)}{d+e x}+12 b^2 e (-2 b B d+5 A b e-3 a B e) \log (a+b x)+12 b^2 e (2 b B d-5 A b e+3 a B e) \log (d+e x)}{6 (b d-a e)^6} \]
((-3*b^2*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^2 - (6*b^2*(b*d - a*e)*(b*B* d - 4*A*b*e + 3*a*B*e))/(a + b*x) + (2*e*(b*d - a*e)^3*(-(B*d) + A*e))/(d + e*x)^3 + (3*e*(b*d - a*e)^2*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x)^2 + (18*b*e*(-(b*d) + a*e)*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x) + 12*b^2*e*(-2 *b*B*d + 5*A*b*e - 3*a*B*e)*Log[a + b*x] + 12*b^2*e*(2*b*B*d - 5*A*b*e + 3 *a*B*e)*Log[d + e*x])/(6*(b*d - a*e)^6)
Time = 0.51 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
| \(\displaystyle \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {2 b^3 e (-3 a B e+5 A b e-2 b B d)}{(a+b x) (b d-a e)^6}+\frac {b^3 (3 a B e-4 A b e+b B d)}{(a+b x)^2 (b d-a e)^5}+\frac {b^3 (A b-a B)}{(a+b x)^3 (b d-a e)^4}-\frac {2 b^2 e^2 (-3 a B e+5 A b e-2 b B d)}{(d+e x) (b d-a e)^6}-\frac {3 b e^2 (-a B e+2 A b e-b B d)}{(d+e x)^2 (b d-a e)^5}-\frac {e^2 (-a B e+3 A b e-2 b B d)}{(d+e x)^3 (b d-a e)^4}-\frac {e^2 (A e-B d)}{(d+e x)^4 (b d-a e)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^2 (3 a B e-4 A b e+b B d)}{(a+b x) (b d-a e)^5}-\frac {b^2 (A b-a B)}{2 (a+b x)^2 (b d-a e)^4}-\frac {2 b^2 e \log (a+b x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}+\frac {2 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}-\frac {3 b e (a B e-2 A b e+b B d)}{(d+e x) (b d-a e)^5}-\frac {e (a B e-3 A b e+2 b B d)}{2 (d+e x)^2 (b d-a e)^4}-\frac {e (B d-A e)}{3 (d+e x)^3 (b d-a e)^3}\) |
-1/2*(b^2*(A*b - a*B))/((b*d - a*e)^4*(a + b*x)^2) - (b^2*(b*B*d - 4*A*b*e + 3*a*B*e))/((b*d - a*e)^5*(a + b*x)) - (e*(B*d - A*e))/(3*(b*d - a*e)^3* (d + e*x)^3) - (e*(2*b*B*d - 3*A*b*e + a*B*e))/(2*(b*d - a*e)^4*(d + e*x)^ 2) - (3*b*e*(b*B*d - 2*A*b*e + a*B*e))/((b*d - a*e)^5*(d + e*x)) - (2*b^2* e*(2*b*B*d - 5*A*b*e + 3*a*B*e)*Log[a + b*x])/(b*d - a*e)^6 + (2*b^2*e*(2* b*B*d - 5*A*b*e + 3*a*B*e)*Log[d + e*x])/(b*d - a*e)^6
3.12.42.3.1 Defintions of rubi rules used
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]))))
Time = 0.87 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {b^{2} \left (4 A b e -3 B a e -B b d \right )}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {\left (A b -B a \right ) b^{2}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}+\frac {2 b^{2} e \left (5 A b e -3 B a e -2 B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {\left (A e -B d \right ) e}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}}+\frac {e \left (3 A b e -B a e -2 B b d \right )}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}-\frac {3 e b \left (2 A b e -B a e -B b d \right )}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {2 b^{2} e \left (5 A b e -3 B a e -2 B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}\) | \(247\) |
| norman | \(\text {Expression too large to display}\) | \(1021\) |
| risch | \(\text {Expression too large to display}\) | \(1271\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1982\) |
-b^2*(4*A*b*e-3*B*a*e-B*b*d)/(a*e-b*d)^5/(b*x+a)-1/2*(A*b-B*a)*b^2/(a*e-b* d)^4/(b*x+a)^2+2*b^2*e*(5*A*b*e-3*B*a*e-2*B*b*d)/(a*e-b*d)^6*ln(b*x+a)-1/3 *(A*e-B*d)*e/(a*e-b*d)^3/(e*x+d)^3+1/2*e*(3*A*b*e-B*a*e-2*B*b*d)/(a*e-b*d) ^4/(e*x+d)^2-3*e*b*(2*A*b*e-B*a*e-B*b*d)/(a*e-b*d)^5/(e*x+d)-2*b^2*e*(5*A* b*e-3*B*a*e-2*B*b*d)/(a*e-b*d)^6*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1845 vs. \(2 (242) = 484\).
Time = 0.31 (sec) , antiderivative size = 1845, normalized size of antiderivative = 7.44 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=\text {Too large to display} \]
-1/6*(2*A*a^5*e^5 + 3*(B*a*b^4 + A*b^5)*d^5 + 2*(22*B*a^2*b^3 - 15*A*a*b^4 )*d^4*e - 4*(9*B*a^3*b^2 + 5*A*a^2*b^3)*d^3*e^2 - 12*(B*a^4*b - 5*A*a^3*b^ 2)*d^2*e^3 + (B*a^5 - 15*A*a^4*b)*d*e^4 + 12*(2*B*b^5*d^2*e^3 + (B*a*b^4 - 5*A*b^5)*d*e^4 - (3*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + 6*(10*B*b^5*d^3*e^2 + (11*B*a*b^4 - 25*A*b^5)*d^2*e^3 - 2*(6*B*a^2*b^3 - 5*A*a*b^4)*d*e^4 - 3 *(3*B*a^3*b^2 - 5*A*a^2*b^3)*e^5)*x^3 + 2*(22*B*b^5*d^4*e + (57*B*a*b^4 - 55*A*b^5)*d^3*e^2 - 6*(B*a^2*b^3 + 10*A*a*b^4)*d^2*e^3 - (67*B*a^3*b^2 - 1 05*A*a^2*b^3)*d*e^4 - 2*(3*B*a^4*b - 5*A*a^3*b^2)*e^5)*x^2 + (6*B*b^5*d^5 + (73*B*a*b^4 - 15*A*b^5)*d^4*e + 16*(3*B*a^2*b^3 - 10*A*a*b^4)*d^3*e^2 - 24*(4*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^3 - 2*(17*B*a^4*b - 30*A*a^3*b^2)*d*e ^4 + (3*B*a^5 - 5*A*a^4*b)*e^5)*x + 12*(2*B*a^2*b^3*d^4*e + (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^3*e^2 + (2*B*b^5*d*e^4 + (3*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (6*B*b^5*d^2*e^3 + (13*B*a*b^4 - 15*A*b^5)*d*e^4 + 2*(3*B*a^2*b^3 - 5*A*a* b^4)*e^5)*x^4 + (6*B*b^5*d^3*e^2 + 3*(7*B*a*b^4 - 5*A*b^5)*d^2*e^3 + 10*(2 *B*a^2*b^3 - 3*A*a*b^4)*d*e^4 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*e^5)*x^3 + (2* B*b^5*d^4*e + 5*(3*B*a*b^4 - A*b^5)*d^3*e^2 + 6*(4*B*a^2*b^3 - 5*A*a*b^4)* d^2*e^3 + 3*(3*B*a^3*b^2 - 5*A*a^2*b^3)*d*e^4)*x^2 + (4*B*a*b^4*d^4*e + 2* (6*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2 + 3*(3*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^3) *x)*log(b*x + a) - 12*(2*B*a^2*b^3*d^4*e + (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^3 *e^2 + (2*B*b^5*d*e^4 + (3*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (6*B*b^5*d^2*e...
Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (245) = 490\).
Time = 4.25 (sec) , antiderivative size = 1975, normalized size of antiderivative = 7.96 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=\text {Too large to display} \]
2*b**2*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)*log(x + (-10*A*a*b**3*e**3 - 10*A* b**4*d*e**2 + 6*B*a**2*b**2*e**3 + 10*B*a*b**3*d*e**2 + 4*B*b**4*d**2*e - 2*a**7*b**2*e**8*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 14*a**6*b **3*d*e**7*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 42*a**5*b**4*d* *2*e**6*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 70*a**4*b**5*d**3* e**5*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 70*a**3*b**6*d**4*e** 4*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 42*a**2*b**7*d**5*e**3*( -5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 14*a*b**8*d**6*e**2*(-5*A*b *e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 2*b**9*d**7*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6)/(-20*A*b**4*e**3 + 12*B*a*b**3*e**3 + 8*B*b**4 *d*e**2))/(a*e - b*d)**6 - 2*b**2*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)*log(x + (-10*A*a*b**3*e**3 - 10*A*b**4*d*e**2 + 6*B*a**2*b**2*e**3 + 10*B*a*b**3* d*e**2 + 4*B*b**4*d**2*e + 2*a**7*b**2*e**8*(-5*A*b*e + 3*B*a*e + 2*B*b*d) /(a*e - b*d)**6 - 14*a**6*b**3*d*e**7*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 42*a**5*b**4*d**2*e**6*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b *d)**6 - 70*a**4*b**5*d**3*e**5*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d) **6 + 70*a**3*b**6*d**4*e**4*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 42*a**2*b**7*d**5*e**3*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 14*a*b**8*d**6*e**2*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 2*b**9 *d**7*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6)/(-20*A*b**4*e**3...
Leaf count of result is larger than twice the leaf count of optimal. 1126 vs. \(2 (242) = 484\).
Time = 0.26 (sec) , antiderivative size = 1126, normalized size of antiderivative = 4.54 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=-\frac {2 \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {2 \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {2 \, A a^{4} e^{4} - 3 \, {\left (B a b^{3} + A b^{4}\right )} d^{4} - {\left (47 \, B a^{2} b^{2} - 27 \, A a b^{3}\right )} d^{3} e - {\left (11 \, B a^{3} b - 47 \, A a^{2} b^{2}\right )} d^{2} e^{2} + {\left (B a^{4} - 13 \, A a^{3} b\right )} d e^{3} - 12 \, {\left (2 \, B b^{4} d e^{3} + {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} e^{4}\right )} x^{4} - 6 \, {\left (10 \, B b^{4} d^{2} e^{2} + {\left (21 \, B a b^{3} - 25 \, A b^{4}\right )} d e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} e^{4}\right )} x^{3} - 2 \, {\left (22 \, B b^{4} d^{3} e + {\left (79 \, B a b^{3} - 55 \, A b^{4}\right )} d^{2} e^{2} + {\left (73 \, B a^{2} b^{2} - 115 \, A a b^{3}\right )} d e^{3} + 2 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} - {\left (6 \, B b^{4} d^{4} + {\left (79 \, B a b^{3} - 15 \, A b^{4}\right )} d^{3} e + {\left (127 \, B a^{2} b^{2} - 175 \, A a b^{3}\right )} d^{2} e^{2} + {\left (31 \, B a^{3} b - 55 \, A a^{2} b^{2}\right )} d e^{3} - {\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} e^{4}\right )} x}{6 \, {\left (a^{2} b^{5} d^{8} - 5 \, a^{3} b^{4} d^{7} e + 10 \, a^{4} b^{3} d^{6} e^{2} - 10 \, a^{5} b^{2} d^{5} e^{3} + 5 \, a^{6} b d^{4} e^{4} - a^{7} d^{3} e^{5} + {\left (b^{7} d^{5} e^{3} - 5 \, a b^{6} d^{4} e^{4} + 10 \, a^{2} b^{5} d^{3} e^{5} - 10 \, a^{3} b^{4} d^{2} e^{6} + 5 \, a^{4} b^{3} d e^{7} - a^{5} b^{2} e^{8}\right )} x^{5} + {\left (3 \, b^{7} d^{6} e^{2} - 13 \, a b^{6} d^{5} e^{3} + 20 \, a^{2} b^{5} d^{4} e^{4} - 10 \, a^{3} b^{4} d^{3} e^{5} - 5 \, a^{4} b^{3} d^{2} e^{6} + 7 \, a^{5} b^{2} d e^{7} - 2 \, a^{6} b e^{8}\right )} x^{4} + {\left (3 \, b^{7} d^{7} e - 9 \, a b^{6} d^{6} e^{2} + a^{2} b^{5} d^{5} e^{3} + 25 \, a^{3} b^{4} d^{4} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{5} + 17 \, a^{5} b^{2} d^{2} e^{6} - a^{6} b d e^{7} - a^{7} e^{8}\right )} x^{3} + {\left (b^{7} d^{8} + a b^{6} d^{7} e - 17 \, a^{2} b^{5} d^{6} e^{2} + 35 \, a^{3} b^{4} d^{5} e^{3} - 25 \, a^{4} b^{3} d^{4} e^{4} - a^{5} b^{2} d^{3} e^{5} + 9 \, a^{6} b d^{2} e^{6} - 3 \, a^{7} d e^{7}\right )} x^{2} + {\left (2 \, a b^{6} d^{8} - 7 \, a^{2} b^{5} d^{7} e + 5 \, a^{3} b^{4} d^{6} e^{2} + 10 \, a^{4} b^{3} d^{5} e^{3} - 20 \, a^{5} b^{2} d^{4} e^{4} + 13 \, a^{6} b d^{3} e^{5} - 3 \, a^{7} d^{2} e^{6}\right )} x\right )}} \]
-2*(2*B*b^3*d*e + (3*B*a*b^2 - 5*A*b^3)*e^2)*log(b*x + a)/(b^6*d^6 - 6*a*b ^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 2*(2*B*b^3*d*e + (3*B*a*b^2 - 5*A*b^3)*e^2)*log (e*x + d)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e ^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 1/6*(2*A*a^4*e^4 - 3* (B*a*b^3 + A*b^4)*d^4 - (47*B*a^2*b^2 - 27*A*a*b^3)*d^3*e - (11*B*a^3*b - 47*A*a^2*b^2)*d^2*e^2 + (B*a^4 - 13*A*a^3*b)*d*e^3 - 12*(2*B*b^4*d*e^3 + ( 3*B*a*b^3 - 5*A*b^4)*e^4)*x^4 - 6*(10*B*b^4*d^2*e^2 + (21*B*a*b^3 - 25*A*b ^4)*d*e^3 + 3*(3*B*a^2*b^2 - 5*A*a*b^3)*e^4)*x^3 - 2*(22*B*b^4*d^3*e + (79 *B*a*b^3 - 55*A*b^4)*d^2*e^2 + (73*B*a^2*b^2 - 115*A*a*b^3)*d*e^3 + 2*(3*B *a^3*b - 5*A*a^2*b^2)*e^4)*x^2 - (6*B*b^4*d^4 + (79*B*a*b^3 - 15*A*b^4)*d^ 3*e + (127*B*a^2*b^2 - 175*A*a*b^3)*d^2*e^2 + (31*B*a^3*b - 55*A*a^2*b^2)* d*e^3 - (3*B*a^4 - 5*A*a^3*b)*e^4)*x)/(a^2*b^5*d^8 - 5*a^3*b^4*d^7*e + 10* a^4*b^3*d^6*e^2 - 10*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^ 7*d^5*e^3 - 5*a*b^6*d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5* a^4*b^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20* a^2*b^5*d^4*e^4 - 10*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^4 + (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 2 5*a^3*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b...
Leaf count of result is larger than twice the leaf count of optimal. 800 vs. \(2 (242) = 484\).
Time = 0.28 (sec) , antiderivative size = 800, normalized size of antiderivative = 3.23 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=-\frac {2 \, {\left (2 \, B b^{4} d e + 3 \, B a b^{3} e^{2} - 5 \, A b^{4} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac {2 \, {\left (2 \, B b^{3} d e^{2} + 3 \, B a b^{2} e^{3} - 5 \, A b^{3} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac {3 \, B a b^{4} d^{5} + 3 \, A b^{5} d^{5} + 44 \, B a^{2} b^{3} d^{4} e - 30 \, A a b^{4} d^{4} e - 36 \, B a^{3} b^{2} d^{3} e^{2} - 20 \, A a^{2} b^{3} d^{3} e^{2} - 12 \, B a^{4} b d^{2} e^{3} + 60 \, A a^{3} b^{2} d^{2} e^{3} + B a^{5} d e^{4} - 15 \, A a^{4} b d e^{4} + 2 \, A a^{5} e^{5} + 12 \, {\left (2 \, B b^{5} d^{2} e^{3} + B a b^{4} d e^{4} - 5 \, A b^{5} d e^{4} - 3 \, B a^{2} b^{3} e^{5} + 5 \, A a b^{4} e^{5}\right )} x^{4} + 6 \, {\left (10 \, B b^{5} d^{3} e^{2} + 11 \, B a b^{4} d^{2} e^{3} - 25 \, A b^{5} d^{2} e^{3} - 12 \, B a^{2} b^{3} d e^{4} + 10 \, A a b^{4} d e^{4} - 9 \, B a^{3} b^{2} e^{5} + 15 \, A a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (22 \, B b^{5} d^{4} e + 57 \, B a b^{4} d^{3} e^{2} - 55 \, A b^{5} d^{3} e^{2} - 6 \, B a^{2} b^{3} d^{2} e^{3} - 60 \, A a b^{4} d^{2} e^{3} - 67 \, B a^{3} b^{2} d e^{4} + 105 \, A a^{2} b^{3} d e^{4} - 6 \, B a^{4} b e^{5} + 10 \, A a^{3} b^{2} e^{5}\right )} x^{2} + {\left (6 \, B b^{5} d^{5} + 73 \, B a b^{4} d^{4} e - 15 \, A b^{5} d^{4} e + 48 \, B a^{2} b^{3} d^{3} e^{2} - 160 \, A a b^{4} d^{3} e^{2} - 96 \, B a^{3} b^{2} d^{2} e^{3} + 120 \, A a^{2} b^{3} d^{2} e^{3} - 34 \, B a^{4} b d e^{4} + 60 \, A a^{3} b^{2} d e^{4} + 3 \, B a^{5} e^{5} - 5 \, A a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{2} {\left (e x + d\right )}^{3}} \]
-2*(2*B*b^4*d*e + 3*B*a*b^3*e^2 - 5*A*b^4*e^2)*log(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2 *e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + 2*(2*B*b^3*d*e^2 + 3*B*a*b^2*e^3 - 5 *A*b^3*e^3)*log(abs(e*x + d))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^4*d^ 4*e^3 - 20*a^3*b^3*d^3*e^4 + 15*a^4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) - 1/6*(3*B*a*b^4*d^5 + 3*A*b^5*d^5 + 44*B*a^2*b^3*d^4*e - 30*A*a*b^4*d^4* e - 36*B*a^3*b^2*d^3*e^2 - 20*A*a^2*b^3*d^3*e^2 - 12*B*a^4*b*d^2*e^3 + 60* A*a^3*b^2*d^2*e^3 + B*a^5*d*e^4 - 15*A*a^4*b*d*e^4 + 2*A*a^5*e^5 + 12*(2*B *b^5*d^2*e^3 + B*a*b^4*d*e^4 - 5*A*b^5*d*e^4 - 3*B*a^2*b^3*e^5 + 5*A*a*b^4 *e^5)*x^4 + 6*(10*B*b^5*d^3*e^2 + 11*B*a*b^4*d^2*e^3 - 25*A*b^5*d^2*e^3 - 12*B*a^2*b^3*d*e^4 + 10*A*a*b^4*d*e^4 - 9*B*a^3*b^2*e^5 + 15*A*a^2*b^3*e^5 )*x^3 + 2*(22*B*b^5*d^4*e + 57*B*a*b^4*d^3*e^2 - 55*A*b^5*d^3*e^2 - 6*B*a^ 2*b^3*d^2*e^3 - 60*A*a*b^4*d^2*e^3 - 67*B*a^3*b^2*d*e^4 + 105*A*a^2*b^3*d* e^4 - 6*B*a^4*b*e^5 + 10*A*a^3*b^2*e^5)*x^2 + (6*B*b^5*d^5 + 73*B*a*b^4*d^ 4*e - 15*A*b^5*d^4*e + 48*B*a^2*b^3*d^3*e^2 - 160*A*a*b^4*d^3*e^2 - 96*B*a ^3*b^2*d^2*e^3 + 120*A*a^2*b^3*d^2*e^3 - 34*B*a^4*b*d*e^4 + 60*A*a^3*b^2*d *e^4 + 3*B*a^5*e^5 - 5*A*a^4*b*e^5)*x)/((b*d - a*e)^6*(b*x + a)^2*(e*x + d )^3)
Time = 2.23 (sec) , antiderivative size = 1030, normalized size of antiderivative = 4.15 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^4} \, dx=\frac {\frac {-B\,a^4\,d\,e^3-2\,A\,a^4\,e^4+11\,B\,a^3\,b\,d^2\,e^2+13\,A\,a^3\,b\,d\,e^3+47\,B\,a^2\,b^2\,d^3\,e-47\,A\,a^2\,b^2\,d^2\,e^2+3\,B\,a\,b^3\,d^4-27\,A\,a\,b^3\,d^3\,e+3\,A\,b^4\,d^4}{6\,\left (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\right )}+\frac {x\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )\,\left (-a^3\,e^3+11\,a^2\,b\,d\,e^2+35\,a\,b^2\,d^2\,e+3\,b^3\,d^3\right )}{6\,\left (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\right )}+\frac {2\,b^3\,e^3\,x^4\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,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}+\frac {b\,x^3\,\left (5\,d\,b^2\,e^2+3\,a\,b\,e^3\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,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}+\frac {b\,x^2\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )\,\left (2\,a^2\,e^3+23\,a\,b\,d\,e^2+11\,b^2\,d^2\,e\right )}{3\,\left (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\right )}}{x^2\,\left (3\,a^2\,d\,e^2+6\,a\,b\,d^2\,e+b^2\,d^3\right )+x^3\,\left (a^2\,e^3+6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )+x\,\left (3\,e\,a^2\,d^2+2\,b\,a\,d^3\right )+x^4\,\left (3\,d\,b^2\,e^2+2\,a\,b\,e^3\right )+a^2\,d^3+b^2\,e^3\,x^5}-\frac {2\,\mathrm {atanh}\left (\frac {\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )\,\left (a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^6\,\left (-10\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+6\,B\,a\,b^2\,e^2\right )}+\frac {2\,b\,e\,x\,\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )\,\left (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\right )}{{\left (a\,e-b\,d\right )}^6\,\left (-10\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+6\,B\,a\,b^2\,e^2\right )}\right )\,\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )}{{\left (a\,e-b\,d\right )}^6} \]
((3*A*b^4*d^4 - 2*A*a^4*e^4 + 3*B*a*b^3*d^4 - B*a^4*d*e^3 + 47*B*a^2*b^2*d ^3*e + 11*B*a^3*b*d^2*e^2 - 47*A*a^2*b^2*d^2*e^2 - 27*A*a*b^3*d^3*e + 13*A *a^3*b*d*e^3)/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2* e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (x*(3*B*a*e - 5*A*b*e + 2*B*b*d)*( 3*b^3*d^3 - a^3*e^3 + 35*a*b^2*d^2*e + 11*a^2*b*d*e^2))/(6*(a^5*e^5 - b^5* d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d* e^4)) + (2*b^3*e^3*x^4*(3*B*a*e - 5*A*b*e + 2*B*b*d))/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (b*x^3*(5*b^2*d*e^2 + 3*a*b*e^3)*(3*B*a*e - 5*A*b*e + 2*B*b*d))/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a ^4*b*d*e^4) + (b*x^2*(3*B*a*e - 5*A*b*e + 2*B*b*d)*(2*a^2*e^3 + 11*b^2*d^2 *e + 23*a*b*d*e^2))/(3*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^ 2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)))/(x^2*(b^2*d^3 + 3*a^2*d*e^2 + 6*a*b*d^2*e) + x^3*(a^2*e^3 + 3*b^2*d^2*e + 6*a*b*d*e^2) + x*(3*a^2*d^2*e + 2*a*b*d^3) + x^4*(3*b^2*d*e^2 + 2*a*b*e^3) + a^2*d^3 + b^2*e^3*x^5) - ( 2*atanh(((2*b^2*e^2*(5*A*b - 3*B*a) - 4*B*b^3*d*e)*(a^6*e^6 - b^6*d^6 - 5* a^2*b^4*d^4*e^2 + 5*a^4*b^2*d^2*e^4 + 4*a*b^5*d^5*e - 4*a^5*b*d*e^5))/((a* e - b*d)^6*(4*B*b^3*d*e - 10*A*b^3*e^2 + 6*B*a*b^2*e^2)) + (2*b*e*x*(2*b^2 *e^2*(5*A*b - 3*B*a) - 4*B*b^3*d*e)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^ 2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/((a*e - b*d)^6...