3.12.32 \(\int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx\) [1132]

3.12.32.1 Optimal result

Integrand size = 20, antiderivative size = 239 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=-\frac {b^3 (A b-a B)}{(b d-a e)^5 (a+b x)}+\frac {B d-A e}{4 (b d-a e)^2 (d+e x)^4}+\frac {b B d-2 A b e+a B e}{3 (b d-a e)^3 (d+e x)^3}+\frac {b (b B d-3 A b e+2 a B e)}{2 (b d-a e)^4 (d+e x)^2}+\frac {b^2 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (d+e x)}+\frac {b^3 (b B d-5 A b e+4 a B e) \log (a+b x)}{(b d-a e)^6}-\frac {b^3 (b B d-5 A b e+4 a B e) \log (d+e x)}{(b d-a e)^6} \]

output

-b^3*(A*b-B*a)/(-a*e+b*d)^5/(b*x+a)+1/4*(-A*e+B*d)/(-a*e+b*d)^2/(e*x+d)^4+ 
1/3*(-2*A*b*e+B*a*e+B*b*d)/(-a*e+b*d)^3/(e*x+d)^3+1/2*b*(-3*A*b*e+2*B*a*e+ 
B*b*d)/(-a*e+b*d)^4/(e*x+d)^2+b^2*(-4*A*b*e+3*B*a*e+B*b*d)/(-a*e+b*d)^5/(e 
*x+d)+b^3*(-5*A*b*e+4*B*a*e+B*b*d)*ln(b*x+a)/(-a*e+b*d)^6-b^3*(-5*A*b*e+4* 
B*a*e+B*b*d)*ln(e*x+d)/(-a*e+b*d)^6
 

3.12.32.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=\frac {-\frac {12 b^3 (A b-a B) (b d-a e)}{a+b x}+\frac {3 (b d-a e)^4 (B d-A e)}{(d+e x)^4}+\frac {4 (b d-a e)^3 (b B d-2 A b e+a B e)}{(d+e x)^3}+\frac {6 b (b d-a e)^2 (b B d-3 A b e+2 a B e)}{(d+e x)^2}+\frac {12 b^2 (b d-a e) (b B d-4 A b e+3 a B e)}{d+e x}+12 b^3 (b B d-5 A b e+4 a B e) \log (a+b x)-12 b^3 (b B d-5 A b e+4 a B e) \log (d+e x)}{12 (b d-a e)^6} \]

input

Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]
 

output

((-12*b^3*(A*b - a*B)*(b*d - a*e))/(a + b*x) + (3*(b*d - a*e)^4*(B*d - A*e 
))/(d + e*x)^4 + (4*(b*d - a*e)^3*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x)^3 + 
 (6*b*(b*d - a*e)^2*(b*B*d - 3*A*b*e + 2*a*B*e))/(d + e*x)^2 + (12*b^2*(b* 
d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e))/(d + e*x) + 12*b^3*(b*B*d - 5*A*b*e 
+ 4*a*B*e)*Log[a + b*x] - 12*b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[d + e*x]) 
/(12*(b*d - a*e)^6)
 

3.12.32.3 Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 239, 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.

\(\displaystyle \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^4 (4 a B e-5 A b e+b B d)}{(a+b x) (b d-a e)^6}+\frac {b^4 (A b-a B)}{(a+b x)^2 (b d-a e)^5}+\frac {b^3 e (-4 a B e+5 A b e-b B d)}{(d+e x) (b d-a e)^6}+\frac {b^2 e (-3 a B e+4 A b e-b B d)}{(d+e x)^2 (b d-a e)^5}+\frac {b e (-2 a B e+3 A b e-b B d)}{(d+e x)^3 (b d-a e)^4}+\frac {e (-a B e+2 A b e-b B d)}{(d+e x)^4 (b d-a e)^3}+\frac {e (A e-B d)}{(d+e x)^5 (b d-a e)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^3 (A b-a B)}{(a+b x) (b d-a e)^5}+\frac {b^3 \log (a+b x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}-\frac {b^3 \log (d+e x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}+\frac {b^2 (3 a B e-4 A b e+b B d)}{(d+e x) (b d-a e)^5}+\frac {b (2 a B e-3 A b e+b B d)}{2 (d+e x)^2 (b d-a e)^4}+\frac {a B e-2 A b e+b B d}{3 (d+e x)^3 (b d-a e)^3}+\frac {B d-A e}{4 (d+e x)^4 (b d-a e)^2}\)

input

Int[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]
 

output

-((b^3*(A*b - a*B))/((b*d - a*e)^5*(a + b*x))) + (B*d - A*e)/(4*(b*d - a*e 
)^2*(d + e*x)^4) + (b*B*d - 2*A*b*e + a*B*e)/(3*(b*d - a*e)^3*(d + e*x)^3) 
 + (b*(b*B*d - 3*A*b*e + 2*a*B*e))/(2*(b*d - a*e)^4*(d + e*x)^2) + (b^2*(b 
*B*d - 4*A*b*e + 3*a*B*e))/((b*d - a*e)^5*(d + e*x)) + (b^3*(b*B*d - 5*A*b 
*e + 4*a*B*e)*Log[a + b*x])/(b*d - a*e)^6 - (b^3*(b*B*d - 5*A*b*e + 4*a*B* 
e)*Log[d + e*x])/(b*d - a*e)^6
 

3.12.32.3.1 Defintions of rubi rules used

rule 86

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]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.12.32.4 Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {b^{3} \left (5 A b e -4 B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {\left (A b -B a \right ) b^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {A e -B d}{4 \left (a e -b d \right )^{2} \left (e x +d \right )^{4}}-\frac {b \left (3 A b e -2 B a e -B b d \right )}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}+\frac {b^{3} \left (5 A b e -4 B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}+\frac {2 A b e -B a e -B b d}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}}+\frac {b^{2} \left (4 A b e -3 B a e -B b d \right )}{\left (a e -b d \right )^{5} \left (e x +d \right )}\)	\(239\)
norman	\(\text {Expression too large to display}\)	\(1011\)
risch	\(\text {Expression too large to display}\)	\(1258\)
parallelrisch	\(\text {Expression too large to display}\)	\(1783\)

input

int((B*x+A)/(b*x+a)^2/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

output

-b^3*(5*A*b*e-4*B*a*e-B*b*d)/(a*e-b*d)^6*ln(b*x+a)+(A*b-B*a)*b^3/(a*e-b*d) 
^5/(b*x+a)-1/4*(A*e-B*d)/(a*e-b*d)^2/(e*x+d)^4-1/2*b*(3*A*b*e-2*B*a*e-B*b* 
d)/(a*e-b*d)^4/(e*x+d)^2+b^3*(5*A*b*e-4*B*a*e-B*b*d)/(a*e-b*d)^6*ln(e*x+d) 
+1/3*(2*A*b*e-B*a*e-B*b*d)/(a*e-b*d)^3/(e*x+d)^3+b^2*(4*A*b*e-3*B*a*e-B*b* 
d)/(a*e-b*d)^5/(e*x+d)
 

3.12.32.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1702 vs. \(2 (232) = 464\).

Time = 0.30 (sec) , antiderivative size = 1702, normalized size of antiderivative = 7.12 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=\text {Too large to display} \]

input

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^5,x, algorithm="fricas")
 

output

-1/12*(3*A*a^5*e^5 - (37*B*a*b^4 - 12*A*b^5)*d^5 + (8*B*a^2*b^3 + 65*A*a*b 
^4)*d^4*e + 12*(3*B*a^3*b^2 - 10*A*a^2*b^3)*d^3*e^2 - 4*(2*B*a^4*b - 15*A* 
a^3*b^2)*d^2*e^3 + (B*a^5 - 20*A*a^4*b)*d*e^4 - 12*(B*b^5*d^2*e^3 + (3*B*a 
*b^4 - 5*A*b^5)*d*e^4 - (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 - 6*(7*B*b^5*d^ 
3*e^2 + (22*B*a*b^4 - 35*A*b^5)*d^2*e^3 - 5*(5*B*a^2*b^3 - 6*A*a*b^4)*d*e^ 
4 - (4*B*a^3*b^2 - 5*A*a^2*b^3)*e^5)*x^3 - 2*(26*B*b^5*d^4*e + (89*B*a*b^4 
 - 130*A*b^5)*d^3*e^2 - 3*(24*B*a^2*b^3 - 25*A*a*b^4)*d^2*e^3 - (47*B*a^3* 
b^2 - 60*A*a^2*b^3)*d*e^4 + (4*B*a^4*b - 5*A*a^3*b^2)*e^5)*x^2 - (25*B*b^5 
*d^5 + (104*B*a*b^4 - 125*A*b^5)*d^4*e - 20*(B*a^2*b^3 + A*a*b^4)*d^3*e^2 
- 4*(34*B*a^3*b^2 - 45*A*a^2*b^3)*d^2*e^3 + (31*B*a^4*b - 40*A*a^3*b^2)*d* 
e^4 - (4*B*a^5 - 5*A*a^4*b)*e^5)*x - 12*(B*a*b^4*d^5 + (4*B*a^2*b^3 - 5*A* 
a*b^4)*d^4*e + (B*b^5*d*e^4 + (4*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (4*B*b^5*d^ 
2*e^3 + (17*B*a*b^4 - 20*A*b^5)*d*e^4 + (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 
 + 2*(3*B*b^5*d^3*e^2 + (14*B*a*b^4 - 15*A*b^5)*d^2*e^3 + 2*(4*B*a^2*b^3 - 
 5*A*a*b^4)*d*e^4)*x^3 + 2*(2*B*b^5*d^4*e + (11*B*a*b^4 - 10*A*b^5)*d^3*e^ 
2 + 3*(4*B*a^2*b^3 - 5*A*a*b^4)*d^2*e^3)*x^2 + (B*b^5*d^5 + (8*B*a*b^4 - 5 
*A*b^5)*d^4*e + 4*(4*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2)*x)*log(b*x + a) + 12* 
(B*a*b^4*d^5 + (4*B*a^2*b^3 - 5*A*a*b^4)*d^4*e + (B*b^5*d*e^4 + (4*B*a*b^4 
 - 5*A*b^5)*e^5)*x^5 + (4*B*b^5*d^2*e^3 + (17*B*a*b^4 - 20*A*b^5)*d*e^4 + 
(4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + 2*(3*B*b^5*d^3*e^2 + (14*B*a*b^4 -...
 

3.12.32.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1877 vs. \(2 (233) = 466\).

Time = 4.16 (sec) , antiderivative size = 1877, normalized size of antiderivative = 7.85 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=\text {Too large to display} \]

input

integrate((B*x+A)/(b*x+a)**2/(e*x+d)**5,x)
 

output

-b**3*(-5*A*b*e + 4*B*a*e + B*b*d)*log(x + (-5*A*a*b**4*e**2 - 5*A*b**5*d* 
e + 4*B*a**2*b**3*e**2 + 5*B*a*b**4*d*e + B*b**5*d**2 - a**7*b**3*e**7*(-5 
*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 7*a**6*b**4*d*e**6*(-5*A*b*e + 
4*B*a*e + B*b*d)/(a*e - b*d)**6 - 21*a**5*b**5*d**2*e**5*(-5*A*b*e + 4*B*a 
*e + B*b*d)/(a*e - b*d)**6 + 35*a**4*b**6*d**3*e**4*(-5*A*b*e + 4*B*a*e + 
B*b*d)/(a*e - b*d)**6 - 35*a**3*b**7*d**4*e**3*(-5*A*b*e + 4*B*a*e + B*b*d 
)/(a*e - b*d)**6 + 21*a**2*b**8*d**5*e**2*(-5*A*b*e + 4*B*a*e + B*b*d)/(a* 
e - b*d)**6 - 7*a*b**9*d**6*e*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 
+ b**10*d**7*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6)/(-10*A*b**5*e**2 
 + 8*B*a*b**4*e**2 + 2*B*b**5*d*e))/(a*e - b*d)**6 + b**3*(-5*A*b*e + 4*B* 
a*e + B*b*d)*log(x + (-5*A*a*b**4*e**2 - 5*A*b**5*d*e + 4*B*a**2*b**3*e**2 
 + 5*B*a*b**4*d*e + B*b**5*d**2 + a**7*b**3*e**7*(-5*A*b*e + 4*B*a*e + B*b 
*d)/(a*e - b*d)**6 - 7*a**6*b**4*d*e**6*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e 
- b*d)**6 + 21*a**5*b**5*d**2*e**5*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d 
)**6 - 35*a**4*b**6*d**3*e**4*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 
+ 35*a**3*b**7*d**4*e**3*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 21* 
a**2*b**8*d**5*e**2*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 7*a*b**9 
*d**6*e*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - b**10*d**7*(-5*A*b*e 
 + 4*B*a*e + B*b*d)/(a*e - b*d)**6)/(-10*A*b**5*e**2 + 8*B*a*b**4*e**2 + 2 
*B*b**5*d*e))/(a*e - b*d)**6 + (-3*A*a**4*e**4 + 17*A*a**3*b*d*e**3 - 4...
 

3.12.32.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1090 vs. \(2 (232) = 464\).

Time = 0.27 (sec) , antiderivative size = 1090, normalized size of antiderivative = 4.56 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=\frac {{\left (B b^{4} d + {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} e\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 {{\left (B b^{4} d + {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} e\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 {3 \, A a^{4} e^{4} + {\left (37 \, B a b^{3} - 12 \, A b^{4}\right )} d^{4} + {\left (29 \, B a^{2} b^{2} - 77 \, A a b^{3}\right )} d^{3} e - {\left (7 \, B a^{3} b - 43 \, A a^{2} b^{2}\right )} d^{2} e^{2} + {\left (B a^{4} - 17 \, A a^{3} b\right )} d e^{3} + 12 \, {\left (B b^{4} d e^{3} + {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} e^{4}\right )} x^{4} + 6 \, {\left (7 \, B b^{4} d^{2} e^{2} + {\left (29 \, B a b^{3} - 35 \, A b^{4}\right )} d e^{3} + {\left (4 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} e^{4}\right )} x^{3} + 2 \, {\left (26 \, B b^{4} d^{3} e + 5 \, {\left (23 \, B a b^{3} - 26 \, A b^{4}\right )} d^{2} e^{2} + {\left (43 \, B a^{2} b^{2} - 55 \, A a b^{3}\right )} d e^{3} - {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + {\left (25 \, B b^{4} d^{4} + {\left (129 \, B a b^{3} - 125 \, A b^{4}\right )} d^{3} e + {\left (109 \, B a^{2} b^{2} - 145 \, A a b^{3}\right )} d^{2} e^{2} - {\left (27 \, B a^{3} b - 35 \, A a^{2} b^{2}\right )} d e^{3} + {\left (4 \, B a^{4} - 5 \, A a^{3} b\right )} e^{4}\right )} x}{12 \, {\left (a b^{5} d^{9} - 5 \, a^{2} b^{4} d^{8} e + 10 \, a^{3} b^{3} d^{7} e^{2} - 10 \, a^{4} b^{2} d^{6} e^{3} + 5 \, a^{5} b d^{5} e^{4} - a^{6} d^{4} e^{5} + {\left (b^{6} d^{5} e^{4} - 5 \, a b^{5} d^{4} e^{5} + 10 \, a^{2} b^{4} d^{3} e^{6} - 10 \, a^{3} b^{3} d^{2} e^{7} + 5 \, a^{4} b^{2} d e^{8} - a^{5} b e^{9}\right )} x^{5} + {\left (4 \, b^{6} d^{6} e^{3} - 19 \, a b^{5} d^{5} e^{4} + 35 \, a^{2} b^{4} d^{4} e^{5} - 30 \, a^{3} b^{3} d^{3} e^{6} + 10 \, a^{4} b^{2} d^{2} e^{7} + a^{5} b d e^{8} - a^{6} e^{9}\right )} x^{4} + 2 \, {\left (3 \, b^{6} d^{7} e^{2} - 13 \, a b^{5} d^{6} e^{3} + 20 \, a^{2} b^{4} d^{5} e^{4} - 10 \, a^{3} b^{3} d^{4} e^{5} - 5 \, a^{4} b^{2} d^{3} e^{6} + 7 \, a^{5} b d^{2} e^{7} - 2 \, a^{6} d e^{8}\right )} x^{3} + 2 \, {\left (2 \, b^{6} d^{8} e - 7 \, a b^{5} d^{7} e^{2} + 5 \, a^{2} b^{4} d^{6} e^{3} + 10 \, a^{3} b^{3} d^{5} e^{4} - 20 \, a^{4} b^{2} d^{4} e^{5} + 13 \, a^{5} b d^{3} e^{6} - 3 \, a^{6} d^{2} e^{7}\right )} x^{2} + {\left (b^{6} d^{9} - a b^{5} d^{8} e - 10 \, a^{2} b^{4} d^{7} e^{2} + 30 \, a^{3} b^{3} d^{6} e^{3} - 35 \, a^{4} b^{2} d^{5} e^{4} + 19 \, a^{5} b d^{4} e^{5} - 4 \, a^{6} d^{3} e^{6}\right )} x\right )}} \]

input

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^5,x, algorithm="maxima")
 

output

(B*b^4*d + (4*B*a*b^3 - 5*A*b^4)*e)*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) - (B*b^4*d + (4*B*a*b^3 - 5*A*b^4)*e)*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) + 1/12*(3*A*a^4*e^4 + (37*B*a*b^3 - 12*A 
*b^4)*d^4 + (29*B*a^2*b^2 - 77*A*a*b^3)*d^3*e - (7*B*a^3*b - 43*A*a^2*b^2) 
*d^2*e^2 + (B*a^4 - 17*A*a^3*b)*d*e^3 + 12*(B*b^4*d*e^3 + (4*B*a*b^3 - 5*A 
*b^4)*e^4)*x^4 + 6*(7*B*b^4*d^2*e^2 + (29*B*a*b^3 - 35*A*b^4)*d*e^3 + (4*B 
*a^2*b^2 - 5*A*a*b^3)*e^4)*x^3 + 2*(26*B*b^4*d^3*e + 5*(23*B*a*b^3 - 26*A* 
b^4)*d^2*e^2 + (43*B*a^2*b^2 - 55*A*a*b^3)*d*e^3 - (4*B*a^3*b - 5*A*a^2*b^ 
2)*e^4)*x^2 + (25*B*b^4*d^4 + (129*B*a*b^3 - 125*A*b^4)*d^3*e + (109*B*a^2 
*b^2 - 145*A*a*b^3)*d^2*e^2 - (27*B*a^3*b - 35*A*a^2*b^2)*d*e^3 + (4*B*a^4 
 - 5*A*a^3*b)*e^4)*x)/(a*b^5*d^9 - 5*a^2*b^4*d^8*e + 10*a^3*b^3*d^7*e^2 - 
10*a^4*b^2*d^6*e^3 + 5*a^5*b*d^5*e^4 - a^6*d^4*e^5 + (b^6*d^5*e^4 - 5*a*b^ 
5*d^4*e^5 + 10*a^2*b^4*d^3*e^6 - 10*a^3*b^3*d^2*e^7 + 5*a^4*b^2*d*e^8 - a^ 
5*b*e^9)*x^5 + (4*b^6*d^6*e^3 - 19*a*b^5*d^5*e^4 + 35*a^2*b^4*d^4*e^5 - 30 
*a^3*b^3*d^3*e^6 + 10*a^4*b^2*d^2*e^7 + a^5*b*d*e^8 - a^6*e^9)*x^4 + 2*(3* 
b^6*d^7*e^2 - 13*a*b^5*d^6*e^3 + 20*a^2*b^4*d^5*e^4 - 10*a^3*b^3*d^4*e^5 - 
 5*a^4*b^2*d^3*e^6 + 7*a^5*b*d^2*e^7 - 2*a^6*d*e^8)*x^3 + 2*(2*b^6*d^8*e - 
 7*a*b^5*d^7*e^2 + 5*a^2*b^4*d^6*e^3 + 10*a^3*b^3*d^5*e^4 - 20*a^4*b^2*...
 

3.12.32.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (232) = 464\).

Time = 0.30 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.47 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=-\frac {{\left (B b^{5} d + 4 \, B a b^{4} e - 5 \, A b^{5} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac {\frac {B a b^{8}}{b x + a} - \frac {A b^{9}}{b x + a}}{b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}} - \frac {25 \, B b^{4} d e^{4} + 52 \, B a b^{3} e^{5} - 77 \, A b^{4} e^{5} + \frac {4 \, {\left (22 \, B b^{6} d^{2} e^{3} + 21 \, B a b^{5} d e^{4} - 65 \, A b^{6} d e^{4} - 43 \, B a^{2} b^{4} e^{5} + 65 \, A a b^{5} e^{5}\right )}}{{\left (b x + a\right )} b} + \frac {12 \, {\left (9 \, B b^{8} d^{3} e^{2} - 2 \, B a b^{7} d^{2} e^{3} - 25 \, A b^{8} d^{2} e^{3} - 23 \, B a^{2} b^{6} d e^{4} + 50 \, A a b^{7} d e^{4} + 16 \, B a^{3} b^{5} e^{5} - 25 \, A a^{2} b^{6} e^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {24 \, {\left (2 \, B b^{10} d^{4} e - 3 \, B a b^{9} d^{3} e^{2} - 5 \, A b^{10} d^{3} e^{2} - 3 \, B a^{2} b^{8} d^{2} e^{3} + 15 \, A a b^{9} d^{2} e^{3} + 7 \, B a^{3} b^{7} d e^{4} - 15 \, A a^{2} b^{8} d e^{4} - 3 \, B a^{4} b^{6} e^{5} + 5 \, A a^{3} b^{7} e^{5}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}^{4}} \]

input

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^5,x, algorithm="giac")
 

output

-(B*b^5*d + 4*B*a*b^4*e - 5*A*b^5*e)*log(abs(b*d/(b*x + a) - a*e/(b*x + a) 
 + e))/(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) + (B*a*b^8/(b*x + a) - 
 A*b^9/(b*x + a))/(b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3* 
b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5) - 1/12*(25*B*b^4*d*e^4 + 52*B 
*a*b^3*e^5 - 77*A*b^4*e^5 + 4*(22*B*b^6*d^2*e^3 + 21*B*a*b^5*d*e^4 - 65*A* 
b^6*d*e^4 - 43*B*a^2*b^4*e^5 + 65*A*a*b^5*e^5)/((b*x + a)*b) + 12*(9*B*b^8 
*d^3*e^2 - 2*B*a*b^7*d^2*e^3 - 25*A*b^8*d^2*e^3 - 23*B*a^2*b^6*d*e^4 + 50* 
A*a*b^7*d*e^4 + 16*B*a^3*b^5*e^5 - 25*A*a^2*b^6*e^5)/((b*x + a)^2*b^2) + 2 
4*(2*B*b^10*d^4*e - 3*B*a*b^9*d^3*e^2 - 5*A*b^10*d^3*e^2 - 3*B*a^2*b^8*d^2 
*e^3 + 15*A*a*b^9*d^2*e^3 + 7*B*a^3*b^7*d*e^4 - 15*A*a^2*b^8*d*e^4 - 3*B*a 
^4*b^6*e^5 + 5*A*a^3*b^7*e^5)/((b*x + a)^3*b^3))/((b*d - a*e)^6*(b*d/(b*x 
+ a) - a*e/(b*x + a) + e)^4)
 

3.12.32.9 Mupad [B] (verification not implemented)

Time = 1.98 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.21 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^5} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (B\,b^4\,d-b^3\,e\,\left (5\,A\,b-4\,B\,a\right )\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {B\,a^4\,d\,e^3+3\,A\,a^4\,e^4-7\,B\,a^3\,b\,d^2\,e^2-17\,A\,a^3\,b\,d\,e^3+29\,B\,a^2\,b^2\,d^3\,e+43\,A\,a^2\,b^2\,d^2\,e^2+37\,B\,a\,b^3\,d^4-77\,A\,a\,b^3\,d^3\,e-12\,A\,b^4\,d^4}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {x\,\left (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )\,\left (a^3\,e^3-7\,a^2\,b\,d\,e^2+29\,a\,b^2\,d^2\,e+25\,b^3\,d^3\right )}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {b^3\,e^3\,x^4\,\left (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {b^2\,x^3\,\left (a\,e^3+7\,b\,d\,e^2\right )\,\left (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {b\,x^2\,\left (4\,B\,a\,e-5\,A\,b\,e+B\,b\,d\right )\,\left (-a^2\,e^3+11\,a\,b\,d\,e^2+26\,b^2\,d^2\,e\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{x^4\,\left (a\,e^4+4\,b\,d\,e^3\right )+a\,d^4+x\,\left (b\,d^4+4\,a\,e\,d^3\right )+x^2\,\left (4\,b\,d^3\,e+6\,a\,d^2\,e^2\right )+x^3\,\left (6\,b\,d^2\,e^2+4\,a\,d\,e^3\right )+b\,e^4\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (b^4\,\left (5\,A\,e-B\,d\right )-4\,B\,a\,b^3\,e\right )}{{\left (a\,e-b\,d\right )}^6} \]

input

int((A + B*x)/((a + b*x)^2*(d + e*x)^5),x)
 

output

(log(a + b*x)*(B*b^4*d - b^3*e*(5*A*b - 4*B*a)))/(a*e - b*d)^6 - ((3*A*a^4 
*e^4 - 12*A*b^4*d^4 + 37*B*a*b^3*d^4 + B*a^4*d*e^3 + 29*B*a^2*b^2*d^3*e - 
7*B*a^3*b*d^2*e^2 + 43*A*a^2*b^2*d^2*e^2 - 77*A*a*b^3*d^3*e - 17*A*a^3*b*d 
*e^3)/(12*(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*(4*B*a*e - 5*A*b*e + B*b*d)*(a^3*e^3 + 
 25*b^3*d^3 + 29*a*b^2*d^2*e - 7*a^2*b*d*e^2))/(12*(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^3*e^3*x^4*(4*B*a*e - 5*A*b*e + 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^2*x^3*( 
a*e^3 + 7*b*d*e^2)*(4*B*a*e - 5*A*b*e + B*b*d))/(2*(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*(4*B*a*e - 5*A*b*e + B*b*d)*(26*b^2*d^2*e - a^2*e^3 + 11*a*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)))/(x^4*(a*e^4 + 4*b*d*e^3) + a*d^4 + x*(b*d^4 + 4 
*a*d^3*e) + x^2*(6*a*d^2*e^2 + 4*b*d^3*e) + x^3*(6*b*d^2*e^2 + 4*a*d*e^3) 
+ b*e^4*x^5) + (log(d + e*x)*(b^4*(5*A*e - B*d) - 4*B*a*b^3*e))/(a*e - b*d 
)^6