\(\int \frac {(e+f x)^2}{(a c+(b c+a d) x+b d x^2)^3} \, dx\) [367]

Optimal result

Integrand size = 29, antiderivative size = 286 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b e-a f)^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {(b e-a f) (3 b d e-2 b c f-a d f)}{(b c-a d)^4 (a+b x)}+\frac {(d e-c f)^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {(d e-c f) (3 b d e-b c f-2 a d f)}{(b c-a d)^4 (c+d x)}+\frac {\left (a^2 d^2 f^2-2 a b d f (3 d e-2 c f)+b^2 \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (a^2 d^2 f^2-2 a b d f (3 d e-2 c f)+b^2 \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.95 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {-\frac {(b c-a d)^2 (b e-a f)^2}{(a+b x)^2}-\frac {2 (b c-a d) (b e-a f) (-3 b d e+2 b c f+a d f)}{a+b x}+\frac {(b c-a d)^2 (d e-c f)^2}{(c+d x)^2}+\frac {2 (b c-a d) (-d e+c f) (-3 b d e+b c f+2 a d f)}{c+d x}+2 \left (a^2 d^2 f^2+2 a b d f (-3 d e+2 c f)+b^2 \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \log (a+b x)-2 \left (a^2 d^2 f^2+2 a b d f (-3 d e+2 c f)+b^2 \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \log (c+d x)}{2 (b c-a d)^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 329, 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.

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

\(\Big \downarrow \) 1141

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

b^3*d^3*(-1/2*(b*e - a*f)^2/(b^3*d^3*(b*c - a*d)^3*(a + b*x)^2) + ((b*e - 
a*f)*(3*b*d*e - 2*b*c*f - a*d*f))/(b^3*d^3*(b*c - a*d)^4*(a + b*x)) + (d*e 
 - c*f)^2/(2*b^3*d^3*(b*c - a*d)^3*(c + d*x)^2) + ((d*e - c*f)*(3*b*d*e - 
b*c*f - 2*a*d*f))/(b^3*d^3*(b*c - a*d)^4*(c + d*x)) + ((a^2*d^2*f^2 - 2*a* 
b*d*f*(3*d*e - 2*c*f) + b^2*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*Log[a + b*x 
])/(b^3*d^3*(b*c - a*d)^5) - ((a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e - 2*c*f) + b 
^2*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*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 = 1.04 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.23

method	result	size
default	\(\frac {a^{2} d \,f^{2}+2 a b c \,f^{2}-4 a b d e f -2 b^{2} c e f +3 d \,e^{2} b^{2}}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {\left (a^{2} d^{2} f^{2}+4 a b c d \,f^{2}-6 a b \,d^{2} e f +b^{2} c^{2} f^{2}-6 b^{2} c d e f +6 d^{2} e^{2} b^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {a^{2} f^{2}-2 a b e f +b^{2} e^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {\left (a^{2} d^{2} f^{2}+4 a b c d \,f^{2}-6 a b \,d^{2} e f +b^{2} c^{2} f^{2}-6 b^{2} c d e f +6 d^{2} e^{2} b^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {c^{2} f^{2}-2 c e f d +d^{2} e^{2}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {2 a c \,f^{2} d -2 a \,d^{2} e f +b \,c^{2} f^{2}-4 b c d e f +3 b \,d^{2} e^{2}}{\left (a d -b c \right )^{4} \left (d x +c \right )}\)	\(353\)
norman	\(\frac {\frac {\left (a^{2} b^{2} d^{4} f^{2}+4 a \,b^{3} c \,d^{3} f^{2}-6 a \,b^{3} d^{4} e f +b^{4} c^{2} d^{2} f^{2}-6 b^{4} c \,d^{3} e f +6 b^{4} d^{4} e^{2}\right ) x^{3}}{d b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {\left (5 a^{3} b^{2} c \,d^{4} f^{2}-2 a^{3} b^{2} d^{5} e f +8 a^{2} b^{3} c^{2} d^{3} f^{2}-16 a^{2} b^{3} c \,d^{4} e f +2 a^{2} b^{3} d^{5} e^{2}+5 a \,b^{4} c^{3} d^{2} f^{2}-16 a \,b^{4} c^{2} d^{3} e f +14 a \,b^{4} c \,d^{4} e^{2}-2 b^{5} c^{3} d^{2} e f +2 b^{5} c^{2} d^{3} e^{2}\right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {6 a^{3} b^{2} c^{2} d^{3} f^{2}-2 a^{3} b^{2} c \,d^{4} e f -a^{3} b^{2} d^{5} e^{2}+6 a^{2} b^{3} c^{3} d^{2} f^{2}-20 a^{2} b^{3} c^{2} d^{3} e f +7 a^{2} b^{3} c \,d^{4} e^{2}-2 a \,b^{4} c^{3} d^{2} e f +7 a \,b^{4} c^{2} d^{3} e^{2}-b^{5} c^{3} d^{2} e^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}+\frac {\left (3 a^{3} b^{2} d^{5} f^{2}+15 a^{2} b^{3} c \,d^{4} f^{2}-18 a^{2} b^{3} d^{5} e f +15 a \,b^{4} c^{2} d^{3} f^{2}-36 a \,b^{4} c \,d^{4} e f +18 a \,b^{4} d^{5} e^{2}+3 b^{5} c^{3} d^{2} f^{2}-18 b^{5} c^{2} d^{3} e f +18 b^{5} c \,d^{4} e^{2}\right ) x^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{\left (d x +c \right )^{2} \left (b x +a \right )^{2}}+\frac {\left (a^{2} d^{2} f^{2}+4 a b c d \,f^{2}-6 a b \,d^{2} e f +b^{2} c^{2} f^{2}-6 b^{2} c d e f +6 d^{2} e^{2} b^{2}\right ) \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {\left (a^{2} d^{2} f^{2}+4 a b c d \,f^{2}-6 a b \,d^{2} e f +b^{2} c^{2} f^{2}-6 b^{2} c d e f +6 d^{2} e^{2} b^{2}\right ) \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\)	\(998\)
risch	\(\text {Expression too large to display}\)	\(1594\)
parallelrisch	\(\text {Expression too large to display}\)	\(2625\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1913 vs. \(2 (282) = 564\).

Time = 0.12 (sec) , antiderivative size = 1913, normalized size of antiderivative = 6.69 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2638 vs. \(2 (270) = 540\).

Time = 5.00 (sec) , antiderivative size = 2638, normalized size of antiderivative = 9.22 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(6*a**3*c**2*d*f**2 - 2*a**3*c*d**2*e*f - a**3*d**3*e**2 + 6*a**2*b*c**3*f 
**2 - 20*a**2*b*c**2*d*e*f + 7*a**2*b*c*d**2*e**2 - 2*a*b**2*c**3*e*f + 7* 
a*b**2*c**2*d*e**2 - b**3*c**3*e**2 + x**3*(2*a**2*b*d**3*f**2 + 8*a*b**2* 
c*d**2*f**2 - 12*a*b**2*d**3*e*f + 2*b**3*c**2*d*f**2 - 12*b**3*c*d**2*e*f 
 + 12*b**3*d**3*e**2) + x**2*(3*a**3*d**3*f**2 + 15*a**2*b*c*d**2*f**2 - 1 
8*a**2*b*d**3*e*f + 15*a*b**2*c**2*d*f**2 - 36*a*b**2*c*d**2*e*f + 18*a*b* 
*2*d**3*e**2 + 3*b**3*c**3*f**2 - 18*b**3*c**2*d*e*f + 18*b**3*c*d**2*e**2 
) + x*(10*a**3*c*d**2*f**2 - 4*a**3*d**3*e*f + 16*a**2*b*c**2*d*f**2 - 32* 
a**2*b*c*d**2*e*f + 4*a**2*b*d**3*e**2 + 10*a*b**2*c**3*f**2 - 32*a*b**2*c 
**2*d*e*f + 28*a*b**2*c*d**2*e**2 - 4*b**3*c**3*e*f + 4*b**3*c**2*d*e**2)) 
/(2*a**6*c**2*d**4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4*d**2 - 8*a**3* 
b**3*c**5*d + 2*a**2*b**4*c**6 + x**4*(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d* 
*5 + 12*a**2*b**4*c**2*d**4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x** 
3*(4*a**5*b*d**6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2*d**4 + 8*a**2*b* 
*4*c**3*d**3 - 12*a*b**5*c**4*d**2 + 4*b**6*c**5*d) + x**2*(2*a**6*d**6 - 
18*a**4*b**2*c**2*d**4 + 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2 + 
 2*b**6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4 + 8*a**4*b**2*c**3* 
d**3 + 8*a**3*b**3*c**4*d**2 - 12*a**2*b**4*c**5*d + 4*a*b**5*c**6)) + (a* 
*2*d**2*f**2 + 4*a*b*c*d*f**2 - 6*a*b*d**2*e*f + b**2*c**2*f**2 - 6*b**2*c 
*d*e*f + 6*b**2*d**2*e**2)*log(x + (-a**6*d**6*(a**2*d**2*f**2 + 4*a*b*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (282) = 564\).

Time = 0.06 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.35 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*b^2*d^2*e^2 - 6*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^ 
2)*f^2)*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) - (6*b^2*d^2*e^2 - 6*(b^2*c*d + a 
*b*d^2)*e*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2)*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*(2*(6*b^3*d^3*e^2 - 6*(b^3*c*d^2 + a*b^2*d^3)*e*f + (b^3*c 
^2*d + 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2)*x^3 - (b^3*c^3 - 7*a*b^2*c^2*d - 7* 
a^2*b*c*d^2 + a^3*d^3)*e^2 - 2*(a*b^2*c^3 + 10*a^2*b*c^2*d + a^3*c*d^2)*e* 
f + 6*(a^2*b*c^3 + a^3*c^2*d)*f^2 + 3*(6*(b^3*c*d^2 + a*b^2*d^3)*e^2 - 6*( 
b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f + (b^3*c^3 + 5*a*b^2*c^2*d + 5* 
a^2*b*c*d^2 + a^3*d^3)*f^2)*x^2 + 2*(2*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b* 
d^3)*e^2 - 2*(b^3*c^3 + 8*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*e*f + (5* 
a*b^2*c^3 + 8*a^2*b*c^2*d + 5*a^3*c*d^2)*f^2)*x)/(a^2*b^4*c^6 - 4*a^3*b^3* 
c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 
 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 
+ 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 
 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3* 
b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4* 
c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^ 
5)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (282) = 564\).

Time = 0.35 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.84 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {{\left (6 \, b^{3} d^{2} e^{2} - 6 \, b^{3} c d e f - 6 \, a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + 4 \, a b^{2} c d f^{2} + a^{2} b d^{2} f^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {{\left (6 \, b^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e f - 6 \, a b d^{3} e f + b^{2} c^{2} d f^{2} + 4 \, a b c d^{2} f^{2} + a^{2} d^{3} f^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} + \frac {12 \, b^{3} d^{3} e^{2} x^{3} - 12 \, b^{3} c d^{2} e f x^{3} - 12 \, a b^{2} d^{3} e f x^{3} + 2 \, b^{3} c^{2} d f^{2} x^{3} + 8 \, a b^{2} c d^{2} f^{2} x^{3} + 2 \, a^{2} b d^{3} f^{2} x^{3} + 18 \, b^{3} c d^{2} e^{2} x^{2} + 18 \, a b^{2} d^{3} e^{2} x^{2} - 18 \, b^{3} c^{2} d e f x^{2} - 36 \, a b^{2} c d^{2} e f x^{2} - 18 \, a^{2} b d^{3} e f x^{2} + 3 \, b^{3} c^{3} f^{2} x^{2} + 15 \, a b^{2} c^{2} d f^{2} x^{2} + 15 \, a^{2} b c d^{2} f^{2} x^{2} + 3 \, a^{3} d^{3} f^{2} x^{2} + 4 \, b^{3} c^{2} d e^{2} x + 28 \, a b^{2} c d^{2} e^{2} x + 4 \, a^{2} b d^{3} e^{2} x - 4 \, b^{3} c^{3} e f x - 32 \, a b^{2} c^{2} d e f x - 32 \, a^{2} b c d^{2} e f x - 4 \, a^{3} d^{3} e f x + 10 \, a b^{2} c^{3} f^{2} x + 16 \, a^{2} b c^{2} d f^{2} x + 10 \, a^{3} c d^{2} f^{2} x - b^{3} c^{3} e^{2} + 7 \, a b^{2} c^{2} d e^{2} + 7 \, a^{2} b c d^{2} e^{2} - a^{3} d^{3} e^{2} - 2 \, a b^{2} c^{3} e f - 20 \, a^{2} b c^{2} d e f - 2 \, a^{3} c d^{2} e f + 6 \, a^{2} b c^{3} f^{2} + 6 \, a^{3} c^{2} d f^{2}}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \] Input:

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

Output:

(6*b^3*d^2*e^2 - 6*b^3*c*d*e*f - 6*a*b^2*d^2*e*f + b^3*c^2*f^2 + 4*a*b^2*c 
*d*f^2 + a^2*b*d^2*f^2)*log(abs(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) - (6*b^2 
*d^3*e^2 - 6*b^2*c*d^2*e*f - 6*a*b*d^3*e*f + b^2*c^2*d*f^2 + 4*a*b*c*d^2*f 
^2 + a^2*d^3*f^2)*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^3* 
d^3*e^2*x^3 - 12*b^3*c*d^2*e*f*x^3 - 12*a*b^2*d^3*e*f*x^3 + 2*b^3*c^2*d*f^ 
2*x^3 + 8*a*b^2*c*d^2*f^2*x^3 + 2*a^2*b*d^3*f^2*x^3 + 18*b^3*c*d^2*e^2*x^2 
 + 18*a*b^2*d^3*e^2*x^2 - 18*b^3*c^2*d*e*f*x^2 - 36*a*b^2*c*d^2*e*f*x^2 - 
18*a^2*b*d^3*e*f*x^2 + 3*b^3*c^3*f^2*x^2 + 15*a*b^2*c^2*d*f^2*x^2 + 15*a^2 
*b*c*d^2*f^2*x^2 + 3*a^3*d^3*f^2*x^2 + 4*b^3*c^2*d*e^2*x + 28*a*b^2*c*d^2* 
e^2*x + 4*a^2*b*d^3*e^2*x - 4*b^3*c^3*e*f*x - 32*a*b^2*c^2*d*e*f*x - 32*a^ 
2*b*c*d^2*e*f*x - 4*a^3*d^3*e*f*x + 10*a*b^2*c^3*f^2*x + 16*a^2*b*c^2*d*f^ 
2*x + 10*a^3*c*d^2*f^2*x - b^3*c^3*e^2 + 7*a*b^2*c^2*d*e^2 + 7*a^2*b*c*d^2 
*e^2 - a^3*d^3*e^2 - 2*a*b^2*c^3*e*f - 20*a^2*b*c^2*d*e*f - 2*a^3*c*d^2*e* 
f + 6*a^2*b*c^3*f^2 + 6*a^3*c^2*d*f^2)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b 
^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(b*d*x^2 + b*c*x + a*d*x + a*c)^2)
 

Mupad [B] (verification not implemented)

Time = 6.37 (sec) , antiderivative size = 1021, normalized size of antiderivative = 3.57 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {\frac {x\,\left (5\,a^3\,c\,d^2\,f^2-2\,a^3\,d^3\,e\,f+8\,a^2\,b\,c^2\,d\,f^2-16\,a^2\,b\,c\,d^2\,e\,f+2\,a^2\,b\,d^3\,e^2+5\,a\,b^2\,c^3\,f^2-16\,a\,b^2\,c^2\,d\,e\,f+14\,a\,b^2\,c\,d^2\,e^2-2\,b^3\,c^3\,e\,f+2\,b^3\,c^2\,d\,e^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}-\frac {-6\,a^3\,c^2\,d\,f^2+2\,a^3\,c\,d^2\,e\,f+a^3\,d^3\,e^2-6\,a^2\,b\,c^3\,f^2+20\,a^2\,b\,c^2\,d\,e\,f-7\,a^2\,b\,c\,d^2\,e^2+2\,a\,b^2\,c^3\,e\,f-7\,a\,b^2\,c^2\,d\,e^2+b^3\,c^3\,e^2}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {3\,x^2\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2\,f^2+4\,a\,b\,c\,d\,f^2-6\,a\,b\,d^2\,e\,f+b^2\,c^2\,f^2-6\,b^2\,c\,d\,e\,f+6\,b^2\,d^2\,e^2\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b\,d\,x^3\,\left (a^2\,d^2\,f^2+4\,a\,b\,c\,d\,f^2-6\,a\,b\,d^2\,e\,f+b^2\,c^2\,f^2-6\,b^2\,c\,d\,e\,f+6\,b^2\,d^2\,e^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+2\,b\,d\,x\right )\,\left (d^2\,\left (a^2\,f^2-6\,a\,b\,e\,f+6\,b^2\,e^2\right )+c\,d\,\left (4\,a\,b\,f^2-6\,b^2\,e\,f\right )+b^2\,c^2\,f^2\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5\,\left (a^2\,d^2\,f^2+4\,a\,b\,c\,d\,f^2-6\,a\,b\,d^2\,e\,f+b^2\,c^2\,f^2-6\,b^2\,c\,d\,e\,f+6\,b^2\,d^2\,e^2\right )}\right )\,\left (d^2\,\left (a^2\,f^2-6\,a\,b\,e\,f+6\,b^2\,e^2\right )+c\,d\,\left (4\,a\,b\,f^2-6\,b^2\,e\,f\right )+b^2\,c^2\,f^2\right )}{{\left (a\,d-b\,c\right )}^5} \] Input:

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

Output:

((x*(5*a*b^2*c^3*f^2 + 2*a^2*b*d^3*e^2 + 5*a^3*c*d^2*f^2 + 2*b^3*c^2*d*e^2 
 - 2*a^3*d^3*e*f - 2*b^3*c^3*e*f + 14*a*b^2*c*d^2*e^2 + 8*a^2*b*c^2*d*f^2 
- 16*a*b^2*c^2*d*e*f - 16*a^2*b*c*d^2*e*f))/(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) - (a^3*d^3*e^2 + b^3*c^3*e^2 - 6 
*a^2*b*c^3*f^2 - 6*a^3*c^2*d*f^2 + 2*a*b^2*c^3*e*f + 2*a^3*c*d^2*e*f - 7*a 
*b^2*c^2*d*e^2 - 7*a^2*b*c*d^2*e^2 + 20*a^2*b*c^2*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)) + (3*x^2*(a*d + 
 b*c)*(a^2*d^2*f^2 + b^2*c^2*f^2 + 6*b^2*d^2*e^2 + 4*a*b*c*d*f^2 - 6*a*b*d 
^2*e*f - 6*b^2*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)) + (b*d*x^3*(a^2*d^2*f^2 + b^2*c^2*f^2 + 6*b^2*d 
^2*e^2 + 4*a*b*c*d*f^2 - 6*a*b*d^2*e*f - 6*b^2*c*d*e*f))/(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) - (2*atanh((((a^5*d^5 + b^5*c^5 + 2*a^2*b^3*c^ 
3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a*b^4*c^4*d - 3*a^4*b*c*d^4)/(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) + 2*b*d*x)*(d^2*( 
a^2*f^2 + 6*b^2*e^2 - 6*a*b*e*f) + c*d*(4*a*b*f^2 - 6*b^2*e*f) + b^2*c^2*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 
))/((a*d - b*c)^5*(a^2*d^2*f^2 + b^2*c^2*f^2 + 6*b^2*d^2*e^2 + 4*a*b*c*d*f 
^2 - 6*a*b*d^2*e*f - 6*b^2*c*d*e*f)))*(d^2*(a^2*f^2 + 6*b^2*e^2 - 6*a*b...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 3731, normalized size of antiderivative = 13.05 \[ \int \frac {(e+f x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*log(a + b*x)*a**5*c**2*d**3*f**2 - 4*log(a + b*x)*a**5*c*d**4*f**2*x 
 - 2*log(a + b*x)*a**5*d**5*f**2*x**2 - 10*log(a + b*x)*a**4*b*c**3*d**2*f 
**2 + 12*log(a + b*x)*a**4*b*c**2*d**3*e*f - 24*log(a + b*x)*a**4*b*c**2*d 
**3*f**2*x + 24*log(a + b*x)*a**4*b*c*d**4*e*f*x - 18*log(a + b*x)*a**4*b* 
c*d**4*f**2*x**2 + 12*log(a + b*x)*a**4*b*d**5*e*f*x**2 - 4*log(a + b*x)*a 
**4*b*d**5*f**2*x**3 - 10*log(a + b*x)*a**3*b**2*c**4*d*f**2 + 24*log(a + 
b*x)*a**3*b**2*c**3*d**2*e*f - 40*log(a + b*x)*a**3*b**2*c**3*d**2*f**2*x 
- 12*log(a + b*x)*a**3*b**2*c**2*d**3*e**2 + 72*log(a + b*x)*a**3*b**2*c** 
2*d**3*e*f*x - 52*log(a + b*x)*a**3*b**2*c**2*d**3*f**2*x**2 - 24*log(a + 
b*x)*a**3*b**2*c*d**4*e**2*x + 72*log(a + b*x)*a**3*b**2*c*d**4*e*f*x**2 - 
 24*log(a + b*x)*a**3*b**2*c*d**4*f**2*x**3 - 12*log(a + b*x)*a**3*b**2*d* 
*5*e**2*x**2 + 24*log(a + b*x)*a**3*b**2*d**5*e*f*x**3 - 2*log(a + b*x)*a* 
*3*b**2*d**5*f**2*x**4 - 2*log(a + b*x)*a**2*b**3*c**5*f**2 + 12*log(a + b 
*x)*a**2*b**3*c**4*d*e*f - 24*log(a + b*x)*a**2*b**3*c**4*d*f**2*x - 12*lo 
g(a + b*x)*a**2*b**3*c**3*d**2*e**2 + 72*log(a + b*x)*a**2*b**3*c**3*d**2* 
e*f*x - 52*log(a + b*x)*a**2*b**3*c**3*d**2*f**2*x**2 - 48*log(a + b*x)*a* 
*2*b**3*c**2*d**3*e**2*x + 120*log(a + b*x)*a**2*b**3*c**2*d**3*e*f*x**2 - 
 40*log(a + b*x)*a**2*b**3*c**2*d**3*f**2*x**3 - 60*log(a + b*x)*a**2*b**3 
*c*d**4*e**2*x**2 + 72*log(a + b*x)*a**2*b**3*c*d**4*e*f*x**3 - 10*log(a + 
 b*x)*a**2*b**3*c*d**4*f**2*x**4 - 24*log(a + b*x)*a**2*b**3*d**5*e**2*...