\(\int \frac {(b+2 c x) (d+e x)^2}{(a+b x+c x^2)^2} \, dx\) [555]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(d+e x)^2}{a+b x+c x^2}-\frac {2 e (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {e^2 \log \left (a+b x+c x^2\right )}{c} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {e^2 (a+b x)-c d (d+2 e x)}{a+x (b+c x)}-\frac {2 e (-2 c d+b e) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+e^2 \log (a+x (b+c x))}{c} \] Input:

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

Output:

((e^2*(a + b*x) - c*d*(d + 2*e*x))/(a + x*(b + c*x)) - (2*e*(-2*c*d + b*e) 
*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e^2*Log[a + 
x*(b + c*x)])/c
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1222, 1142, 1083, 219, 1103}

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.

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

\(\Big \downarrow \) 1222

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

-((d + e*x)^2/(a + b*x + c*x^2)) + 2*e*(-(((2*c*d - b*e)*ArcTanh[(b + 2*c* 
x)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c])) + (e*Log[a + b*x + c*x^2])/( 
2*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 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1222

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

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.26

method	result	size
default	\(\frac {\frac {e \left (b e -2 c d \right ) x}{c}+\frac {a \,e^{2}-c \,d^{2}}{c}}{c \,x^{2}+b x +a}+2 e \left (\frac {e \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (d -\frac {b e}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\)	\(110\)
risch	\(\frac {\frac {e \left (b e -2 c d \right ) x}{c}+\frac {a \,e^{2}-c \,d^{2}}{c}}{c \,x^{2}+b x +a}+\frac {4 e^{2} \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}-2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a}{4 a c -b^{2}}-\frac {e^{2} \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}-2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2}}{\left (4 a c -b^{2}\right ) c}+\frac {e \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}-2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{\left (4 a c -b^{2}\right ) c}+\frac {4 e^{2} \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}+2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a}{4 a c -b^{2}}-\frac {e^{2} \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}+2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2}}{\left (4 a c -b^{2}\right ) c}-\frac {e \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}+2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{\left (4 a c -b^{2}\right ) c}\)	\(700\)

Input:

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

Output:

(e*(b*e-2*c*d)/c*x+(a*e^2-c*d^2)/c)/(c*x^2+b*x+a)+2*e*(1/2*e*ln(c*x^2+b*x+ 
a)/c+2*(d-1/2*b/c*e)/(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. 277 vs. \(2 (83) = 166\).

Time = 0.09 (sec) , antiderivative size = 573, normalized size of antiderivative = 6.59 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\left [-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2} + {\left (2 \, a c d e - a b e^{2} + {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + {\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} x - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} e^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{a b^{2} c - 4 \, a^{2} c^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} x}, -\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2} + 2 \, {\left (2 \, a c d e - a b e^{2} + {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + {\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} x - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} e^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{a b^{2} c - 4 \, a^{2} c^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} x}\right ] \] Input:

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

Output:

[-((b^2*c - 4*a*c^2)*d^2 - (a*b^2 - 4*a^2*c)*e^2 + (2*a*c*d*e - a*b*e^2 + 
(2*c^2*d*e - b*c*e^2)*x^2 + (2*b*c*d*e - b^2*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*(b^2*c - 4*a*c^2)*d*e - (b^3 - 4*a*b*c)*e^2)*x - ((b^2* 
c - 4*a*c^2)*e^2*x^2 + (b^3 - 4*a*b*c)*e^2*x + (a*b^2 - 4*a^2*c)*e^2)*log( 
c*x^2 + b*x + a))/(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c 
- 4*a*b*c^2)*x), -((b^2*c - 4*a*c^2)*d^2 - (a*b^2 - 4*a^2*c)*e^2 + 2*(2*a* 
c*d*e - a*b*e^2 + (2*c^2*d*e - b*c*e^2)*x^2 + (2*b*c*d*e - b^2*e^2)*x)*sqr 
t(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2 
*(b^2*c - 4*a*c^2)*d*e - (b^3 - 4*a*b*c)*e^2)*x - ((b^2*c - 4*a*c^2)*e^2*x 
^2 + (b^3 - 4*a*b*c)*e^2*x + (a*b^2 - 4*a^2*c)*e^2)*log(c*x^2 + b*x + a))/ 
(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (78) = 156\).

Time = 1.88 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.91 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\left (\frac {e^{2}}{c} - \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e^{2}}{c} - \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{c \left (4 a c - b^{2}\right )}\right ) + 4 a e^{2} + b^{2} \left (\frac {e^{2}}{c} - \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{c \left (4 a c - b^{2}\right )}\right ) - 2 b d e}{2 b e^{2} - 4 c d e} \right )} + \left (\frac {e^{2}}{c} + \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e^{2}}{c} + \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{c \left (4 a c - b^{2}\right )}\right ) + 4 a e^{2} + b^{2} \left (\frac {e^{2}}{c} + \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{c \left (4 a c - b^{2}\right )}\right ) - 2 b d e}{2 b e^{2} - 4 c d e} \right )} + \frac {a e^{2} - c d^{2} + x \left (b e^{2} - 2 c d e\right )}{a c + b c x + c^{2} x^{2}} \] Input:

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

Output:

(e**2/c - e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(c*(4*a*c - b**2)))*log(x + 
(-4*a*c*(e**2/c - e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(c*(4*a*c - b**2))) 
+ 4*a*e**2 + b**2*(e**2/c - e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(c*(4*a*c 
- b**2))) - 2*b*d*e)/(2*b*e**2 - 4*c*d*e)) + (e**2/c + e*sqrt(-4*a*c + b** 
2)*(b*e - 2*c*d)/(c*(4*a*c - b**2)))*log(x + (-4*a*c*(e**2/c + e*sqrt(-4*a 
*c + b**2)*(b*e - 2*c*d)/(c*(4*a*c - b**2))) + 4*a*e**2 + b**2*(e**2/c + e 
*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(c*(4*a*c - b**2))) - 2*b*d*e)/(2*b*e** 
2 - 4*c*d*e)) + (a*e**2 - c*d**2 + x*(b*e**2 - 2*c*d*e))/(a*c + b*c*x + c* 
*2*x**2)
 

Maxima [F(-2)]

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

integrate((2*c*x+b)*(e*x+d)^2/(c*x^2+b*x+a)^2,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 [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.33 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^{2} \log \left (c x^{2} + b x + a\right )}{c} + \frac {2 \, {\left (2 \, c d e - b e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} - \frac {\frac {{\left (2 \, c d e - b e^{2}\right )} x}{c} + \frac {c d^{2} - a e^{2}}{c}}{c x^{2} + b x + a} \] Input:

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

Output:

e^2*log(c*x^2 + b*x + a)/c + 2*(2*c*d*e - b*e^2)*arctan((2*c*x + b)/sqrt(- 
b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c) - ((2*c*d*e - b*e^2)*x/c + (c*d^2 - a 
*e^2)/c)/(c*x^2 + b*x + a)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.85 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {a\,e^2}{c^2\,x^2+b\,c\,x+a\,c}-\frac {d^2}{c\,x^2+b\,x+a}+\frac {b\,e^2\,x}{c^2\,x^2+b\,c\,x+a\,c}-\frac {2\,d\,e\,x}{c\,x^2+b\,x+a}-\frac {b^2\,e^2\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}+\frac {4\,d\,e\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {2\,b\,e^2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}}+\frac {4\,a\,c\,e^2\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c} \] Input:

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

Output:

(a*e^2)/(a*c + c^2*x^2 + b*c*x) - d^2/(a + b*x + c*x^2) + (b*e^2*x)/(a*c + 
 c^2*x^2 + b*c*x) - (2*d*e*x)/(a + b*x + c*x^2) - (b^2*e^2*log(a + b*x + c 
*x^2))/(4*a*c^2 - b^2*c) + (4*d*e*atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4* 
a*c - b^2)^(1/2)))/(4*a*c - b^2)^(1/2) - (2*b*e^2*atan(b/(4*a*c - b^2)^(1/ 
2) + (2*c*x)/(4*a*c - b^2)^(1/2)))/(c*(4*a*c - b^2)^(1/2)) + (4*a*c*e^2*lo 
g(a + b*x + c*x^2))/(4*a*c^2 - b^2*c)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 517, normalized size of antiderivative = 5.94 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} e^{2}+4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c d e -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} e^{2} x +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c d e x -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c \,e^{2} x^{2}+4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{2} d e \,x^{2}+4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b c \,e^{2}-\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{3} e^{2}+4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{2} c \,e^{2} x +4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a b \,c^{2} e^{2} x^{2}-\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{4} e^{2} x -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{3} c \,e^{2} x^{2}+8 a^{2} c^{2} d e -2 a \,b^{2} c d e -4 a b \,c^{2} d^{2}-4 a b \,c^{2} e^{2} x^{2}+8 a \,c^{3} d e \,x^{2}+b^{3} c \,d^{2}+b^{3} c \,e^{2} x^{2}-2 b^{2} c^{2} d e \,x^{2}}{b c \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:

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

Output:

( - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*e**2 
+ 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d*e - 2* 
sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*e**2*x + 4*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*d*e*x - 2*sqr 
t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*e**2*x**2 + 4* 
sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**2*d*e*x**2 + 
4*log(a + b*x + c*x**2)*a**2*b*c*e**2 - log(a + b*x + c*x**2)*a*b**3*e**2 
+ 4*log(a + b*x + c*x**2)*a*b**2*c*e**2*x + 4*log(a + b*x + c*x**2)*a*b*c* 
*2*e**2*x**2 - log(a + b*x + c*x**2)*b**4*e**2*x - log(a + b*x + c*x**2)*b 
**3*c*e**2*x**2 + 8*a**2*c**2*d*e - 2*a*b**2*c*d*e - 4*a*b*c**2*d**2 - 4*a 
*b*c**2*e**2*x**2 + 8*a*c**3*d*e*x**2 + b**3*c*d**2 + b**3*c*e**2*x**2 - 2 
*b**2*c**2*d*e*x**2)/(b*c*(4*a**2*c - a*b**2 + 4*a*b*c*x + 4*a*c**2*x**2 - 
 b**3*x - b**2*c*x**2))