\(\int \frac {a+b x}{(d+e x)^4 (a^2+2 a b x+b^2 x^2)^3} \, dx\) [62]

Optimal result

Integrand size = 31, antiderivative size = 222 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {b^3}{4 (b d-a e)^4 (a+b x)^4}+\frac {4 b^3 e}{3 (b d-a e)^5 (a+b x)^3}-\frac {5 b^3 e^2}{(b d-a e)^6 (a+b x)^2}+\frac {20 b^3 e^3}{(b d-a e)^7 (a+b x)}+\frac {e^4}{3 (b d-a e)^5 (d+e x)^3}+\frac {5 b e^4}{2 (b d-a e)^6 (d+e x)^2}+\frac {15 b^2 e^4}{(b d-a e)^7 (d+e x)}+\frac {35 b^3 e^4 \log (a+b x)}{(b d-a e)^8}-\frac {35 b^3 e^4 \log (d+e x)}{(b d-a e)^8} \] Output:

-1/4*b^3/(-a*e+b*d)^4/(b*x+a)^4+4/3*b^3*e/(-a*e+b*d)^5/(b*x+a)^3-5*b^3*e^2 
/(-a*e+b*d)^6/(b*x+a)^2+20*b^3*e^3/(-a*e+b*d)^7/(b*x+a)+1/3*e^4/(-a*e+b*d) 
^5/(e*x+d)^3+5/2*b*e^4/(-a*e+b*d)^6/(e*x+d)^2+15*b^2*e^4/(-a*e+b*d)^7/(e*x 
+d)+35*b^3*e^4*ln(b*x+a)/(-a*e+b*d)^8-35*b^3*e^4*ln(e*x+d)/(-a*e+b*d)^8
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {3 b^3 (b d-a e)^4}{(a+b x)^4}+\frac {16 b^3 e (b d-a e)^3}{(a+b x)^3}-\frac {60 b^3 e^2 (b d-a e)^2}{(a+b x)^2}+\frac {240 b^3 e^3 (b d-a e)}{a+b x}+\frac {4 e^4 (b d-a e)^3}{(d+e x)^3}+\frac {30 b e^4 (b d-a e)^2}{(d+e x)^2}+\frac {180 b^2 e^4 (b d-a e)}{d+e x}+420 b^3 e^4 \log (a+b x)-420 b^3 e^4 \log (d+e x)}{12 (b d-a e)^8} \] Input:

Integrate[(a + b*x)/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

((-3*b^3*(b*d - a*e)^4)/(a + b*x)^4 + (16*b^3*e*(b*d - a*e)^3)/(a + b*x)^3 
 - (60*b^3*e^2*(b*d - a*e)^2)/(a + b*x)^2 + (240*b^3*e^3*(b*d - a*e))/(a + 
 b*x) + (4*e^4*(b*d - a*e)^3)/(d + e*x)^3 + (30*b*e^4*(b*d - a*e)^2)/(d + 
e*x)^2 + (180*b^2*e^4*(b*d - a*e))/(d + e*x) + 420*b^3*e^4*Log[a + b*x] - 
420*b^3*e^4*Log[d + e*x])/(12*(b*d - a*e)^8)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 222, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^5 (d+e x)^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {1}{(a+b x)^5 (d+e x)^4}dx\)

\(\Big \downarrow \) 54

\(\displaystyle \int \left (\frac {35 b^4 e^4}{(a+b x) (b d-a e)^8}-\frac {20 b^4 e^3}{(a+b x)^2 (b d-a e)^7}+\frac {10 b^4 e^2}{(a+b x)^3 (b d-a e)^6}-\frac {4 b^4 e}{(a+b x)^4 (b d-a e)^5}+\frac {b^4}{(a+b x)^5 (b d-a e)^4}-\frac {35 b^3 e^5}{(d+e x) (b d-a e)^8}-\frac {15 b^2 e^5}{(d+e x)^2 (b d-a e)^7}-\frac {5 b e^5}{(d+e x)^3 (b d-a e)^6}-\frac {e^5}{(d+e x)^4 (b d-a e)^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {35 b^3 e^4 \log (a+b x)}{(b d-a e)^8}-\frac {35 b^3 e^4 \log (d+e x)}{(b d-a e)^8}+\frac {20 b^3 e^3}{(a+b x) (b d-a e)^7}-\frac {5 b^3 e^2}{(a+b x)^2 (b d-a e)^6}+\frac {4 b^3 e}{3 (a+b x)^3 (b d-a e)^5}-\frac {b^3}{4 (a+b x)^4 (b d-a e)^4}+\frac {15 b^2 e^4}{(d+e x) (b d-a e)^7}+\frac {5 b e^4}{2 (d+e x)^2 (b d-a e)^6}+\frac {e^4}{3 (d+e x)^3 (b d-a e)^5}\)

Input:

Int[(a + b*x)/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

-1/4*b^3/((b*d - a*e)^4*(a + b*x)^4) + (4*b^3*e)/(3*(b*d - a*e)^5*(a + b*x 
)^3) - (5*b^3*e^2)/((b*d - a*e)^6*(a + b*x)^2) + (20*b^3*e^3)/((b*d - a*e) 
^7*(a + b*x)) + e^4/(3*(b*d - a*e)^5*(d + e*x)^3) + (5*b*e^4)/(2*(b*d - a* 
e)^6*(d + e*x)^2) + (15*b^2*e^4)/((b*d - a*e)^7*(d + e*x)) + (35*b^3*e^4*L 
og[a + b*x])/(b*d - a*e)^8 - (35*b^3*e^4*Log[d + e*x])/(b*d - a*e)^8
 

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 = 1.48 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {b^{3}}{4 \left (a e -b d \right )^{4} \left (b x +a \right )^{4}}+\frac {35 b^{3} e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{8}}-\frac {20 b^{3} e^{3}}{\left (a e -b d \right )^{7} \left (b x +a \right )}-\frac {5 b^{3} e^{2}}{\left (a e -b d \right )^{6} \left (b x +a \right )^{2}}-\frac {4 b^{3} e}{3 \left (a e -b d \right )^{5} \left (b x +a \right )^{3}}-\frac {e^{4}}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{3}}-\frac {35 b^{3} e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{8}}-\frac {15 e^{4} b^{2}}{\left (a e -b d \right )^{7} \left (e x +d \right )}+\frac {5 e^{4} b}{2 \left (a e -b d \right )^{6} \left (e x +d \right )^{2}}\)	\(215\)
risch	\(\text {Expression too large to display}\)	\(1265\)
parallelrisch	\(\text {Expression too large to display}\)	\(1452\)
norman	\(\text {Expression too large to display}\)	\(1509\)

Input:

int((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/4*b^3/(a*e-b*d)^4/(b*x+a)^4+35*b^3/(a*e-b*d)^8*e^4*ln(b*x+a)-20*b^3/(a* 
e-b*d)^7*e^3/(b*x+a)-5*b^3/(a*e-b*d)^6*e^2/(b*x+a)^2-4/3*b^3/(a*e-b*d)^5*e 
/(b*x+a)^3-1/3*e^4/(a*e-b*d)^5/(e*x+d)^3-35*b^3/(a*e-b*d)^8*e^4*ln(e*x+d)- 
15*e^4/(a*e-b*d)^7*b^2/(e*x+d)+5/2*e^4/(a*e-b*d)^6*b/(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2090 vs. \(2 (214) = 428\).

Time = 0.12 (sec) , antiderivative size = 2090, normalized size of antiderivative = 9.41 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

-1/12*(3*b^7*d^7 - 28*a*b^6*d^6*e + 126*a^2*b^5*d^5*e^2 - 420*a^3*b^4*d^4* 
e^3 + 105*a^4*b^3*d^3*e^4 + 252*a^5*b^2*d^2*e^5 - 42*a^6*b*d*e^6 + 4*a^7*e 
^7 - 420*(b^7*d*e^6 - a*b^6*e^7)*x^6 - 210*(5*b^7*d^2*e^5 + 2*a*b^6*d*e^6 
- 7*a^2*b^5*e^7)*x^5 - 70*(11*b^7*d^3*e^4 + 42*a*b^6*d^2*e^5 - 27*a^2*b^5* 
d*e^6 - 26*a^3*b^4*e^7)*x^4 - 35*(3*b^7*d^4*e^3 + 76*a*b^6*d^3*e^4 + 54*a^ 
2*b^5*d^2*e^5 - 108*a^3*b^4*d*e^6 - 25*a^4*b^3*e^7)*x^3 + 21*(b^7*d^5*e^2 
- 20*a*b^6*d^4*e^3 - 150*a^2*b^5*d^3*e^4 + 60*a^3*b^4*d^2*e^5 + 105*a^4*b^ 
3*d*e^6 + 4*a^5*b^2*e^7)*x^2 - 7*(b^7*d^6*e - 12*a*b^6*d^5*e^2 + 90*a^2*b^ 
5*d^4*e^3 + 180*a^3*b^4*d^3*e^4 - 225*a^4*b^3*d^2*e^5 - 36*a^5*b^2*d*e^6 + 
 2*a^6*b*e^7)*x - 420*(b^7*e^7*x^7 + a^4*b^3*d^3*e^4 + (3*b^7*d*e^6 + 4*a* 
b^6*e^7)*x^6 + 3*(b^7*d^2*e^5 + 4*a*b^6*d*e^6 + 2*a^2*b^5*e^7)*x^5 + (b^7* 
d^3*e^4 + 12*a*b^6*d^2*e^5 + 18*a^2*b^5*d*e^6 + 4*a^3*b^4*e^7)*x^4 + (4*a* 
b^6*d^3*e^4 + 18*a^2*b^5*d^2*e^5 + 12*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 3 
*(2*a^2*b^5*d^3*e^4 + 4*a^3*b^4*d^2*e^5 + a^4*b^3*d*e^6)*x^2 + (4*a^3*b^4* 
d^3*e^4 + 3*a^4*b^3*d^2*e^5)*x)*log(b*x + a) + 420*(b^7*e^7*x^7 + a^4*b^3* 
d^3*e^4 + (3*b^7*d*e^6 + 4*a*b^6*e^7)*x^6 + 3*(b^7*d^2*e^5 + 4*a*b^6*d*e^6 
 + 2*a^2*b^5*e^7)*x^5 + (b^7*d^3*e^4 + 12*a*b^6*d^2*e^5 + 18*a^2*b^5*d*e^6 
 + 4*a^3*b^4*e^7)*x^4 + (4*a*b^6*d^3*e^4 + 18*a^2*b^5*d^2*e^5 + 12*a^3*b^4 
*d*e^6 + a^4*b^3*e^7)*x^3 + 3*(2*a^2*b^5*d^3*e^4 + 4*a^3*b^4*d^2*e^5 + a^4 
*b^3*d*e^6)*x^2 + (4*a^3*b^4*d^3*e^4 + 3*a^4*b^3*d^2*e^5)*x)*log(e*x + ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2009 vs. \(2 (202) = 404\).

Time = 5.72 (sec) , antiderivative size = 2009, normalized size of antiderivative = 9.05 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

-35*b**3*e**4*log(x + (-35*a**9*b**3*e**13/(a*e - b*d)**8 + 315*a**8*b**4* 
d*e**12/(a*e - b*d)**8 - 1260*a**7*b**5*d**2*e**11/(a*e - b*d)**8 + 2940*a 
**6*b**6*d**3*e**10/(a*e - b*d)**8 - 4410*a**5*b**7*d**4*e**9/(a*e - b*d)* 
*8 + 4410*a**4*b**8*d**5*e**8/(a*e - b*d)**8 - 2940*a**3*b**9*d**6*e**7/(a 
*e - b*d)**8 + 1260*a**2*b**10*d**7*e**6/(a*e - b*d)**8 - 315*a*b**11*d**8 
*e**5/(a*e - b*d)**8 + 35*a*b**3*e**5 + 35*b**12*d**9*e**4/(a*e - b*d)**8 
+ 35*b**4*d*e**4)/(70*b**4*e**5))/(a*e - b*d)**8 + 35*b**3*e**4*log(x + (3 
5*a**9*b**3*e**13/(a*e - b*d)**8 - 315*a**8*b**4*d*e**12/(a*e - b*d)**8 + 
1260*a**7*b**5*d**2*e**11/(a*e - b*d)**8 - 2940*a**6*b**6*d**3*e**10/(a*e 
- b*d)**8 + 4410*a**5*b**7*d**4*e**9/(a*e - b*d)**8 - 4410*a**4*b**8*d**5* 
e**8/(a*e - b*d)**8 + 2940*a**3*b**9*d**6*e**7/(a*e - b*d)**8 - 1260*a**2* 
b**10*d**7*e**6/(a*e - b*d)**8 + 315*a*b**11*d**8*e**5/(a*e - b*d)**8 + 35 
*a*b**3*e**5 - 35*b**12*d**9*e**4/(a*e - b*d)**8 + 35*b**4*d*e**4)/(70*b** 
4*e**5))/(a*e - b*d)**8 + (-4*a**6*e**6 + 38*a**5*b*d*e**5 - 214*a**4*b**2 
*d**2*e**4 - 319*a**3*b**3*d**3*e**3 + 101*a**2*b**4*d**4*e**2 - 25*a*b**5 
*d**5*e + 3*b**6*d**6 - 420*b**6*e**6*x**6 + x**5*(-1470*a*b**5*e**6 - 105 
0*b**6*d*e**5) + x**4*(-1820*a**2*b**4*e**6 - 3710*a*b**5*d*e**5 - 770*b** 
6*d**2*e**4) + x**3*(-875*a**3*b**3*e**6 - 4655*a**2*b**4*d*e**5 - 2765*a* 
b**5*d**2*e**4 - 105*b**6*d**3*e**3) + x**2*(-84*a**4*b**2*e**6 - 2289*a** 
3*b**3*d*e**5 - 3549*a**2*b**4*d**2*e**4 - 399*a*b**5*d**3*e**3 + 21*b*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1587 vs. \(2 (214) = 428\).

Time = 0.14 (sec) , antiderivative size = 1587, normalized size of antiderivative = 7.15 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
 

Output:

35*b^3*e^4*log(b*x + a)/(b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56 
*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^ 
2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8) - 35*b^3*e^4*log(e*x + d)/(b^8*d^8 - 8*a* 
b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 
 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8) + 1/12 
*(420*b^6*e^6*x^6 - 3*b^6*d^6 + 25*a*b^5*d^5*e - 101*a^2*b^4*d^4*e^2 + 319 
*a^3*b^3*d^3*e^3 + 214*a^4*b^2*d^2*e^4 - 38*a^5*b*d*e^5 + 4*a^6*e^6 + 210* 
(5*b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 70*(11*b^6*d^2*e^4 + 53*a*b^5*d*e^5 + 26 
*a^2*b^4*e^6)*x^4 + 35*(3*b^6*d^3*e^3 + 79*a*b^5*d^2*e^4 + 133*a^2*b^4*d*e 
^5 + 25*a^3*b^3*e^6)*x^3 - 21*(b^6*d^4*e^2 - 19*a*b^5*d^3*e^3 - 169*a^2*b^ 
4*d^2*e^4 - 109*a^3*b^3*d*e^5 - 4*a^4*b^2*e^6)*x^2 + 7*(b^6*d^5*e - 11*a*b 
^5*d^4*e^2 + 79*a^2*b^4*d^3*e^3 + 259*a^3*b^3*d^2*e^4 + 34*a^4*b^2*d*e^5 - 
 2*a^5*b*e^6)*x)/(a^4*b^7*d^10 - 7*a^5*b^6*d^9*e + 21*a^6*b^5*d^8*e^2 - 35 
*a^7*b^4*d^7*e^3 + 35*a^8*b^3*d^6*e^4 - 21*a^9*b^2*d^5*e^5 + 7*a^10*b*d^4* 
e^6 - a^11*d^3*e^7 + (b^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*b^9*d^5*e^5 
 - 35*a^3*b^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*b^6*d^2*e^8 + 7*a^6*b^ 
5*d*e^9 - a^7*b^4*e^10)*x^7 + (3*b^11*d^8*e^2 - 17*a*b^10*d^7*e^3 + 35*a^2 
*b^9*d^6*e^4 - 21*a^3*b^8*d^5*e^5 - 35*a^4*b^7*d^4*e^6 + 77*a^5*b^6*d^3*e^ 
7 - 63*a^6*b^5*d^2*e^8 + 25*a^7*b^4*d*e^9 - 4*a^8*b^3*e^10)*x^6 + 3*(b^11* 
d^9*e - 3*a*b^10*d^8*e^2 - 5*a^2*b^9*d^7*e^3 + 35*a^3*b^8*d^6*e^4 - 63*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (214) = 428\).

Time = 0.68 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.22 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {35 \, b^{4} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{9} d^{8} - 8 \, a b^{8} d^{7} e + 28 \, a^{2} b^{7} d^{6} e^{2} - 56 \, a^{3} b^{6} d^{5} e^{3} + 70 \, a^{4} b^{5} d^{4} e^{4} - 56 \, a^{5} b^{4} d^{3} e^{5} + 28 \, a^{6} b^{3} d^{2} e^{6} - 8 \, a^{7} b^{2} d e^{7} + a^{8} b e^{8}} - \frac {35 \, b^{3} e^{5} \log \left ({\left | e x + d \right |}\right )}{b^{8} d^{8} e - 8 \, a b^{7} d^{7} e^{2} + 28 \, a^{2} b^{6} d^{6} e^{3} - 56 \, a^{3} b^{5} d^{5} e^{4} + 70 \, a^{4} b^{4} d^{4} e^{5} - 56 \, a^{5} b^{3} d^{3} e^{6} + 28 \, a^{6} b^{2} d^{2} e^{7} - 8 \, a^{7} b d e^{8} + a^{8} e^{9}} - \frac {3 \, b^{7} d^{7} - 28 \, a b^{6} d^{6} e + 126 \, a^{2} b^{5} d^{5} e^{2} - 420 \, a^{3} b^{4} d^{4} e^{3} + 105 \, a^{4} b^{3} d^{3} e^{4} + 252 \, a^{5} b^{2} d^{2} e^{5} - 42 \, a^{6} b d e^{6} + 4 \, a^{7} e^{7} - 420 \, {\left (b^{7} d e^{6} - a b^{6} e^{7}\right )} x^{6} - 210 \, {\left (5 \, b^{7} d^{2} e^{5} + 2 \, a b^{6} d e^{6} - 7 \, a^{2} b^{5} e^{7}\right )} x^{5} - 70 \, {\left (11 \, b^{7} d^{3} e^{4} + 42 \, a b^{6} d^{2} e^{5} - 27 \, a^{2} b^{5} d e^{6} - 26 \, a^{3} b^{4} e^{7}\right )} x^{4} - 35 \, {\left (3 \, b^{7} d^{4} e^{3} + 76 \, a b^{6} d^{3} e^{4} + 54 \, a^{2} b^{5} d^{2} e^{5} - 108 \, a^{3} b^{4} d e^{6} - 25 \, a^{4} b^{3} e^{7}\right )} x^{3} + 21 \, {\left (b^{7} d^{5} e^{2} - 20 \, a b^{6} d^{4} e^{3} - 150 \, a^{2} b^{5} d^{3} e^{4} + 60 \, a^{3} b^{4} d^{2} e^{5} + 105 \, a^{4} b^{3} d e^{6} + 4 \, a^{5} b^{2} e^{7}\right )} x^{2} - 7 \, {\left (b^{7} d^{6} e - 12 \, a b^{6} d^{5} e^{2} + 90 \, a^{2} b^{5} d^{4} e^{3} + 180 \, a^{3} b^{4} d^{3} e^{4} - 225 \, a^{4} b^{3} d^{2} e^{5} - 36 \, a^{5} b^{2} d e^{6} + 2 \, a^{6} b e^{7}\right )} x}{12 \, {\left (b d - a e\right )}^{8} {\left (b x + a\right )}^{4} {\left (e x + d\right )}^{3}} \] Input:

integrate((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

35*b^4*e^4*log(abs(b*x + a))/(b^9*d^8 - 8*a*b^8*d^7*e + 28*a^2*b^7*d^6*e^2 
 - 56*a^3*b^6*d^5*e^3 + 70*a^4*b^5*d^4*e^4 - 56*a^5*b^4*d^3*e^5 + 28*a^6*b 
^3*d^2*e^6 - 8*a^7*b^2*d*e^7 + a^8*b*e^8) - 35*b^3*e^5*log(abs(e*x + d))/( 
b^8*d^8*e - 8*a*b^7*d^7*e^2 + 28*a^2*b^6*d^6*e^3 - 56*a^3*b^5*d^5*e^4 + 70 
*a^4*b^4*d^4*e^5 - 56*a^5*b^3*d^3*e^6 + 28*a^6*b^2*d^2*e^7 - 8*a^7*b*d*e^8 
 + a^8*e^9) - 1/12*(3*b^7*d^7 - 28*a*b^6*d^6*e + 126*a^2*b^5*d^5*e^2 - 420 
*a^3*b^4*d^4*e^3 + 105*a^4*b^3*d^3*e^4 + 252*a^5*b^2*d^2*e^5 - 42*a^6*b*d* 
e^6 + 4*a^7*e^7 - 420*(b^7*d*e^6 - a*b^6*e^7)*x^6 - 210*(5*b^7*d^2*e^5 + 2 
*a*b^6*d*e^6 - 7*a^2*b^5*e^7)*x^5 - 70*(11*b^7*d^3*e^4 + 42*a*b^6*d^2*e^5 
- 27*a^2*b^5*d*e^6 - 26*a^3*b^4*e^7)*x^4 - 35*(3*b^7*d^4*e^3 + 76*a*b^6*d^ 
3*e^4 + 54*a^2*b^5*d^2*e^5 - 108*a^3*b^4*d*e^6 - 25*a^4*b^3*e^7)*x^3 + 21* 
(b^7*d^5*e^2 - 20*a*b^6*d^4*e^3 - 150*a^2*b^5*d^3*e^4 + 60*a^3*b^4*d^2*e^5 
 + 105*a^4*b^3*d*e^6 + 4*a^5*b^2*e^7)*x^2 - 7*(b^7*d^6*e - 12*a*b^6*d^5*e^ 
2 + 90*a^2*b^5*d^4*e^3 + 180*a^3*b^4*d^3*e^4 - 225*a^4*b^3*d^2*e^5 - 36*a^ 
5*b^2*d*e^6 + 2*a^6*b*e^7)*x)/((b*d - a*e)^8*(b*x + a)^4*(e*x + d)^3)
 

Mupad [B] (verification not implemented)

Time = 11.85 (sec) , antiderivative size = 1469, normalized size of antiderivative = 6.62 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((a + b*x)/((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
 

Output:

(70*b^3*e^4*atanh((a^8*e^8 - b^8*d^8 - 14*a^2*b^6*d^6*e^2 + 14*a^3*b^5*d^5 
*e^3 - 14*a^5*b^3*d^3*e^5 + 14*a^6*b^2*d^2*e^6 + 6*a*b^7*d^7*e - 6*a^7*b*d 
*e^7)/(a*e - b*d)^8 + (2*b*e*x*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 3 
5*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6* 
e - 7*a^6*b*d*e^6))/(a*e - b*d)^8))/(a*e - b*d)^8 - ((4*a^6*e^6 - 3*b^6*d^ 
6 - 101*a^2*b^4*d^4*e^2 + 319*a^3*b^3*d^3*e^3 + 214*a^4*b^2*d^2*e^4 + 25*a 
*b^5*d^5*e - 38*a^5*b*d*e^5)/(12*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 
 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^ 
6*e - 7*a^6*b*d*e^6)) + (35*b^6*e^6*x^6)/(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d 
^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7* 
a*b^6*d^6*e - 7*a^6*b*d*e^6) + (7*e^2*x^2*(4*a^4*b^2*e^4 - b^6*d^4 + 109*a 
^3*b^3*d*e^3 + 169*a^2*b^4*d^2*e^2 + 19*a*b^5*d^3*e))/(4*(a^7*e^7 - b^7*d^ 
7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5* 
b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6)) + (7*e*x*(b^6*d^5 - 2*a^5*b* 
e^5 + 34*a^4*b^2*d*e^4 + 79*a^2*b^4*d^3*e^2 + 259*a^3*b^3*d^2*e^3 - 11*a*b 
^5*d^4*e))/(12*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^ 
3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^ 
6)) + (35*e^2*x^3*(3*b^6*d^3*e + 25*a^3*b^3*e^4 + 79*a*b^5*d^2*e^2 + 133*a 
^2*b^4*d*e^3))/(12*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^ 
4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 3155, normalized size of antiderivative = 14.21 \[ \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(1680*log(a + b*x)*a**5*b**3*d**3*e**5 + 5040*log(a + b*x)*a**5*b**3*d**2* 
e**6*x + 5040*log(a + b*x)*a**5*b**3*d*e**7*x**2 + 1680*log(a + b*x)*a**5* 
b**3*e**8*x**3 + 1260*log(a + b*x)*a**4*b**4*d**4*e**4 + 10500*log(a + b*x 
)*a**4*b**4*d**3*e**5*x + 23940*log(a + b*x)*a**4*b**4*d**2*e**6*x**2 + 21 
420*log(a + b*x)*a**4*b**4*d*e**7*x**3 + 6720*log(a + b*x)*a**4*b**4*e**8* 
x**4 + 5040*log(a + b*x)*a**3*b**5*d**4*e**4*x + 25200*log(a + b*x)*a**3*b 
**5*d**3*e**5*x**2 + 45360*log(a + b*x)*a**3*b**5*d**2*e**6*x**3 + 35280*l 
og(a + b*x)*a**3*b**5*d*e**7*x**4 + 10080*log(a + b*x)*a**3*b**5*e**8*x**5 
 + 7560*log(a + b*x)*a**2*b**6*d**4*e**4*x**2 + 29400*log(a + b*x)*a**2*b* 
*6*d**3*e**5*x**3 + 42840*log(a + b*x)*a**2*b**6*d**2*e**6*x**4 + 27720*lo 
g(a + b*x)*a**2*b**6*d*e**7*x**5 + 6720*log(a + b*x)*a**2*b**6*e**8*x**6 + 
 5040*log(a + b*x)*a*b**7*d**4*e**4*x**3 + 16800*log(a + b*x)*a*b**7*d**3* 
e**5*x**4 + 20160*log(a + b*x)*a*b**7*d**2*e**6*x**5 + 10080*log(a + b*x)* 
a*b**7*d*e**7*x**6 + 1680*log(a + b*x)*a*b**7*e**8*x**7 + 1260*log(a + b*x 
)*b**8*d**4*e**4*x**4 + 3780*log(a + b*x)*b**8*d**3*e**5*x**5 + 3780*log(a 
 + b*x)*b**8*d**2*e**6*x**6 + 1260*log(a + b*x)*b**8*d*e**7*x**7 - 1680*lo 
g(d + e*x)*a**5*b**3*d**3*e**5 - 5040*log(d + e*x)*a**5*b**3*d**2*e**6*x - 
 5040*log(d + e*x)*a**5*b**3*d*e**7*x**2 - 1680*log(d + e*x)*a**5*b**3*e** 
8*x**3 - 1260*log(d + e*x)*a**4*b**4*d**4*e**4 - 10500*log(d + e*x)*a**4*b 
**4*d**3*e**5*x - 23940*log(d + e*x)*a**4*b**4*d**2*e**6*x**2 - 21420*l...