\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^5} \, dx\) [304]

Optimal result

Integrand size = 31, antiderivative size = 189 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=\frac {b^4 B x}{e^5}+\frac {(b d-a e)^4 (B d-A e)}{4 e^6 (d+e x)^4}-\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e)}{3 e^6 (d+e x)^3}+\frac {b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{e^6 (d+e x)^2}-\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e)}{e^6 (d+e x)}-\frac {b^3 (5 b B d-A b e-4 a B e) \log (d+e x)}{e^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.79 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=-\frac {a^4 e^4 (3 A e+B (d+4 e x))+4 a^3 b e^3 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )-4 a b^3 e \left (-3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-b^4 \left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+12 b^3 (5 b B d-A b e-4 a B e) (d+e x)^4 \log (d+e x)}{12 e^6 (d+e x)^4} \] Input:

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

Output:

-1/12*(a^4*e^4*(3*A*e + B*(d + 4*e*x)) + 4*a^3*b*e^3*(A*e*(d + 4*e*x) + B* 
(d^2 + 4*d*e*x + 6*e^2*x^2)) + 6*a^2*b^2*e^2*(A*e*(d^2 + 4*d*e*x + 6*e^2*x 
^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) - 4*a*b^3*e*(-3*A*e 
*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + B*d*(25*d^3 + 88*d^2*e*x + 
108*d*e^2*x^2 + 48*e^3*x^3)) - b^4*(A*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2 
*x^2 + 48*e^3*x^3) - B*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^ 
3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + 12*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*( 
d + e*x)^4*Log[d + e*x])/(e^6*(d + e*x)^4)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 189, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^5}dx}{b^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^5}dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1184

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]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(185)=370\).

Time = 1.20 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.22

method	result	size
norman	\(\frac {\frac {B \,b^{4} x^{5}}{e}-\frac {3 a^{4} A \,e^{5}+4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}+12 A a \,b^{3} d^{3} e^{2}-25 A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}+18 B \,a^{2} b^{2} d^{3} e^{2}-100 B a \,b^{3} d^{4} e +125 B \,b^{4} d^{5}}{12 e^{6}}-\frac {2 \left (2 A a \,b^{3} e^{2}-2 A \,b^{4} d e +3 B \,e^{2} a^{2} b^{2}-8 B a \,b^{3} d e +10 B \,b^{4} d^{2}\right ) x^{3}}{e^{3}}-\frac {\left (3 A \,a^{2} b^{2} e^{3}+6 A a \,b^{3} d \,e^{2}-9 A \,b^{4} d^{2} e +2 B \,e^{3} a^{3} b +9 B \,a^{2} b^{2} d \,e^{2}-36 B a \,b^{3} d^{2} e +45 B \,b^{4} d^{3}\right ) x^{2}}{e^{4}}-\frac {\left (4 A \,a^{3} b \,e^{4}+6 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-22 A \,b^{4} d^{3} e +B \,e^{4} a^{4}+4 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-88 B a \,b^{3} d^{3} e +110 B \,b^{4} d^{4}\right ) x}{3 e^{5}}}{\left (e x +d \right )^{4}}+\frac {b^{3} \left (A b e +4 B a e -5 B b d \right ) \ln \left (e x +d \right )}{e^{6}}\)	\(420\)
default	\(\frac {b^{4} B x}{e^{5}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,e^{4} a^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 B \,b^{4} d^{4}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {a^{4} A \,e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -B \,b^{4} d^{5}}{4 e^{6} \left (e x +d \right )^{4}}+\frac {b^{3} \left (A b e +4 B a e -5 B b d \right ) \ln \left (e x +d \right )}{e^{6}}-\frac {2 b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 B \,e^{2} a^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{e^{6} \left (e x +d \right )}-\frac {b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{e^{6} \left (e x +d \right )^{2}}\)	\(421\)
risch	\(\frac {b^{4} B x}{e^{5}}+\frac {\left (-4 A a \,b^{3} e^{4}+4 A \,b^{4} d \,e^{3}-6 B \,e^{4} a^{2} b^{2}+16 B a \,b^{3} d \,e^{3}-10 B \,b^{4} d^{2} e^{2}\right ) x^{3}-e b \left (3 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}-9 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}+9 B \,a^{2} b d \,e^{2}-36 B a \,b^{2} d^{2} e +25 B \,b^{3} d^{3}\right ) x^{2}+\left (-\frac {4}{3} A \,a^{3} b \,e^{4}-2 A \,a^{2} b^{2} d \,e^{3}-4 A a \,b^{3} d^{2} e^{2}+\frac {22}{3} A \,b^{4} d^{3} e -\frac {1}{3} B \,e^{4} a^{4}-\frac {4}{3} B \,a^{3} b d \,e^{3}-6 B \,a^{2} b^{2} d^{2} e^{2}+\frac {88}{3} B a \,b^{3} d^{3} e -\frac {65}{3} B \,b^{4} d^{4}\right ) x -\frac {3 a^{4} A \,e^{5}+4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}+12 A a \,b^{3} d^{3} e^{2}-25 A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}+18 B \,a^{2} b^{2} d^{3} e^{2}-100 B a \,b^{3} d^{4} e +77 B \,b^{4} d^{5}}{12 e}}{e^{5} \left (e x +d \right )^{4}}+\frac {b^{4} \ln \left (e x +d \right ) A}{e^{5}}+\frac {4 b^{3} \ln \left (e x +d \right ) B a}{e^{5}}-\frac {5 b^{4} \ln \left (e x +d \right ) B d}{e^{6}}\)	\(433\)
parallelrisch	\(\frac {12 A \ln \left (e x +d \right ) x^{4} b^{4} e^{5}+192 B \ln \left (e x +d \right ) x a \,b^{3} d^{3} e^{2}+100 B a \,b^{3} d^{4} e +192 B \ln \left (e x +d \right ) x^{3} a \,b^{3} d \,e^{4}-6 A \,a^{2} b^{2} d^{2} e^{3}-12 A a \,b^{3} d^{3} e^{2}-18 B \,a^{2} b^{2} d^{3} e^{2}+288 B \ln \left (e x +d \right ) x^{2} a \,b^{3} d^{2} e^{3}-4 A \,a^{3} b d \,e^{4}-60 B \ln \left (e x +d \right ) b^{4} d^{5}-4 B x \,a^{4} e^{5}+12 B \,x^{5} b^{4} e^{5}+12 A \ln \left (e x +d \right ) b^{4} d^{4} e -4 B \,a^{3} b \,d^{2} e^{3}-48 A \,x^{3} a \,b^{3} e^{5}+48 A \,x^{3} b^{4} d \,e^{4}-72 B \,x^{3} a^{2} b^{2} e^{5}-240 B \,x^{3} b^{4} d^{2} e^{3}-36 A \,x^{2} a^{2} b^{2} e^{5}+108 A \,x^{2} b^{4} d^{2} e^{3}-24 B \,x^{2} a^{3} b \,e^{5}-540 B \,x^{2} b^{4} d^{3} e^{2}-16 A x \,a^{3} b \,e^{5}+88 A x \,b^{4} d^{3} e^{2}-440 B x \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+25 A \,b^{4} d^{4} e +72 A \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{3}+48 B \ln \left (e x +d \right ) x^{4} a \,b^{3} e^{5}-60 B \ln \left (e x +d \right ) x^{4} b^{4} d \,e^{4}+48 A \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{4}-240 B \ln \left (e x +d \right ) x^{3} b^{4} d^{2} e^{3}-16 B x \,a^{3} b d \,e^{4}-72 B x \,a^{2} b^{2} d^{2} e^{3}+352 B x a \,b^{3} d^{3} e^{2}-360 B \ln \left (e x +d \right ) x^{2} b^{4} d^{3} e^{2}-48 A x a \,b^{3} d^{2} e^{3}+192 B \,x^{3} a \,b^{3} d \,e^{4}-72 A \,x^{2} a \,b^{3} d \,e^{4}-108 B \,x^{2} a^{2} b^{2} d \,e^{4}+432 B \,x^{2} a \,b^{3} d^{2} e^{3}-24 A x \,a^{2} b^{2} d \,e^{4}+48 B \ln \left (e x +d \right ) a \,b^{3} d^{4} e +48 A \ln \left (e x +d \right ) x \,b^{4} d^{3} e^{2}-240 B \ln \left (e x +d \right ) x \,b^{4} d^{4} e -3 a^{4} A \,e^{5}-125 B \,b^{4} d^{5}}{12 e^{6} \left (e x +d \right )^{4}}\)	\(715\)

Input:

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

Output:

(B*b^4/e*x^5-1/12*(3*A*a^4*e^5+4*A*a^3*b*d*e^4+6*A*a^2*b^2*d^2*e^3+12*A*a* 
b^3*d^3*e^2-25*A*b^4*d^4*e+B*a^4*d*e^4+4*B*a^3*b*d^2*e^3+18*B*a^2*b^2*d^3* 
e^2-100*B*a*b^3*d^4*e+125*B*b^4*d^5)/e^6-2*(2*A*a*b^3*e^2-2*A*b^4*d*e+3*B* 
a^2*b^2*e^2-8*B*a*b^3*d*e+10*B*b^4*d^2)/e^3*x^3-(3*A*a^2*b^2*e^3+6*A*a*b^3 
*d*e^2-9*A*b^4*d^2*e+2*B*a^3*b*e^3+9*B*a^2*b^2*d*e^2-36*B*a*b^3*d^2*e+45*B 
*b^4*d^3)/e^4*x^2-1/3*(4*A*a^3*b*e^4+6*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2- 
22*A*b^4*d^3*e+B*a^4*e^4+4*B*a^3*b*d*e^3+18*B*a^2*b^2*d^2*e^2-88*B*a*b^3*d 
^3*e+110*B*b^4*d^4)/e^5*x)/(e*x+d)^4+1/e^6*b^3*(A*b*e+4*B*a*e-5*B*b*d)*ln( 
e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (185) = 370\).

Time = 0.08 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.19 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=\frac {12 \, B b^{4} e^{5} x^{5} + 48 \, B b^{4} d e^{4} x^{4} - 77 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 25 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 24 \, {\left (2 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 12 \, {\left (21 \, B b^{4} d^{3} e^{2} - 9 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} - 4 \, {\left (62 \, B b^{4} d^{4} e - 22 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 12 \, {\left (5 \, B b^{4} d^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + {\left (5 \, B b^{4} d e^{4} - {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B b^{4} d^{2} e^{3} - {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B b^{4} d^{3} e^{2} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B b^{4} d^{4} e - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \] Input:

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

Output:

1/12*(12*B*b^4*e^5*x^5 + 48*B*b^4*d*e^4*x^4 - 77*B*b^4*d^5 - 3*A*a^4*e^5 + 
 25*(4*B*a*b^3 + A*b^4)*d^4*e - 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2 
*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - (B*a^4 + 4*A*a^3*b)*d*e^4 - 24*(2*B*b^4* 
d^2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 
 - 12*(21*B*b^4*d^3*e^2 - 9*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 
 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 - 4*(62*B*b^4*d^4*e 
 - 22*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 
2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x - 12*(5*B*b 
^4*d^5 - (4*B*a*b^3 + A*b^4)*d^4*e + (5*B*b^4*d*e^4 - (4*B*a*b^3 + A*b^4)* 
e^5)*x^4 + 4*(5*B*b^4*d^2*e^3 - (4*B*a*b^3 + A*b^4)*d*e^4)*x^3 + 6*(5*B*b^ 
4*d^3*e^2 - (4*B*a*b^3 + A*b^4)*d^2*e^3)*x^2 + 4*(5*B*b^4*d^4*e - (4*B*a*b 
^3 + A*b^4)*d^3*e^2)*x)*log(e*x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8* 
x^2 + 4*d^3*e^7*x + d^4*e^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (189) = 378\).

Time = 34.15 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.74 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=\frac {B b^{4} x}{e^{5}} + \frac {b^{3} \left (A b e + 4 B a e - 5 B b d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- 3 A a^{4} e^{5} - 4 A a^{3} b d e^{4} - 6 A a^{2} b^{2} d^{2} e^{3} - 12 A a b^{3} d^{3} e^{2} + 25 A b^{4} d^{4} e - B a^{4} d e^{4} - 4 B a^{3} b d^{2} e^{3} - 18 B a^{2} b^{2} d^{3} e^{2} + 100 B a b^{3} d^{4} e - 77 B b^{4} d^{5} + x^{3} \left (- 48 A a b^{3} e^{5} + 48 A b^{4} d e^{4} - 72 B a^{2} b^{2} e^{5} + 192 B a b^{3} d e^{4} - 120 B b^{4} d^{2} e^{3}\right ) + x^{2} \left (- 36 A a^{2} b^{2} e^{5} - 72 A a b^{3} d e^{4} + 108 A b^{4} d^{2} e^{3} - 24 B a^{3} b e^{5} - 108 B a^{2} b^{2} d e^{4} + 432 B a b^{3} d^{2} e^{3} - 300 B b^{4} d^{3} e^{2}\right ) + x \left (- 16 A a^{3} b e^{5} - 24 A a^{2} b^{2} d e^{4} - 48 A a b^{3} d^{2} e^{3} + 88 A b^{4} d^{3} e^{2} - 4 B a^{4} e^{5} - 16 B a^{3} b d e^{4} - 72 B a^{2} b^{2} d^{2} e^{3} + 352 B a b^{3} d^{3} e^{2} - 260 B b^{4} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \] Input:

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

Output:

B*b**4*x/e**5 + b**3*(A*b*e + 4*B*a*e - 5*B*b*d)*log(d + e*x)/e**6 + (-3*A 
*a**4*e**5 - 4*A*a**3*b*d*e**4 - 6*A*a**2*b**2*d**2*e**3 - 12*A*a*b**3*d** 
3*e**2 + 25*A*b**4*d**4*e - B*a**4*d*e**4 - 4*B*a**3*b*d**2*e**3 - 18*B*a* 
*2*b**2*d**3*e**2 + 100*B*a*b**3*d**4*e - 77*B*b**4*d**5 + x**3*(-48*A*a*b 
**3*e**5 + 48*A*b**4*d*e**4 - 72*B*a**2*b**2*e**5 + 192*B*a*b**3*d*e**4 - 
120*B*b**4*d**2*e**3) + x**2*(-36*A*a**2*b**2*e**5 - 72*A*a*b**3*d*e**4 + 
108*A*b**4*d**2*e**3 - 24*B*a**3*b*e**5 - 108*B*a**2*b**2*d*e**4 + 432*B*a 
*b**3*d**2*e**3 - 300*B*b**4*d**3*e**2) + x*(-16*A*a**3*b*e**5 - 24*A*a**2 
*b**2*d*e**4 - 48*A*a*b**3*d**2*e**3 + 88*A*b**4*d**3*e**2 - 4*B*a**4*e**5 
 - 16*B*a**3*b*d*e**4 - 72*B*a**2*b**2*d**2*e**3 + 352*B*a*b**3*d**3*e**2 
- 260*B*b**4*d**4*e))/(12*d**4*e**6 + 48*d**3*e**7*x + 72*d**2*e**8*x**2 + 
 48*d*e**9*x**3 + 12*e**10*x**4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (185) = 370\).

Time = 0.05 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.33 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=\frac {B b^{4} x}{e^{5}} - \frac {77 \, B b^{4} d^{5} + 3 \, A a^{4} e^{5} - 25 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 24 \, {\left (5 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 12 \, {\left (25 \, B b^{4} d^{3} e^{2} - 9 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 4 \, {\left (65 \, B b^{4} d^{4} e - 22 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} - \frac {{\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \] Input:

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

Output:

B*b^4*x/e^5 - 1/12*(77*B*b^4*d^5 + 3*A*a^4*e^5 - 25*(4*B*a*b^3 + A*b^4)*d^ 
4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^ 
2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4 + 24*(5*B*b^4*d^2*e^3 - 2*(4*B*a*b^3 + A 
*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 12*(25*B*b^4*d^3*e^2 - 
9*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a 
^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 4*(65*B*b^4*d^4*e - 22*(4*B*a*b^3 + A*b^4)* 
d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 2*(2*B*a^3*b + 3*A*a^2*b^2 
)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8* 
x^2 + 4*d^3*e^7*x + d^4*e^6) - (5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*log(e*x 
 + d)/e^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (185) = 370\).

Time = 0.14 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.45 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(e*x + d)*B*b^4/e^6 + (5*B*b^4*d - 4*B*a*b^3*e - A*b^4*e)*log(abs(e*x + d) 
/((e*x + d)^2*abs(e)))/e^6 - 1/12*(120*B*b^4*d^2*e^22/(e*x + d) - 60*B*b^4 
*d^3*e^22/(e*x + d)^2 + 20*B*b^4*d^4*e^22/(e*x + d)^3 - 3*B*b^4*d^5*e^22/( 
e*x + d)^4 - 192*B*a*b^3*d*e^23/(e*x + d) - 48*A*b^4*d*e^23/(e*x + d) + 14 
4*B*a*b^3*d^2*e^23/(e*x + d)^2 + 36*A*b^4*d^2*e^23/(e*x + d)^2 - 64*B*a*b^ 
3*d^3*e^23/(e*x + d)^3 - 16*A*b^4*d^3*e^23/(e*x + d)^3 + 12*B*a*b^3*d^4*e^ 
23/(e*x + d)^4 + 3*A*b^4*d^4*e^23/(e*x + d)^4 + 72*B*a^2*b^2*e^24/(e*x + d 
) + 48*A*a*b^3*e^24/(e*x + d) - 108*B*a^2*b^2*d*e^24/(e*x + d)^2 - 72*A*a* 
b^3*d*e^24/(e*x + d)^2 + 72*B*a^2*b^2*d^2*e^24/(e*x + d)^3 + 48*A*a*b^3*d^ 
2*e^24/(e*x + d)^3 - 18*B*a^2*b^2*d^3*e^24/(e*x + d)^4 - 12*A*a*b^3*d^3*e^ 
24/(e*x + d)^4 + 24*B*a^3*b*e^25/(e*x + d)^2 + 36*A*a^2*b^2*e^25/(e*x + d) 
^2 - 32*B*a^3*b*d*e^25/(e*x + d)^3 - 48*A*a^2*b^2*d*e^25/(e*x + d)^3 + 12* 
B*a^3*b*d^2*e^25/(e*x + d)^4 + 18*A*a^2*b^2*d^2*e^25/(e*x + d)^4 + 4*B*a^4 
*e^26/(e*x + d)^3 + 16*A*a^3*b*e^26/(e*x + d)^3 - 3*B*a^4*d*e^26/(e*x + d) 
^4 - 12*A*a^3*b*d*e^26/(e*x + d)^4 + 3*A*a^4*e^27/(e*x + d)^4)/e^28
 

Mupad [B] (verification not implemented)

Time = 11.80 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.44 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (A\,b^4\,e-5\,B\,b^4\,d+4\,B\,a\,b^3\,e\right )}{e^6}-\frac {x^3\,\left (6\,B\,a^2\,b^2\,e^4-16\,B\,a\,b^3\,d\,e^3+4\,A\,a\,b^3\,e^4+10\,B\,b^4\,d^2\,e^2-4\,A\,b^4\,d\,e^3\right )+\frac {B\,a^4\,d\,e^4+3\,A\,a^4\,e^5+4\,B\,a^3\,b\,d^2\,e^3+4\,A\,a^3\,b\,d\,e^4+18\,B\,a^2\,b^2\,d^3\,e^2+6\,A\,a^2\,b^2\,d^2\,e^3-100\,B\,a\,b^3\,d^4\,e+12\,A\,a\,b^3\,d^3\,e^2+77\,B\,b^4\,d^5-25\,A\,b^4\,d^4\,e}{12\,e}+x\,\left (\frac {B\,a^4\,e^4}{3}+\frac {4\,B\,a^3\,b\,d\,e^3}{3}+\frac {4\,A\,a^3\,b\,e^4}{3}+6\,B\,a^2\,b^2\,d^2\,e^2+2\,A\,a^2\,b^2\,d\,e^3-\frac {88\,B\,a\,b^3\,d^3\,e}{3}+4\,A\,a\,b^3\,d^2\,e^2+\frac {65\,B\,b^4\,d^4}{3}-\frac {22\,A\,b^4\,d^3\,e}{3}\right )+x^2\,\left (2\,B\,a^3\,b\,e^4+9\,B\,a^2\,b^2\,d\,e^3+3\,A\,a^2\,b^2\,e^4-36\,B\,a\,b^3\,d^2\,e^2+6\,A\,a\,b^3\,d\,e^3+25\,B\,b^4\,d^3\,e-9\,A\,b^4\,d^2\,e^2\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,b^4\,x}{e^5} \] Input:

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

Output:

(log(d + e*x)*(A*b^4*e - 5*B*b^4*d + 4*B*a*b^3*e))/e^6 - (x^3*(4*A*a*b^3*e 
^4 - 4*A*b^4*d*e^3 + 6*B*a^2*b^2*e^4 + 10*B*b^4*d^2*e^2 - 16*B*a*b^3*d*e^3 
) + (3*A*a^4*e^5 + 77*B*b^4*d^5 - 25*A*b^4*d^4*e + B*a^4*d*e^4 + 12*A*a*b^ 
3*d^3*e^2 + 4*B*a^3*b*d^2*e^3 + 6*A*a^2*b^2*d^2*e^3 + 18*B*a^2*b^2*d^3*e^2 
 + 4*A*a^3*b*d*e^4 - 100*B*a*b^3*d^4*e)/(12*e) + x*((B*a^4*e^4)/3 + (65*B* 
b^4*d^4)/3 + (4*A*a^3*b*e^4)/3 - (22*A*b^4*d^3*e)/3 + 4*A*a*b^3*d^2*e^2 + 
2*A*a^2*b^2*d*e^3 + 6*B*a^2*b^2*d^2*e^2 - (88*B*a*b^3*d^3*e)/3 + (4*B*a^3* 
b*d*e^3)/3) + x^2*(2*B*a^3*b*e^4 + 25*B*b^4*d^3*e + 3*A*a^2*b^2*e^4 - 9*A* 
b^4*d^2*e^2 - 36*B*a*b^3*d^2*e^2 + 9*B*a^2*b^2*d*e^3 + 6*A*a*b^3*d*e^3))/( 
d^4*e^5 + e^9*x^4 + 4*d^3*e^6*x + 4*d*e^8*x^3 + 6*d^2*e^7*x^2) + (B*b^4*x) 
/e^5
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.30 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^5} \, dx=\frac {-65 b^{5} d^{6}+60 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{5} e -240 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{5} e x -20 a^{4} b d \,e^{5} x -40 a^{3} b^{2} d^{2} e^{4} x -60 a^{3} b^{2} d \,e^{5} x^{2}+200 a \,b^{4} d^{4} e^{2} x +180 a \,b^{4} d^{3} e^{3} x^{2}-60 a \,b^{4} d \,e^{5} x^{4}-10 a^{3} b^{2} d^{3} e^{3}+65 a \,b^{4} d^{5} e +30 a^{2} b^{3} e^{6} x^{4}-360 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{4} e^{2} x^{2}-60 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{2} e^{4} x^{4}+240 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{2} e^{4} x^{3}+360 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{3} e^{3} x^{2}-3 a^{5} d \,e^{5}-5 a^{4} b \,d^{2} e^{4}+240 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{4} e^{2} x -60 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{6}-240 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{3} e^{3} x^{3}-200 b^{5} d^{5} e x -180 b^{5} d^{4} e^{2} x^{2}+60 b^{5} d^{2} e^{4} x^{4}+12 b^{5} d \,e^{5} x^{5}+60 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d \,e^{5} x^{4}}{12 d \,e^{6} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )} \] Input:

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

Output:

(60*log(d + e*x)*a*b**4*d**5*e + 240*log(d + e*x)*a*b**4*d**4*e**2*x + 360 
*log(d + e*x)*a*b**4*d**3*e**3*x**2 + 240*log(d + e*x)*a*b**4*d**2*e**4*x* 
*3 + 60*log(d + e*x)*a*b**4*d*e**5*x**4 - 60*log(d + e*x)*b**5*d**6 - 240* 
log(d + e*x)*b**5*d**5*e*x - 360*log(d + e*x)*b**5*d**4*e**2*x**2 - 240*lo 
g(d + e*x)*b**5*d**3*e**3*x**3 - 60*log(d + e*x)*b**5*d**2*e**4*x**4 - 3*a 
**5*d*e**5 - 5*a**4*b*d**2*e**4 - 20*a**4*b*d*e**5*x - 10*a**3*b**2*d**3*e 
**3 - 40*a**3*b**2*d**2*e**4*x - 60*a**3*b**2*d*e**5*x**2 + 30*a**2*b**3*e 
**6*x**4 + 65*a*b**4*d**5*e + 200*a*b**4*d**4*e**2*x + 180*a*b**4*d**3*e** 
3*x**2 - 60*a*b**4*d*e**5*x**4 - 65*b**5*d**6 - 200*b**5*d**5*e*x - 180*b* 
*5*d**4*e**2*x**2 + 60*b**5*d**2*e**4*x**4 + 12*b**5*d*e**5*x**5)/(12*d*e* 
*6*(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4))