\(\int \frac {(d+e x)^3}{(a+b x+c x^2)^3} \, dx\) [487]

Optimal result

Integrand size = 20, antiderivative size = 158 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {3 b^3 c d e^2-b^4 e^3+b^2 c e \left (-9 c d^2+5 a e^2+6 c d e x\right )+4 c^2 \left (-4 a^2 e^3+3 c^2 d^3 x+3 a c d e^2 x\right )+6 b c^2 \left (c d^2 (d-3 e x)+a e^2 (d-e x)\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {-b^3 e^3 x+b^2 e^2 (-a e+3 c d x)+2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )+b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))^2}+\frac {12 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \] Input:

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

Output:

((3*b^3*c*d*e^2 - b^4*e^3 + b^2*c*e*(-9*c*d^2 + 5*a*e^2 + 6*c*d*e*x) + 4*c 
^2*(-4*a^2*e^3 + 3*c^2*d^3*x + 3*a*c*d*e^2*x) + 6*b*c^2*(c*d^2*(d - 3*e*x) 
 + a*e^2*(d - e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^3*e^3* 
x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d 
+ e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x)))/(c^2*(-b^2 + 4*a*c) 
*(a + x*(b + c*x))^2) + (12*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*ArcTa 
n[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1156, 1153, 1083, 219}

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.

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

\(\Big \downarrow \) 1156

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

\(\Big \downarrow \) 1153

\(\displaystyle -\frac {3 (2 c d-b e) \left (-\frac {2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {3 (2 c d-b e) \left (\frac {4 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {3 (2 c d-b e) \left (\frac {4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (a e^2-b d e+c d^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

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

Output:

-1/2*((b + 2*c*x)*(d + e*x)^3)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*(2 
*c*d - b*e)*(-(((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)* 
(a + b*x + c*x^2))) + (4*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(b + 2*c*x)/Sqrt[ 
b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(2*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1153

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - 
b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m - 2)*(a + b*x + 
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
&& LtQ[p, -1]
 

rule 1156

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* 
(b^2 - 4*a*c))), x] + Simp[m*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a*c)))   Int[ 
(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(484\) vs. \(2(150)=300\).

Time = 0.97 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.07

method	result	size
default	\(\frac {-\frac {3 c \left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) x^{3}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {\left (16 a^{2} c^{2} e^{3}+a \,b^{2} c \,e^{3}-18 a b \,c^{2} d \,e^{2}+e^{3} b^{4}-9 b^{3} c d \,e^{2}+27 b^{2} c^{2} d^{2} e -18 b \,c^{3} d^{3}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c}-\frac {\left (5 a^{2} b c \,e^{3}+6 a^{2} c^{2} d \,e^{2}+a \,b^{3} e^{3}-15 a \,b^{2} c d \,e^{2}+15 a b \,c^{2} d^{2} e -10 a \,c^{3} d^{3}+3 b^{3} c \,d^{2} e -2 b^{2} c^{2} d^{3}\right ) x}{c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {8 a^{3} c \,e^{3}+a^{2} b^{2} e^{3}-18 a^{2} b c d \,e^{2}+24 a^{2} c^{2} d^{2} e +3 a \,b^{2} c \,d^{2} e -10 d^{3} a \,c^{2} b +b^{3} c \,d^{3}}{2 c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 \left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \sqrt {4 a c -b^{2}}}\)	\(485\)
risch	\(\text {Expression too large to display}\)	\(1170\)

Input:

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

Output:

(-3*c*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/(16*a^2*c^2-8* 
a*b^2*c+b^4)*x^3-1/2*(16*a^2*c^2*e^3+a*b^2*c*e^3-18*a*b*c^2*d*e^2+b^4*e^3- 
9*b^3*c*d*e^2+27*b^2*c^2*d^2*e-18*b*c^3*d^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/c* 
x^2-1/c*(5*a^2*b*c*e^3+6*a^2*c^2*d*e^2+a*b^3*e^3-15*a*b^2*c*d*e^2+15*a*b*c 
^2*d^2*e-10*a*c^3*d^3+3*b^3*c*d^2*e-2*b^2*c^2*d^3)/(16*a^2*c^2-8*a*b^2*c+b 
^4)*x-1/2/c*(8*a^3*c*e^3+a^2*b^2*e^3-18*a^2*b*c*d*e^2+24*a^2*c^2*d^2*e+3*a 
*b^2*c*d^2*e-10*a*b*c^2*d^3+b^3*c*d^3)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+ 
b*x+a)^2-6*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/(16*a^2*c 
^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1032 vs. \(2 (150) = 300\).

Time = 0.13 (sec) , antiderivative size = 2084, normalized size of antiderivative = 13.19 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*((b^5*c - 14*a*b^3*c^2 + 40*a^2*b*c^3)*d^3 + 3*(a*b^4*c + 4*a^2*b^2* 
c^2 - 32*a^3*c^3)*d^2*e - 18*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^2 + (a^2*b^4 + 
4*a^3*b^2*c - 32*a^4*c^2)*e^3 - 6*(2*(b^2*c^4 - 4*a*c^5)*d^3 - 3*(b^3*c^3 
- 4*a*b*c^4)*d^2*e + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d*e^2 - (a*b^3*c^ 
2 - 4*a^2*b*c^3)*e^3)*x^3 - (18*(b^3*c^3 - 4*a*b*c^4)*d^3 - 27*(b^4*c^2 - 
4*a*b^2*c^3)*d^2*e + 9*(b^5*c - 2*a*b^3*c^2 - 8*a^2*b*c^3)*d*e^2 - (b^6 - 
3*a*b^4*c + 12*a^2*b^2*c^2 - 64*a^3*c^3)*e^3)*x^2 + 6*(2*a^2*c^3*d^3 - 3*a 
^2*b*c^2*d^2*e - a^3*b*c*e^3 + (2*c^5*d^3 - 3*b*c^4*d^2*e - a*b*c^3*e^3 + 
(b^2*c^3 + 2*a*c^4)*d*e^2)*x^4 + (a^2*b^2*c + 2*a^3*c^2)*d*e^2 + 2*(2*b*c^ 
4*d^3 - 3*b^2*c^3*d^2*e - a*b^2*c^2*e^3 + (b^3*c^2 + 2*a*b*c^3)*d*e^2)*x^3 
 + (2*(b^2*c^3 + 2*a*c^4)*d^3 - 3*(b^3*c^2 + 2*a*b*c^3)*d^2*e + (b^4*c + 4 
*a*b^2*c^2 + 4*a^2*c^3)*d*e^2 - (a*b^3*c + 2*a^2*b*c^2)*e^3)*x^2 + 2*(2*a* 
b*c^3*d^3 - 3*a*b^2*c^2*d^2*e - a^2*b^2*c*e^3 + (a*b^3*c + 2*a^2*b*c^2)*d* 
e^2)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^ 
2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - 2*(2*(b^4*c^2 + a*b^2*c^3 - 2 
0*a^2*c^4)*d^3 - 3*(b^5*c + a*b^3*c^2 - 20*a^2*b*c^3)*d^2*e + 3*(5*a*b^4*c 
 - 22*a^2*b^2*c^2 + 8*a^3*c^3)*d*e^2 - (a*b^5 + a^2*b^3*c - 20*a^3*b*c^2)* 
e^3)*x)/(a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5*c^4 + (b^6*c 
^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^4 + 2*(b^7*c^2 - 12*a*b 
^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^3 + (b^8*c - 10*a*b^6*c^2 + 2...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (148) = 296\).

Time = 4.79 (sec) , antiderivative size = 1180, normalized size of antiderivative = 7.47 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

3*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)*log(x 
 + (-192*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d* 
e + c*d**2) + 144*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)* 
(a*e**2 - b*d*e + c*d**2) - 36*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 
2*c*d)*(a*e**2 - b*d*e + c*d**2) + 3*a*b**2*e**3 - 6*a*b*c*d*e**2 + 3*b**6 
*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2) - 3*b* 
*3*d*e**2 + 9*b**2*c*d**2*e - 6*b*c**2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e 
**2 - 6*b**2*c*d*e**2 + 18*b*c**2*d**2*e - 12*c**3*d**3)) - 3*sqrt(-1/(4*a 
*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)*log(x + (192*a**3*c 
**3*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2) - 1 
44*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d*e 
 + c*d**2) + 36*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(b*e - 2*c*d)*(a*e**2 
- b*d*e + c*d**2) + 3*a*b**2*e**3 - 6*a*b*c*d*e**2 - 3*b**6*sqrt(-1/(4*a*c 
 - b**2)**5)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2) - 3*b**3*d*e**2 + 9*b 
**2*c*d**2*e - 6*b*c**2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e**2 - 6*b**2*c* 
d*e**2 + 18*b*c**2*d**2*e - 12*c**3*d**3)) + (-8*a**3*c*e**3 - a**2*b**2*e 
**3 + 18*a**2*b*c*d*e**2 - 24*a**2*c**2*d**2*e - 3*a*b**2*c*d**2*e + 10*a* 
b*c**2*d**3 - b**3*c*d**3 + x**3*(-6*a*b*c**2*e**3 + 12*a*c**3*d*e**2 + 6* 
b**2*c**2*d*e**2 - 18*b*c**3*d**2*e + 12*c**4*d**3) + x**2*(-16*a**2*c**2* 
e**3 - a*b**2*c*e**3 + 18*a*b*c**2*d*e**2 - b**4*e**3 + 9*b**3*c*d*e**2...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (150) = 300\).

Time = 0.34 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.89 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\frac {6 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} e x^{3} + 6 \, b^{2} c^{2} d e^{2} x^{3} + 12 \, a c^{3} d e^{2} x^{3} - 6 \, a b c^{2} e^{3} x^{3} + 18 \, b c^{3} d^{3} x^{2} - 27 \, b^{2} c^{2} d^{2} e x^{2} + 9 \, b^{3} c d e^{2} x^{2} + 18 \, a b c^{2} d e^{2} x^{2} - b^{4} e^{3} x^{2} - a b^{2} c e^{3} x^{2} - 16 \, a^{2} c^{2} e^{3} x^{2} + 4 \, b^{2} c^{2} d^{3} x + 20 \, a c^{3} d^{3} x - 6 \, b^{3} c d^{2} e x - 30 \, a b c^{2} d^{2} e x + 30 \, a b^{2} c d e^{2} x - 12 \, a^{2} c^{2} d e^{2} x - 2 \, a b^{3} e^{3} x - 10 \, a^{2} b c e^{3} x - b^{3} c d^{3} + 10 \, a b c^{2} d^{3} - 3 \, a b^{2} c d^{2} e - 24 \, a^{2} c^{2} d^{2} e + 18 \, a^{2} b c d e^{2} - a^{2} b^{2} e^{3} - 8 \, a^{3} c e^{3}}{2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \] Input:

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

Output:

6*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)*arctan((2* 
c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4 
*a*c)) + 1/2*(12*c^4*d^3*x^3 - 18*b*c^3*d^2*e*x^3 + 6*b^2*c^2*d*e^2*x^3 + 
12*a*c^3*d*e^2*x^3 - 6*a*b*c^2*e^3*x^3 + 18*b*c^3*d^3*x^2 - 27*b^2*c^2*d^2 
*e*x^2 + 9*b^3*c*d*e^2*x^2 + 18*a*b*c^2*d*e^2*x^2 - b^4*e^3*x^2 - a*b^2*c* 
e^3*x^2 - 16*a^2*c^2*e^3*x^2 + 4*b^2*c^2*d^3*x + 20*a*c^3*d^3*x - 6*b^3*c* 
d^2*e*x - 30*a*b*c^2*d^2*e*x + 30*a*b^2*c*d*e^2*x - 12*a^2*c^2*d*e^2*x - 2 
*a*b^3*e^3*x - 10*a^2*b*c*e^3*x - b^3*c*d^3 + 10*a*b*c^2*d^3 - 3*a*b^2*c*d 
^2*e - 24*a^2*c^2*d^2*e + 18*a^2*b*c*d*e^2 - a^2*b^2*e^3 - 8*a^3*c*e^3)/(( 
b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*(c*x^2 + b*x + a)^2)
 

Mupad [B] (verification not implemented)

Time = 6.06 (sec) , antiderivative size = 636, normalized size of antiderivative = 4.03 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\frac {6\,\mathrm {atan}\left (\frac {\left (\frac {3\,\left (b\,e-2\,c\,d\right )\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {6\,c\,x\,\left (b\,e-2\,c\,d\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{3\,b^2\,d\,e^2-9\,b\,c\,d^2\,e-3\,a\,b\,e^3+6\,c^2\,d^3+6\,a\,c\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {8\,a^3\,c\,e^3+a^2\,b^2\,e^3-18\,a^2\,b\,c\,d\,e^2+24\,a^2\,c^2\,d^2\,e+3\,a\,b^2\,c\,d^2\,e-10\,a\,b\,c^2\,d^3+b^3\,c\,d^3}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (16\,a^2\,c^2\,e^3+a\,b^2\,c\,e^3-18\,a\,b\,c^2\,d\,e^2+b^4\,e^3-9\,b^3\,c\,d\,e^2+27\,b^2\,c^2\,d^2\,e-18\,b\,c^3\,d^3\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {3\,c\,x^3\,\left (b^2\,d\,e^2-3\,b\,c\,d^2\,e-a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x\,\left (5\,a^2\,b\,c\,e^3+6\,a^2\,c^2\,d\,e^2+a\,b^3\,e^3-15\,a\,b^2\,c\,d\,e^2+15\,a\,b\,c^2\,d^2\,e-10\,a\,c^3\,d^3+3\,b^3\,c\,d^2\,e-2\,b^2\,c^2\,d^3\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \] Input:

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

Output:

(6*atan((((3*(b*e - 2*c*d)*(b^5 + 16*a^2*b*c^2 - 8*a*b^3*c)*(a*e^2 + c*d^2 
 - b*d*e))/((4*a*c - b^2)^(5/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (6*c*x*( 
b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(5/2))*(b^4 + 16*a^2*c 
^2 - 8*a*b^2*c))/(6*c^2*d^3 + 3*b^2*d*e^2 - 3*a*b*e^3 + 6*a*c*d*e^2 - 9*b* 
c*d^2*e))*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(5/2) - ((8 
*a^3*c*e^3 + b^3*c*d^3 + a^2*b^2*e^3 + 24*a^2*c^2*d^2*e - 10*a*b*c^2*d^3 + 
 3*a*b^2*c*d^2*e - 18*a^2*b*c*d*e^2)/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) 
+ (x^2*(b^4*e^3 - 18*b*c^3*d^3 + 16*a^2*c^2*e^3 + 27*b^2*c^2*d^2*e + a*b^2 
*c*e^3 - 9*b^3*c*d*e^2 - 18*a*b*c^2*d*e^2))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b 
^2*c)) - (3*c*x^3*(2*c^2*d^3 + b^2*d*e^2 - a*b*e^3 + 2*a*c*d*e^2 - 3*b*c*d 
^2*e))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (x*(a*b^3*e^3 - 10*a*c^3*d^3 - 2*b 
^2*c^2*d^3 + 6*a^2*c^2*d*e^2 + 5*a^2*b*c*e^3 + 3*b^3*c*d^2*e + 15*a*b*c^2* 
d^2*e - 15*a*b^2*c*d*e^2))/(c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(2*a*c 
 + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2233, normalized size of antiderivative = 14.13 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c 
*e**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b* 
c**2*d*e**2 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
**2*b**3*c*d*e**2 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*a**2*b**3*c*e**3*x - 36*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* 
c - b**2))*a**2*b**2*c**2*d**2*e + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/ 
sqrt(4*a*c - b**2))*a**2*b**2*c**2*d*e**2*x - 24*sqrt(4*a*c - b**2)*atan(( 
b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**2*e**3*x**2 + 24*sqrt(4*a*c - 
b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**3*d**3 + 48*sqrt(4*a* 
c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**3*d*e**2*x**2 + 2 
4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*d*e**2* 
x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*e* 
*3*x**2 - 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b** 
3*c**2*d**2*e*x + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a*b**3*c**2*d*e**2*x**2 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a*b**3*c**2*e**3*x**3 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c* 
x)/sqrt(4*a*c - b**2))*a*b**2*c**3*d**3*x - 72*sqrt(4*a*c - b**2)*atan((b 
+ 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**3*d**2*e*x**2 + 48*sqrt(4*a*c - b** 
2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**3*d*e**2*x**3 - 12*sqrt( 
4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**3*e**3*x**...