\(\int \frac {(e+f x)^3}{(a c+(b c+a d) x+b d x^2)^3} \, dx\) [366]

Optimal result

Integrand size = 29, antiderivative size = 243 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b e-a f)^3}{2 b (b c-a d)^3 (a+b x)^2}+\frac {3 (b e-a f)^2 (d e-c f)}{(b c-a d)^4 (a+b x)}+\frac {(d e-c f)^3}{2 d (b c-a d)^3 (c+d x)^2}+\frac {3 (b e-a f) (d e-c f)^2}{(b c-a d)^4 (c+d x)}+\frac {3 (b e-a f) (d e-c f) (2 b d e-b c f-a d f) \log (a+b x)}{(b c-a d)^5}-\frac {3 (b e-a f) (d e-c f) (2 b d e-b c f-a d f) \log (c+d x)}{(b c-a d)^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.98 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {1}{2} \left (-\frac {(b e-a f)^3}{b (b c-a d)^3 (a+b x)^2}+\frac {6 (b e-a f)^2 (d e-c f)}{(b c-a d)^4 (a+b x)}-\frac {(d e-c f)^3}{d (-b c+a d)^3 (c+d x)^2}+\frac {6 (b e-a f) (d e-c f)^2}{(b c-a d)^4 (c+d x)}+\frac {6 (b e-a f) (-d e+c f) (-2 b d e+b c f+a d f) \log (a+b x)}{(b c-a d)^5}-\frac {6 (b e-a f) (-d e+c f) (-2 b d e+b c f+a d f) \log (c+d x)}{(b c-a d)^5}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1141, 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.

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

\(\Big \downarrow \) 1141

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1141

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[ 
(d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 
1] ||  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 
0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 2009

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

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}}{2 \left (a d -b c \right )^{3} b \left (b x +a \right )^{2}}-\frac {3 \left (a^{2} c \,f^{3}-a^{2} d e \,f^{2}-2 a b c e \,f^{2}+2 a b d \,e^{2} f +b^{2} c \,e^{2} f -b^{2} d \,e^{3}\right )}{\left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {3 \left (a^{2} c d \,f^{3}-a^{2} d^{2} e \,f^{2}+a b \,c^{2} f^{3}-4 a b c d e \,f^{2}+3 a b \,d^{2} e^{2} f -b^{2} c^{2} e \,f^{2}+3 b^{2} c d \,e^{2} f -2 b^{2} d^{2} e^{3}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {-c^{3} f^{3}+3 c^{2} d e \,f^{2}-3 c \,e^{2} f \,d^{2}+e^{3} d^{3}}{2 \left (a d -b c \right )^{3} d \left (d x +c \right )^{2}}-\frac {3 \left (a \,c^{2} f^{3}-2 a c e \,f^{2} d +a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -b \,d^{2} e^{3}\right )}{\left (a d -b c \right )^{4} \left (d x +c \right )}-\frac {3 \left (a^{2} c d \,f^{3}-a^{2} d^{2} e \,f^{2}+a b \,c^{2} f^{3}-4 a b c d e \,f^{2}+3 a b \,d^{2} e^{2} f -b^{2} c^{2} e \,f^{2}+3 b^{2} c d \,e^{2} f -2 b^{2} d^{2} e^{3}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}\)	\(474\)
norman	\(\text {Expression too large to display}\)	\(1299\)
risch	\(\text {Expression too large to display}\)	\(2257\)
parallelrisch	\(\text {Expression too large to display}\)	\(3709\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2669 vs. \(2 (239) = 478\).

Time = 0.13 (sec) , antiderivative size = 2669, normalized size of antiderivative = 10.98 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*((b^5*c^4*d - 8*a*b^4*c^3*d^2 + 8*a^3*b^2*c*d^4 - a^4*b*d^5)*e^3 + 3* 
(a*b^4*c^4*d + 9*a^2*b^3*c^3*d^2 - 9*a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*e^2*f 
- 18*(a^2*b^3*c^4*d - a^4*b*c^2*d^3)*e*f^2 + (a^2*b^3*c^5 + 9*a^3*b^2*c^4* 
d - 9*a^4*b*c^3*d^2 - a^5*c^2*d^3)*f^3 - 6*(2*(b^5*c*d^4 - a*b^4*d^5)*e^3 
- 3*(b^5*c^2*d^3 - a^2*b^3*d^5)*e^2*f + (b^5*c^3*d^2 + 3*a*b^4*c^2*d^3 - 3 
*a^2*b^3*c*d^4 - a^3*b^2*d^5)*e*f^2 - (a*b^4*c^3*d^2 - a^3*b^2*c*d^4)*f^3) 
*x^3 - (18*(b^5*c^2*d^3 - a^2*b^3*d^5)*e^3 - 27*(b^5*c^3*d^2 + a*b^4*c^2*d 
^3 - a^2*b^3*c*d^4 - a^3*b^2*d^5)*e^2*f + 9*(b^5*c^4*d + 4*a*b^4*c^3*d^2 - 
 4*a^3*b^2*c*d^4 - a^4*b*d^5)*e*f^2 - (b^5*c^5 + 4*a*b^4*c^4*d + 19*a^2*b^ 
3*c^3*d^2 - 19*a^3*b^2*c^2*d^3 - 4*a^4*b*c*d^4 - a^5*d^5)*f^3)*x^2 - 2*(2* 
(b^5*c^3*d^2 + 6*a*b^4*c^2*d^3 - 6*a^2*b^3*c*d^4 - a^3*b^2*d^5)*e^3 - 3*(b 
^5*c^4*d + 7*a*b^4*c^3*d^2 - 7*a^3*b^2*c*d^4 - a^4*b*d^5)*e^2*f + 3*(5*a*b 
^4*c^4*d + 3*a^2*b^3*c^3*d^2 - 3*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4)*e*f^2 - 
(a*b^4*c^5 + 7*a^2*b^3*c^4*d - 7*a^4*b*c^2*d^3 - a^5*c*d^4)*f^3)*x - 6*(2* 
a^2*b^3*c^2*d^3*e^3 + (2*b^5*d^5*e^3 - 3*(b^5*c*d^4 + a*b^4*d^5)*e^2*f + ( 
b^5*c^2*d^3 + 4*a*b^4*c*d^4 + a^2*b^3*d^5)*e*f^2 - (a*b^4*c^2*d^3 + a^2*b^ 
3*c*d^4)*f^3)*x^4 - 3*(a^2*b^3*c^3*d^2 + a^3*b^2*c^2*d^3)*e^2*f + (a^2*b^3 
*c^4*d + 4*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*e*f^2 - (a^3*b^2*c^4*d + a^4*b 
*c^3*d^2)*f^3 + 2*(2*(b^5*c*d^4 + a*b^4*d^5)*e^3 - 3*(b^5*c^2*d^3 + 2*a*b^ 
4*c*d^4 + a^2*b^3*d^5)*e^2*f + (b^5*c^3*d^2 + 5*a*b^4*c^2*d^3 + 5*a^2*b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2564 vs. \(2 (214) = 428\).

Time = 12.01 (sec) , antiderivative size = 2564, normalized size of antiderivative = 10.55 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(-a**4*c**2*d**2*f**3 - 10*a**3*b*c**3*d*f**3 + 18*a**3*b*c**2*d**2*e*f**2 
 - 3*a**3*b*c*d**3*e**2*f - a**3*b*d**4*e**3 - a**2*b**2*c**4*f**3 + 18*a* 
*2*b**2*c**3*d*e*f**2 - 30*a**2*b**2*c**2*d**2*e**2*f + 7*a**2*b**2*c*d**3 
*e**3 - 3*a*b**3*c**3*d*e**2*f + 7*a*b**3*c**2*d**2*e**3 - b**4*c**3*d*e** 
3 + x**3*(-6*a**2*b**2*c*d**3*f**3 + 6*a**2*b**2*d**4*e*f**2 - 6*a*b**3*c* 
*2*d**2*f**3 + 24*a*b**3*c*d**3*e*f**2 - 18*a*b**3*d**4*e**2*f + 6*b**4*c* 
*2*d**2*e*f**2 - 18*b**4*c*d**3*e**2*f + 12*b**4*d**4*e**3) + x**2*(-a**4* 
d**4*f**3 - 5*a**3*b*c*d**3*f**3 + 9*a**3*b*d**4*e*f**2 - 24*a**2*b**2*c** 
2*d**2*f**3 + 45*a**2*b**2*c*d**3*e*f**2 - 27*a**2*b**2*d**4*e**2*f - 5*a* 
b**3*c**3*d*f**3 + 45*a*b**3*c**2*d**2*e*f**2 - 54*a*b**3*c*d**3*e**2*f + 
18*a*b**3*d**4*e**3 - b**4*c**4*f**3 + 9*b**4*c**3*d*e*f**2 - 27*b**4*c**2 
*d**2*e**2*f + 18*b**4*c*d**3*e**3) + x*(-2*a**4*c*d**3*f**3 - 16*a**3*b*c 
**2*d**2*f**3 + 30*a**3*b*c*d**3*e*f**2 - 6*a**3*b*d**4*e**2*f - 16*a**2*b 
**2*c**3*d*f**3 + 48*a**2*b**2*c**2*d**2*e*f**2 - 48*a**2*b**2*c*d**3*e**2 
*f + 4*a**2*b**2*d**4*e**3 - 2*a*b**3*c**4*f**3 + 30*a*b**3*c**3*d*e*f**2 
- 48*a*b**3*c**2*d**2*e**2*f + 28*a*b**3*c*d**3*e**3 - 6*b**4*c**3*d*e**2* 
f + 4*b**4*c**2*d**2*e**3))/(2*a**6*b*c**2*d**5 - 8*a**5*b**2*c**3*d**4 + 
12*a**4*b**3*c**4*d**3 - 8*a**3*b**4*c**5*d**2 + 2*a**2*b**5*c**6*d + x**4 
*(2*a**4*b**3*d**7 - 8*a**3*b**4*c*d**6 + 12*a**2*b**5*c**2*d**5 - 8*a*b** 
6*c**3*d**4 + 2*b**7*c**4*d**3) + x**3*(4*a**5*b**2*d**7 - 12*a**4*b**3...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (239) = 478\).

Time = 0.07 (sec) , antiderivative size = 1243, normalized size of antiderivative = 5.12 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3*(2*b^2*d^2*e^3 - 3*(b^2*c*d + a*b*d^2)*e^2*f + (b^2*c^2 + 4*a*b*c*d + a^ 
2*d^2)*e*f^2 - (a*b*c^2 + a^2*c*d)*f^3)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^ 
4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 
 3*(2*b^2*d^2*e^3 - 3*(b^2*c*d + a*b*d^2)*e^2*f + (b^2*c^2 + 4*a*b*c*d + a 
^2*d^2)*e*f^2 - (a*b*c^2 + a^2*c*d)*f^3)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c 
^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) 
- 1/2*((b^4*c^3*d - 7*a*b^3*c^2*d^2 - 7*a^2*b^2*c*d^3 + a^3*b*d^4)*e^3 + 3 
*(a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*e^2*f - 18*(a^2*b^2*c^3* 
d + a^3*b*c^2*d^2)*e*f^2 + (a^2*b^2*c^4 + 10*a^3*b*c^3*d + a^4*c^2*d^2)*f^ 
3 - 6*(2*b^4*d^4*e^3 - 3*(b^4*c*d^3 + a*b^3*d^4)*e^2*f + (b^4*c^2*d^2 + 4* 
a*b^3*c*d^3 + a^2*b^2*d^4)*e*f^2 - (a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*f^3)*x^ 
3 - (18*(b^4*c*d^3 + a*b^3*d^4)*e^3 - 27*(b^4*c^2*d^2 + 2*a*b^3*c*d^3 + a^ 
2*b^2*d^4)*e^2*f + 9*(b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3* 
b*d^4)*e*f^2 - (b^4*c^4 + 5*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d 
^3 + a^4*d^4)*f^3)*x^2 - 2*(2*(b^4*c^2*d^2 + 7*a*b^3*c*d^3 + a^2*b^2*d^4)* 
e^3 - 3*(b^4*c^3*d + 8*a*b^3*c^2*d^2 + 8*a^2*b^2*c*d^3 + a^3*b*d^4)*e^2*f 
+ 3*(5*a*b^3*c^3*d + 8*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d^3)*e*f^2 - (a*b^3*c^4 
 + 8*a^2*b^2*c^3*d + 8*a^3*b*c^2*d^2 + a^4*c*d^3)*f^3)*x)/(a^2*b^5*c^6*d - 
 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5 
 + (b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1179 vs. \(2 (239) = 478\).

Time = 0.37 (sec) , antiderivative size = 1179, normalized size of antiderivative = 4.85 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

3*(2*b^3*d^2*e^3 - 3*b^3*c*d*e^2*f - 3*a*b^2*d^2*e^2*f + b^3*c^2*e*f^2 + 4 
*a*b^2*c*d*e*f^2 + a^2*b*d^2*e*f^2 - a*b^2*c^2*f^3 - a^2*b*c*d*f^3)*log(ab 
s(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2 
*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) - 3*(2*b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f 
 - 3*a*b*d^3*e^2*f + b^2*c^2*d*e*f^2 + 4*a*b*c*d^2*e*f^2 + a^2*d^3*e*f^2 - 
 a*b*c^2*d*f^3 - a^2*c*d^2*f^3)*log(abs(d*x + c))/(b^5*c^5*d - 5*a*b^4*c^4 
*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6) 
+ 1/2*(12*b^4*d^4*e^3*x^3 - 18*b^4*c*d^3*e^2*f*x^3 - 18*a*b^3*d^4*e^2*f*x^ 
3 + 6*b^4*c^2*d^2*e*f^2*x^3 + 24*a*b^3*c*d^3*e*f^2*x^3 + 6*a^2*b^2*d^4*e*f 
^2*x^3 - 6*a*b^3*c^2*d^2*f^3*x^3 - 6*a^2*b^2*c*d^3*f^3*x^3 + 18*b^4*c*d^3* 
e^3*x^2 + 18*a*b^3*d^4*e^3*x^2 - 27*b^4*c^2*d^2*e^2*f*x^2 - 54*a*b^3*c*d^3 
*e^2*f*x^2 - 27*a^2*b^2*d^4*e^2*f*x^2 + 9*b^4*c^3*d*e*f^2*x^2 + 45*a*b^3*c 
^2*d^2*e*f^2*x^2 + 45*a^2*b^2*c*d^3*e*f^2*x^2 + 9*a^3*b*d^4*e*f^2*x^2 - b^ 
4*c^4*f^3*x^2 - 5*a*b^3*c^3*d*f^3*x^2 - 24*a^2*b^2*c^2*d^2*f^3*x^2 - 5*a^3 
*b*c*d^3*f^3*x^2 - a^4*d^4*f^3*x^2 + 4*b^4*c^2*d^2*e^3*x + 28*a*b^3*c*d^3* 
e^3*x + 4*a^2*b^2*d^4*e^3*x - 6*b^4*c^3*d*e^2*f*x - 48*a*b^3*c^2*d^2*e^2*f 
*x - 48*a^2*b^2*c*d^3*e^2*f*x - 6*a^3*b*d^4*e^2*f*x + 30*a*b^3*c^3*d*e*f^2 
*x + 48*a^2*b^2*c^2*d^2*e*f^2*x + 30*a^3*b*c*d^3*e*f^2*x - 2*a*b^3*c^4*f^3 
*x - 16*a^2*b^2*c^3*d*f^3*x - 16*a^3*b*c^2*d^2*f^3*x - 2*a^4*c*d^3*f^3*x - 
 b^4*c^3*d*e^3 + 7*a*b^3*c^2*d^2*e^3 + 7*a^2*b^2*c*d^3*e^3 - a^3*b*d^4*...
 

Mupad [B] (verification not implemented)

Time = 6.81 (sec) , antiderivative size = 1312, normalized size of antiderivative = 5.40 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

- ((a^3*b*d^4*e^3 + b^4*c^3*d*e^3 + a^2*b^2*c^4*f^3 + a^4*c^2*d^2*f^3 + 10 
*a^3*b*c^3*d*f^3 - 7*a*b^3*c^2*d^2*e^3 - 7*a^2*b^2*c*d^3*e^3 - 18*a^2*b^2* 
c^3*d*e*f^2 - 18*a^3*b*c^2*d^2*e*f^2 + 30*a^2*b^2*c^2*d^2*e^2*f + 3*a*b^3* 
c^3*d*e^2*f + 3*a^3*b*c*d^3*e^2*f)/(2*b*d*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c 
^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x*(a*b^3*c^4*f^3 + a^4*c*d^3*f 
^3 - 2*a^2*b^2*d^4*e^3 - 2*b^4*c^2*d^2*e^3 - 14*a*b^3*c*d^3*e^3 + 3*a^3*b* 
d^4*e^2*f + 3*b^4*c^3*d*e^2*f + 8*a^2*b^2*c^3*d*f^3 + 8*a^3*b*c^2*d^2*f^3 
+ 24*a*b^3*c^2*d^2*e^2*f + 24*a^2*b^2*c*d^3*e^2*f - 24*a^2*b^2*c^2*d^2*e*f 
^2 - 15*a*b^3*c^3*d*e*f^2 - 15*a^3*b*c*d^3*e*f^2))/(b*d*(a^4*d^4 + b^4*c^4 
 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x^2*(a^4*d^4*f^3 
 + b^4*c^4*f^3 - 18*a*b^3*d^4*e^3 - 18*b^4*c*d^3*e^3 + 24*a^2*b^2*c^2*d^2* 
f^3 + 5*a*b^3*c^3*d*f^3 + 5*a^3*b*c*d^3*f^3 - 9*a^3*b*d^4*e*f^2 - 9*b^4*c^ 
3*d*e*f^2 + 27*a^2*b^2*d^4*e^2*f + 27*b^4*c^2*d^2*e^2*f - 45*a*b^3*c^2*d^2 
*e*f^2 - 45*a^2*b^2*c*d^3*e*f^2 + 54*a*b^3*c*d^3*e^2*f))/(2*b*d*(a^4*d^4 + 
 b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (3*b*d*x^ 
3*(2*b^2*d^2*e^3 + a^2*d^2*e*f^2 + b^2*c^2*e*f^2 - a*b*c^2*f^3 - a^2*c*d*f 
^3 - 3*a*b*d^2*e^2*f - 3*b^2*c*d*e^2*f + 4*a*b*c*d*e*f^2))/(a^4*d^4 + b^4* 
c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(x*(2*a*b*c^2 + 
2*a^2*c*d) + x^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*a*b*d^2 + 2*b^2* 
c*d) + a^2*c^2 + b^2*d^2*x^4) - (6*atanh((3*(a*f - b*e)*(c*f - d*e)*((a...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 5214, normalized size of antiderivative = 21.46 \[ \int \frac {(e+f x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*log(a + b*x)*a**5*b*c**3*d**3*f**3 - 6*log(a + b*x)*a**5*b*c**2*d**4*e* 
f**2 + 12*log(a + b*x)*a**5*b*c**2*d**4*f**3*x - 12*log(a + b*x)*a**5*b*c* 
d**5*e*f**2*x + 6*log(a + b*x)*a**5*b*c*d**5*f**3*x**2 - 6*log(a + b*x)*a* 
*5*b*d**6*e*f**2*x**2 + 12*log(a + b*x)*a**4*b**2*c**4*d**2*f**3 - 30*log( 
a + b*x)*a**4*b**2*c**3*d**3*e*f**2 + 36*log(a + b*x)*a**4*b**2*c**3*d**3* 
f**3*x + 18*log(a + b*x)*a**4*b**2*c**2*d**4*e**2*f - 72*log(a + b*x)*a**4 
*b**2*c**2*d**4*e*f**2*x + 36*log(a + b*x)*a**4*b**2*c**2*d**4*f**3*x**2 + 
 36*log(a + b*x)*a**4*b**2*c*d**5*e**2*f*x - 54*log(a + b*x)*a**4*b**2*c*d 
**5*e*f**2*x**2 + 12*log(a + b*x)*a**4*b**2*c*d**5*f**3*x**3 + 18*log(a + 
b*x)*a**4*b**2*d**6*e**2*f*x**2 - 12*log(a + b*x)*a**4*b**2*d**6*e*f**2*x* 
*3 + 6*log(a + b*x)*a**3*b**3*c**5*d*f**3 - 30*log(a + b*x)*a**3*b**3*c**4 
*d**2*e*f**2 + 36*log(a + b*x)*a**3*b**3*c**4*d**2*f**3*x + 36*log(a + b*x 
)*a**3*b**3*c**3*d**3*e**2*f - 120*log(a + b*x)*a**3*b**3*c**3*d**3*e*f**2 
*x + 60*log(a + b*x)*a**3*b**3*c**3*d**3*f**3*x**2 - 12*log(a + b*x)*a**3* 
b**3*c**2*d**4*e**3 + 108*log(a + b*x)*a**3*b**3*c**2*d**4*e**2*f*x - 156* 
log(a + b*x)*a**3*b**3*c**2*d**4*e*f**2*x**2 + 36*log(a + b*x)*a**3*b**3*c 
**2*d**4*f**3*x**3 - 24*log(a + b*x)*a**3*b**3*c*d**5*e**3*x + 108*log(a + 
 b*x)*a**3*b**3*c*d**5*e**2*f*x**2 - 72*log(a + b*x)*a**3*b**3*c*d**5*e*f* 
*2*x**3 + 6*log(a + b*x)*a**3*b**3*c*d**5*f**3*x**4 - 12*log(a + b*x)*a**3 
*b**3*d**6*e**3*x**2 + 36*log(a + b*x)*a**3*b**3*d**6*e**2*f*x**3 - 6*l...