\(\int \frac {a+b x}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^3} \, dx\) [60]

Optimal result

Integrand size = 31, antiderivative size = 159 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {b}{4 (b d-a e)^2 (a+b x)^4}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^3}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x)^2}+\frac {4 b e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^5 (d+e x)}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {3 b (b d-a e)^4}{(a+b x)^4}+\frac {8 b e (b d-a e)^3}{(a+b x)^3}-\frac {18 b e^2 (b d-a e)^2}{(a+b x)^2}+\frac {48 b e^3 (b d-a e)}{a+b x}+\frac {12 e^4 (b d-a e)}{d+e x}+60 b e^4 \log (a+b x)-60 b e^4 \log (d+e x)}{12 (b d-a e)^6} \] Input:

Integrate[(a + b*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

((-3*b*(b*d - a*e)^4)/(a + b*x)^4 + (8*b*e*(b*d - a*e)^3)/(a + b*x)^3 - (1 
8*b*e^2*(b*d - a*e)^2)/(a + b*x)^2 + (48*b*e^3*(b*d - a*e))/(a + b*x) + (1 
2*e^4*(b*d - a*e))/(d + e*x) + 60*b*e^4*Log[a + b*x] - 60*b*e^4*Log[d + e* 
x])/(12*(b*d - a*e)^6)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 159, 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.

\(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^2} \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^5 (d+e x)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {1}{(a+b x)^5 (d+e x)^2}dx\)

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

-1/4*b/((b*d - a*e)^2*(a + b*x)^4) + (2*b*e)/(3*(b*d - a*e)^3*(a + b*x)^3) 
 - (3*b*e^2)/(2*(b*d - a*e)^4*(a + b*x)^2) + (4*b*e^3)/((b*d - a*e)^5*(a + 
 b*x)) + e^4/((b*d - a*e)^5*(d + e*x)) + (5*b*e^4*Log[a + b*x])/(b*d - a*e 
)^6 - (5*b*e^4*Log[d + e*x])/(b*d - a*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 54

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])
 

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 [A] (verified)

Time = 1.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {b}{4 \left (a e -b d \right )^{2} \left (b x +a \right )^{4}}+\frac {5 b \,e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {4 b \,e^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {3 b \,e^{2}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}-\frac {2 b e}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}-\frac {e^{4}}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}\)	\(155\)
risch	\(\frac {-\frac {5 b^{4} e^{4} x^{4}}{e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {5 b^{3} \left (7 a e +b d \right ) e^{3} x^{3}}{2 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {5 b^{2} e^{2} \left (26 e^{2} a^{2}+11 a b d e -b^{2} d^{2}\right ) x^{2}}{6 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {5 \left (25 e^{3} a^{3}+29 a^{2} b d \,e^{2}-7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) b e x}{12 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {12 a^{4} e^{4}+77 a^{3} b d \,e^{3}-43 a^{2} b^{2} d^{2} e^{2}+17 a \,b^{3} d^{3} e -3 b^{4} d^{4}}{12 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}+\frac {5 b \,e^{4} \ln \left (-b x -a \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}\)	\(709\)
parallelrisch	\(\frac {180 x^{3} a \,b^{8} d \,e^{5}+150 x^{2} a^{2} b^{7} d \,e^{5}+120 x^{2} a \,b^{8} d^{2} e^{4}-20 x \,a^{3} b^{6} d \,e^{5}+180 x \,a^{2} b^{7} d^{2} e^{4}-40 x a \,b^{8} d^{3} e^{3}+240 \ln \left (b x +a \right ) x^{4} a \,b^{8} e^{6}+60 \ln \left (b x +a \right ) x^{4} b^{9} d \,e^{5}-240 \ln \left (e x +d \right ) x^{4} a \,b^{8} e^{6}-60 \ln \left (e x +d \right ) x^{4} b^{9} d \,e^{5}+360 \ln \left (b x +a \right ) x^{3} a^{2} b^{7} e^{6}-360 \ln \left (e x +d \right ) x^{3} a^{2} b^{7} e^{6}+240 \ln \left (b x +a \right ) x^{2} a^{3} b^{6} e^{6}-240 \ln \left (e x +d \right ) x^{2} a^{3} b^{6} e^{6}+60 \ln \left (b x +a \right ) x \,a^{4} b^{5} e^{6}-60 \ln \left (e x +d \right ) x \,a^{4} b^{5} e^{6}+60 \ln \left (b x +a \right ) a^{4} b^{5} d \,e^{5}-60 \ln \left (e x +d \right ) a^{4} b^{5} d \,e^{5}+240 \ln \left (b x +a \right ) x^{3} a \,b^{8} d \,e^{5}-240 \ln \left (e x +d \right ) x^{3} a \,b^{8} d \,e^{5}+360 \ln \left (b x +a \right ) x^{2} a^{2} b^{7} d \,e^{5}-360 \ln \left (e x +d \right ) x^{2} a^{2} b^{7} d \,e^{5}+240 \ln \left (b x +a \right ) x \,a^{3} b^{6} d \,e^{5}-240 \ln \left (e x +d \right ) x \,a^{3} b^{6} d \,e^{5}-260 x^{2} a^{3} b^{6} e^{6}-10 x^{2} b^{9} d^{3} e^{3}-125 x \,a^{4} b^{5} e^{6}+5 x \,b^{9} d^{4} e^{2}-60 x^{4} a \,b^{8} e^{6}+60 x^{4} b^{9} d \,e^{5}-210 x^{3} a^{2} b^{7} e^{6}+30 x^{3} b^{9} d^{2} e^{4}+60 \ln \left (b x +a \right ) x^{5} b^{9} e^{6}-60 \ln \left (e x +d \right ) x^{5} b^{9} e^{6}-65 a^{4} b^{5} d \,e^{5}+120 a^{3} b^{6} d^{2} e^{4}-60 a^{2} b^{7} d^{3} e^{3}+20 a \,b^{8} d^{4} e^{2}-3 b^{9} d^{5} e -12 a^{5} b^{4} e^{6}}{12 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (e x +d \right ) b^{4} e}\)	\(757\)
norman	\(\frac {-\frac {5 b^{5} e^{4} x^{5}}{e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {5 \left (-77 a^{3} b^{6} e^{5}-51 a^{2} b^{7} d \,e^{4}+9 a \,b^{8} d^{2} e^{3}-b^{9} d^{3} e^{2}\right ) x^{2}}{12 b^{4} e \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {a \left (-12 a^{4} b^{5} e^{5}-77 a^{3} b^{6} d \,e^{4}+43 a^{2} b^{7} d^{2} e^{3}-17 a \,b^{8} d^{3} e^{2}+3 b^{9} d^{4} e \right )}{12 e \,b^{5} \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {5 \left (-9 e^{5} a \,b^{6}-d \,e^{4} b^{7}\right ) x^{4}}{2 b^{2} e \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {5 \left (-47 a^{2} b^{6} e^{5}-14 a \,b^{7} d \,e^{4}+b^{8} d^{2} e^{3}\right ) x^{3}}{6 e \,b^{3} \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {\left (-137 a^{4} b^{6} e^{5}-222 a^{3} b^{7} d \,e^{4}+78 a^{2} b^{8} d^{2} e^{3}-22 a \,b^{9} d^{3} e^{2}+3 b^{10} d^{4} e \right ) x}{12 e \,b^{5} \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}}{\left (b x +a \right )^{5} \left (e x +d \right )}+\frac {5 b \,e^{4} \ln \left (b x +a \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}\)	\(874\)

Input:

int((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (153) = 306\).

Time = 0.13 (sec) , antiderivative size = 1083, normalized size of antiderivative = 6.81 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

-1/12*(3*b^5*d^5 - 20*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e 
^3 + 65*a^4*b*d*e^4 + 12*a^5*e^5 - 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^ 
5*d^2*e^3 + 6*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 + 10*(b^5*d^3*e^2 - 12*a*b^ 
4*d^2*e^3 - 15*a^2*b^3*d*e^4 + 26*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 8*a*b^ 
4*d^3*e^2 + 36*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b 
^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^ 
4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3* 
b^2*d*e^4 + a^4*b*e^5)*x)*log(b*x + a) + 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + ( 
b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 2*( 
3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*lo 
g(e*x + d))/(a^4*b^6*d^7 - 6*a^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b 
^3*d^4*e^3 + 15*a^8*b^2*d^3*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (b^10*d^6 
*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^ 
6*d^2*e^5 - 6*a^5*b^5*d*e^6 + a^6*b^4*e^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e 
 - 9*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 65*a^4*b^6*d^3*e^4 + 54*a^5*b^ 
5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 + 2*(2*a*b^9*d^7 - 9*a^2 
*b^8*d^6*e + 12*a^3*b^7*d^5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*a^5*b^5*d^3*e^4 + 
 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 + 2*(3*a^2*b^8 
*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5*a^6* 
b^4*d^3*e^4 + 12*a^7*b^3*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 +...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (144) = 288\).

Time = 2.54 (sec) , antiderivative size = 1178, normalized size of antiderivative = 7.41 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

-5*b*e**4*log(x + (-5*a**7*b*e**11/(a*e - b*d)**6 + 35*a**6*b**2*d*e**10/( 
a*e - b*d)**6 - 105*a**5*b**3*d**2*e**9/(a*e - b*d)**6 + 175*a**4*b**4*d** 
3*e**8/(a*e - b*d)**6 - 175*a**3*b**5*d**4*e**7/(a*e - b*d)**6 + 105*a**2* 
b**6*d**5*e**6/(a*e - b*d)**6 - 35*a*b**7*d**6*e**5/(a*e - b*d)**6 + 5*a*b 
*e**5 + 5*b**8*d**7*e**4/(a*e - b*d)**6 + 5*b**2*d*e**4)/(10*b**2*e**5))/( 
a*e - b*d)**6 + 5*b*e**4*log(x + (5*a**7*b*e**11/(a*e - b*d)**6 - 35*a**6* 
b**2*d*e**10/(a*e - b*d)**6 + 105*a**5*b**3*d**2*e**9/(a*e - b*d)**6 - 175 
*a**4*b**4*d**3*e**8/(a*e - b*d)**6 + 175*a**3*b**5*d**4*e**7/(a*e - b*d)* 
*6 - 105*a**2*b**6*d**5*e**6/(a*e - b*d)**6 + 35*a*b**7*d**6*e**5/(a*e - b 
*d)**6 + 5*a*b*e**5 - 5*b**8*d**7*e**4/(a*e - b*d)**6 + 5*b**2*d*e**4)/(10 
*b**2*e**5))/(a*e - b*d)**6 + (-12*a**4*e**4 - 77*a**3*b*d*e**3 + 43*a**2* 
b**2*d**2*e**2 - 17*a*b**3*d**3*e + 3*b**4*d**4 - 60*b**4*e**4*x**4 + x**3 
*(-210*a*b**3*e**4 - 30*b**4*d*e**3) + x**2*(-260*a**2*b**2*e**4 - 110*a*b 
**3*d*e**3 + 10*b**4*d**2*e**2) + x*(-125*a**3*b*e**4 - 145*a**2*b**2*d*e* 
*3 + 35*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(12*a**9*d*e**5 - 60*a**8*b*d** 
2*e**4 + 120*a**7*b**2*d**3*e**3 - 120*a**6*b**3*d**4*e**2 + 60*a**5*b**4* 
d**5*e - 12*a**4*b**5*d**6 + x**5*(12*a**5*b**4*e**6 - 60*a**4*b**5*d*e**5 
 + 120*a**3*b**6*d**2*e**4 - 120*a**2*b**7*d**3*e**3 + 60*a*b**8*d**4*e**2 
 - 12*b**9*d**5*e) + x**4*(48*a**6*b**3*e**6 - 228*a**5*b**4*d*e**5 + 420* 
a**4*b**5*d**2*e**4 - 360*a**3*b**6*d**3*e**3 + 120*a**2*b**7*d**4*e**2...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 858 vs. \(2 (153) = 306\).

Time = 0.08 (sec) , antiderivative size = 858, normalized size of antiderivative = 5.40 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
 

Output:

5*b*e^4*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) - 5*b*e^4*lo 
g(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*(60*b^4*e^4*x^4 
 - 3*b^4*d^4 + 17*a*b^3*d^3*e - 43*a^2*b^2*d^2*e^2 + 77*a^3*b*d*e^3 + 12*a 
^4*e^4 + 30*(b^4*d*e^3 + 7*a*b^3*e^4)*x^3 - 10*(b^4*d^2*e^2 - 11*a*b^3*d*e 
^3 - 26*a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e - 7*a*b^3*d^2*e^2 + 29*a^2*b^2*d*e 
^3 + 25*a^3*b*e^4)*x)/(a^4*b^5*d^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 
- 10*a^7*b^2*d^3*e^3 + 5*a^8*b*d^2*e^4 - a^9*d*e^5 + (b^9*d^5*e - 5*a*b^8* 
d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5* 
b^4*e^6)*x^5 + (b^9*d^6 - a*b^8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^ 
3*e^3 - 35*a^4*b^5*d^2*e^4 + 19*a^5*b^4*d*e^5 - 4*a^6*b^3*e^6)*x^4 + 2*(2* 
a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d^3*e^3 - 20* 
a^5*b^4*d^2*e^4 + 13*a^6*b^3*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 
 - 13*a^3*b^6*d^5*e + 20*a^4*b^5*d^4*e^2 - 10*a^5*b^4*d^3*e^3 - 5*a^6*b^3* 
d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^6 - 19*a^4*b^5 
*d^5*e + 35*a^5*b^4*d^4*e^2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^ 
8*b*d*e^5 - a^9*e^6)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (153) = 306\).

Time = 0.20 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.35 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{9}}{{\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )} {\left (e x + d\right )}} + \frac {5 \, b e^{5} \log \left ({\left | b - \frac {b d}{e x + d} + \frac {a e}{e x + d} \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} + \frac {77 \, b^{5} e^{4} - \frac {260 \, {\left (b^{5} d e^{5} - a b^{4} e^{6}\right )}}{{\left (e x + d\right )} e} + \frac {300 \, {\left (b^{5} d^{2} e^{6} - 2 \, a b^{4} d e^{7} + a^{2} b^{3} e^{8}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}\right )}}{{\left (e x + d\right )}^{3} e^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (b - \frac {b d}{e x + d} + \frac {a e}{e x + d}\right )}^{4}} \] Input:

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

e^9/((b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^3*d^3*e^7 - 10*a^3*b^2*d^2* 
e^8 + 5*a^4*b*d*e^9 - a^5*e^10)*(e*x + d)) + 5*b*e^5*log(abs(b - b*d/(e*x 
+ d) + a*e/(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) + 1/12 
*(77*b^5*e^4 - 260*(b^5*d*e^5 - a*b^4*e^6)/((e*x + d)*e) + 300*(b^5*d^2*e^ 
6 - 2*a*b^4*d*e^7 + a^2*b^3*e^8)/((e*x + d)^2*e^2) - 120*(b^5*d^3*e^7 - 3* 
a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10)/((e*x + d)^3*e^3))/((b*d - 
 a*e)^6*(b - b*d/(e*x + d) + a*e/(e*x + d))^4)
 

Mupad [B] (verification not implemented)

Time = 11.37 (sec) , antiderivative size = 763, normalized size of antiderivative = 4.80 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {10\,b\,e^4\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {12\,a^4\,e^4+77\,a^3\,b\,d\,e^3-43\,a^2\,b^2\,d^2\,e^2+17\,a\,b^3\,d^3\,e-3\,b^4\,d^4}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e\,x\,\left (25\,a^3\,b\,e^3+29\,a^2\,b^2\,d\,e^2-7\,a\,b^3\,d^2\,e+b^4\,d^3\right )}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {5\,e^3\,x^3\,\left (d\,b^4+7\,a\,e\,b^3\right )}{2\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e^2\,x^2\,\left (26\,a^2\,b^2\,e^2+11\,a\,b^3\,d\,e-b^4\,d^2\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{x^4\,\left (d\,b^4+4\,a\,e\,b^3\right )+a^4\,d+x\,\left (e\,a^4+4\,b\,d\,a^3\right )+x^2\,\left (4\,e\,a^3\,b+6\,d\,a^2\,b^2\right )+x^3\,\left (6\,e\,a^2\,b^2+4\,d\,a\,b^3\right )+b^4\,e\,x^5} \] Input:

int((a + b*x)/((d + e*x)^2*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1706, normalized size of antiderivative = 10.73 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(240*log(a + b*x)*a**5*b*d*e**5 + 240*log(a + b*x)*a**5*b*e**6*x + 60*log( 
a + b*x)*a**4*b**2*d**2*e**4 + 1020*log(a + b*x)*a**4*b**2*d*e**5*x + 960* 
log(a + b*x)*a**4*b**2*e**6*x**2 + 240*log(a + b*x)*a**3*b**3*d**2*e**4*x 
+ 1680*log(a + b*x)*a**3*b**3*d*e**5*x**2 + 1440*log(a + b*x)*a**3*b**3*e* 
*6*x**3 + 360*log(a + b*x)*a**2*b**4*d**2*e**4*x**2 + 1320*log(a + b*x)*a* 
*2*b**4*d*e**5*x**3 + 960*log(a + b*x)*a**2*b**4*e**6*x**4 + 240*log(a + b 
*x)*a*b**5*d**2*e**4*x**3 + 480*log(a + b*x)*a*b**5*d*e**5*x**4 + 240*log( 
a + b*x)*a*b**5*e**6*x**5 + 60*log(a + b*x)*b**6*d**2*e**4*x**4 + 60*log(a 
 + b*x)*b**6*d*e**5*x**5 - 240*log(d + e*x)*a**5*b*d*e**5 - 240*log(d + e* 
x)*a**5*b*e**6*x - 60*log(d + e*x)*a**4*b**2*d**2*e**4 - 1020*log(d + e*x) 
*a**4*b**2*d*e**5*x - 960*log(d + e*x)*a**4*b**2*e**6*x**2 - 240*log(d + e 
*x)*a**3*b**3*d**2*e**4*x - 1680*log(d + e*x)*a**3*b**3*d*e**5*x**2 - 1440 
*log(d + e*x)*a**3*b**3*e**6*x**3 - 360*log(d + e*x)*a**2*b**4*d**2*e**4*x 
**2 - 1320*log(d + e*x)*a**2*b**4*d*e**5*x**3 - 960*log(d + e*x)*a**2*b**4 
*e**6*x**4 - 240*log(d + e*x)*a*b**5*d**2*e**4*x**3 - 480*log(d + e*x)*a*b 
**5*d*e**5*x**4 - 240*log(d + e*x)*a*b**5*e**6*x**5 - 60*log(d + e*x)*b**6 
*d**2*e**4*x**4 - 60*log(d + e*x)*b**6*d*e**5*x**5 - 48*a**6*e**6 - 212*a* 
*5*b*d*e**5 - 440*a**5*b*e**6*x + 355*a**4*b**2*d**2*e**4 - 25*a**4*b**2*d 
*e**5*x - 800*a**4*b**2*e**6*x**2 - 120*a**3*b**3*d**3*e**3 + 460*a**3*b** 
3*d**2*e**4*x + 460*a**3*b**3*d*e**5*x**2 - 480*a**3*b**3*e**6*x**3 + 2...