\(\int \frac {A+B x+C x^2}{(a d+(b d+a e) x+(c d+b e) x^2+c e x^3)^2} \, dx\) [128]

Optimal result

Integrand size = 43, antiderivative size = 676 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx=-\frac {e \left (b^2 \left (2 C d^2-e (B d-2 A e)\right )-b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )+2 \left (A c \left (c d^2-3 a e^2\right )+a \left (a C e^2-c d (3 C d-4 B e)\right )\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {A \left (b c d-b^2 e+2 a c e\right )-a (2 B c d-b C d-b B e+2 a C e)-((B c-b C) (b d-2 a e)-(A c-a C) (2 c d-b e)) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {\left (b^4 e^3 (B d-2 A e)+6 b^2 c e \left (B c d^3+2 a C d^2 e-a B d e^2+2 a A e^3\right )-b^3 e \left (a e^2 (2 C d-B e)+c d \left (2 C d^2+3 B d e-4 A e^2\right )\right )-2 b c \left (4 A c d e \left (c d^2+3 a e^2\right )+B \left (c^2 d^4+3 a^2 e^4\right )\right )+4 c \left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e \left (e^2 (b B d-2 a C d-2 A b e+a B e)+c d \left (2 C d^2-e (3 B d-4 A e)\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e \left (e^2 (b B d-2 a C d-2 A b e+a B e)+c d \left (2 C d^2-e (3 B d-4 A e)\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \] Output:

-e*(b^2*(2*C*d^2-e*(-2*A*e+B*d))-b*(2*(A*c+C*a)*d*e+B*(a*e^2+c*d^2))+2*A*c 
*(-3*a*e^2+c*d^2)+2*a*(a*C*e^2-c*d*(-4*B*e+3*C*d)))/(-4*a*c+b^2)/(a*e^2-b* 
d*e+c*d^2)^2/(e*x+d)-(A*(2*a*c*e-b^2*e+b*c*d)-a*(-B*b*e+2*B*c*d+2*C*a*e-C* 
b*d)-((B*c-C*b)*(-2*a*e+b*d)-(A*c-C*a)*(-b*e+2*c*d))*x)/(-4*a*c+b^2)/(a*e^ 
2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)+(b^4*e^3*(-2*A*e+B*d)+6*b^2*c*e*(2*A* 
a*e^3-B*a*d*e^2+B*c*d^3+2*C*a*d^2*e)-b^3*e*(a*e^2*(-B*e+2*C*d)+c*d*(-4*A*e 
^2+3*B*d*e+2*C*d^2))-2*b*c*(4*A*c*d*e*(3*a*e^2+c*d^2)+B*(3*a^2*e^4+c^2*d^4 
))+4*c*(A*c*(-3*a^2*e^4+6*a*c*d^2*e^2+c^2*d^4)+a*(a^2*C*e^4+c^2*d^3*(-2*B* 
e+C*d)-6*a*c*d*e^2*(-B*e+C*d))))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4 
*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^3+e*(e^2*(-2*A*b*e+B*a*e+B*b*d-2*C*a*d 
)+c*d*(2*C*d^2-e*(-4*A*e+3*B*d)))*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3-1/2*e*(e 
^2*(-2*A*b*e+B*a*e+B*b*d-2*C*a*d)+c*d*(2*C*d^2-e*(-4*A*e+3*B*d)))*ln(c*x^2 
+b*x+a)/(a*e^2-b*d*e+c*d^2)^3
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 661, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx=-\frac {e \left (C d^2+e (-B d+A e)\right )}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {b c (B c-b C) d^2 x-a^2 e (b C e+2 c (-2 C d+B e+C e x))-A \left (b^3 e^2+b^2 c e (-2 d+e x)+b c \left (-3 a e^2+c d (d-2 e x)\right )+2 c^2 \left (c d^2 x+a e (2 d-e x)\right )\right )+a \left (c C d (-b d+2 c d x+2 b e x)+B \left (b^2 e^2+2 c^2 d (d-2 e x)+b c e (-2 d+e x)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {\left (b^4 e^3 (B d-2 A e)+6 b^2 c e \left (B c d^3+2 a C d^2 e-a B d e^2+2 a A e^3\right )+b^3 \left (a e^3 (-2 C d+B e)+c d e \left (-2 C d^2-3 B d e+4 A e^2\right )\right )-2 b c \left (4 A c d e \left (c d^2+3 a e^2\right )+B \left (c^2 d^4+3 a^2 e^4\right )\right )+4 c \left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)+6 a c d e^2 (-C d+B e)\right )\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {e \left (2 c C d^3+c d e (-3 B d+4 A e)+e^2 (b B d-2 a C d-2 A b e+a B e)\right ) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}-\frac {e \left (2 c C d^3+c d e (-3 B d+4 A e)+e^2 (b B d-2 a C d-2 A b e+a B e)\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^3} \] Input:

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

Output:

-((e*(C*d^2 + e*(-(B*d) + A*e)))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x))) 
 + (b*c*(B*c - b*C)*d^2*x - a^2*e*(b*C*e + 2*c*(-2*C*d + B*e + C*e*x)) - A 
*(b^3*e^2 + b^2*c*e*(-2*d + e*x) + b*c*(-3*a*e^2 + c*d*(d - 2*e*x)) + 2*c^ 
2*(c*d^2*x + a*e*(2*d - e*x))) + a*(c*C*d*(-(b*d) + 2*c*d*x + 2*b*e*x) + B 
*(b^2*e^2 + 2*c^2*d*(d - 2*e*x) + b*c*e*(-2*d + e*x))))/((b^2 - 4*a*c)*(c* 
d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) - ((b^4*e^3*(B*d - 2*A*e) + 6 
*b^2*c*e*(B*c*d^3 + 2*a*C*d^2*e - a*B*d*e^2 + 2*a*A*e^3) + b^3*(a*e^3*(-2* 
C*d + B*e) + c*d*e*(-2*C*d^2 - 3*B*d*e + 4*A*e^2)) - 2*b*c*(4*A*c*d*e*(c*d 
^2 + 3*a*e^2) + B*(c^2*d^4 + 3*a^2*e^4)) + 4*c*(A*c*(c^2*d^4 + 6*a*c*d^2*e 
^2 - 3*a^2*e^4) + a*(a^2*C*e^4 + c^2*d^3*(C*d - 2*B*e) + 6*a*c*d*e^2*(-(C* 
d) + B*e))))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2) 
*(-(c*d^2) + e*(b*d - a*e))^3) + (e*(2*c*C*d^3 + c*d*e*(-3*B*d + 4*A*e) + 
e^2*(b*B*d - 2*a*C*d - 2*A*b*e + a*B*e))*Log[d + e*x])/(c*d^2 + e*(-(b*d) 
+ a*e))^3 - (e*(2*c*C*d^3 + c*d*e*(-3*B*d + 4*A*e) + e^2*(b*B*d - 2*a*C*d 
- 2*A*b*e + a*B*e))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)
 

Rubi [A] (verified)

Time = 3.13 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2462, 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.

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

\(\Big \downarrow \) 2462

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((e*(C*d^2 - e*(B*d - A*e)))/((c*d^2 - b*d*e + a*e^2)^2*(d + e*x))) + (A* 
(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2)) - a*(C*(b*c* 
d^2 - 4*a*c*d*e + a*b*e^2) - B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))) 
+ c*(2*b*(A*c + a*C)*d*e - 2*a^2*C*e^2 + 2*a*c*d*(C*d - 2*B*e) - 2*A*c*(c* 
d^2 - a*e^2) + b*B*(c*d^2 + a*e^2) - b^2*(C*d^2 + A*e^2))*x)/((b^2 - 4*a*c 
)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (2*c*(2*b*(A*c + a*C)*d*e 
 - 2*a^2*C*e^2 + 2*a*c*d*(C*d - 2*B*e) - 2*A*c*(c*d^2 - a*e^2) + b*B*(c*d^ 
2 + a*e^2) - b^2*(C*d^2 + A*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/ 
((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^2) + ((b*e^3*(b*B*d - 2*a*C*d 
 - 2*A*b*e + a*B*e) - 2*c^2*(C*d^4 - d^2*e*(2*B*d - 3*A*e)) - c*e^2*(3*b*d 
*(B*d - 2*A*e) - 2*a*(3*C*d^2 - 2*B*d*e + A*e^2)))*ArcTanh[(b + 2*c*x)/Sqr 
t[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^3) + (e*(2*c*C 
*d^3 - c*d*e*(3*B*d - 4*A*e) + e^2*(b*B*d - 2*a*C*d - 2*A*b*e + a*B*e))*Lo 
g[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A 
*e) + e^2*(b*B*d - 2*a*C*d - 2*A*b*e + a*B*e))*Log[a + b*x + c*x^2])/(2*(c 
*d^2 - b*d*e + a*e^2)^3)
 

Defintions of rubi rules used

rule 2009

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

rule 2462

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u*Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1352\) vs. \(2(669)=1338\).

Time = 1.06 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1353\)
risch	\(\text {Expression too large to display}\)	\(1359690\)

Input:

int((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x,method=_RETU 
RNVERBOSE)
 

Output:

-1/(a*e^2-b*d*e+c*d^2)^3*((c*(2*A*a^2*c*e^4-A*a*b^2*e^4+A*b^3*d*e^3-3*A*b^ 
2*c*d^2*e^2+4*A*b*c^2*d^3*e-2*A*c^3*d^4+B*a^2*b*e^4-4*B*a^2*c*d*e^3-B*a*b^ 
2*d*e^3+6*B*a*b*c*d^2*e^2-4*B*a*c^2*d^3*e-B*b^2*c*d^3*e+B*b*c^2*d^4-2*C*a^ 
3*e^4+4*C*a^2*b*d*e^3-3*C*a*b^2*d^2*e^2+2*C*a*c^2*d^4+C*b^3*d^3*e-C*b^2*c* 
d^4)/(4*a*c-b^2)*x+(3*A*a^2*b*c*e^4-4*A*a^2*c^2*d*e^3-A*a*b^3*e^4-A*a*b^2* 
c*d*e^3+6*A*a*b*c^2*d^2*e^2-4*A*a*c^3*d^3*e+A*b^4*d*e^3-3*A*b^3*c*d^2*e^2+ 
3*A*b^2*c^2*d^3*e-A*b*c^3*d^4-2*B*a^3*c*e^4+B*a^2*b^2*e^4-B*a*b^3*d*e^3+3* 
B*a*b^2*c*d^2*e^2-4*B*a*b*c^2*d^3*e+2*B*a*c^3*d^4-C*a^3*b*e^4+4*C*a^3*c*d* 
e^3+C*a^2*b^2*d*e^3-6*C*a^2*b*c*d^2*e^2+4*C*a^2*c^2*d^3*e+C*a*b^2*c*d^3*e- 
C*a*b*c^2*d^4)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)*(1/2*(-8*A*a*b*c^2 
*e^4+16*A*a*c^3*d*e^3+2*A*b^3*c*e^4-4*A*b^2*c^2*d*e^3+4*B*a^2*c^2*e^4-B*a* 
b^2*c*e^4+4*B*a*b*c^2*d*e^3-12*B*a*c^3*d^2*e^2-B*b^3*c*d*e^3+3*B*b^2*c^2*d 
^2*e^2-8*C*a^2*c^2*d*e^3+2*C*a*b^2*c*d*e^3+8*C*a*c^3*d^3*e-2*C*b^2*c^2*d^3 
*e)/c*ln(c*x^2+b*x+a)+2*(6*A*a^2*c^2*e^4-2*C*a^3*c*e^4-2*C*a*c^3*d^4+20*A* 
a*b*c^2*d*e^3+5*B*a*b^2*c*d*e^3-6*B*a*b*c^2*d^2*e^2-4*C*a^2*b*c*d*e^3+4*C* 
a*b*c^2*d^3*e-B*a*b^3*e^4-B*b^4*d*e^3-12*A*a*c^3*d^2*e^2+3*B*b^3*c*d^2*e^2 
-3*B*b^2*c^2*d^3*e+12*C*a^2*c^2*d^2*e^2+2*C*a*b^3*d*e^3-10*A*a*b^2*c*e^4-4 
*A*b^3*c*d*e^3+4*A*b*c^3*d^3*e+5*B*a^2*b*c*e^4-12*B*a^2*c^2*d*e^3+4*B*a*c^ 
3*d^3*e+B*b*c^3*d^4-6*C*a*b^2*c*d^2*e^2-2*A*c^4*d^4-1/2*(-8*A*a*b*c^2*e^4+ 
16*A*a*c^3*d*e^3+2*A*b^3*c*e^4-4*A*b^2*c^2*d*e^3+4*B*a^2*c^2*e^4-B*a*b^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x, algor 
ithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x, algor 
ithm="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. 1517 vs. \(2 (671) = 1342\).

Time = 0.15 (sec) , antiderivative size = 1517, normalized size of antiderivative = 2.24 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x, algor 
ithm="giac")
 

Output:

-1/2*(2*C*c*d^3*e - 3*B*c*d^2*e^2 - 2*C*a*d*e^3 + B*b*d*e^3 + 4*A*c*d*e^3 
+ B*a*e^4 - 2*A*b*e^4)*log(c*x^2 + b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b 
^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d 
^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) + (2*C*c*d^3*e^2 - 3*B 
*c*d^2*e^3 - 2*C*a*d*e^4 + B*b*d*e^4 + 4*A*c*d*e^4 + B*a*e^5 - 2*A*b*e^5)* 
log(abs(e*x + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2 
*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e 
^5 - 3*a^2*b*d*e^6 + a^3*e^7) - (4*C*a*c^3*d^4 - 2*B*b*c^3*d^4 + 4*A*c^4*d 
^4 - 2*C*b^3*c*d^3*e + 6*B*b^2*c^2*d^3*e - 8*B*a*c^3*d^3*e - 8*A*b*c^3*d^3 
*e + 12*C*a*b^2*c*d^2*e^2 - 3*B*b^3*c*d^2*e^2 - 24*C*a^2*c^2*d^2*e^2 + 24* 
A*a*c^3*d^2*e^2 - 2*C*a*b^3*d*e^3 + B*b^4*d*e^3 - 6*B*a*b^2*c*d*e^3 + 4*A* 
b^3*c*d*e^3 + 24*B*a^2*c^2*d*e^3 - 24*A*a*b*c^2*d*e^3 + B*a*b^3*e^4 - 2*A* 
b^4*e^4 + 4*C*a^3*c*e^4 - 6*B*a^2*b*c*e^4 + 12*A*a*b^2*c*e^4 - 12*A*a^2*c^ 
2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3*d^6 - 4*a*c^4*d^6 
- 3*b^3*c^2*d^5*e + 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b^2*c^2*d^4*e 
^2 - 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24*a^2*b*c^2*d 
^3*e^3 + 3*a*b^4*d^2*e^4 - 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^ 
2*b^3*d*e^5 + 12*a^3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*sqrt(-b^2 + 4* 
a*c)) - (2*C*b^2*c*d^2*e*x^2 - 6*C*a*c^2*d^2*e*x^2 - B*b*c^2*d^2*e*x^2 + 2 
*A*c^3*d^2*e*x^2 - 2*C*a*b*c*d*e^2*x^2 - B*b^2*c*d*e^2*x^2 + 8*B*a*c^2*...
 

Mupad [B] (verification not implemented)

Time = 17.01 (sec) , antiderivative size = 26278, normalized size of antiderivative = 38.87 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((A*a*b^2*e^3 + A*b*c^2*d^3 - 4*A*a^2*c*e^3 - 2*B*a*c^2*d^3 + A*b^3*d*e^2 
+ C*a*b*c*d^3 + 4*A*a*c^2*d^2*e - 2*B*a*b^2*d*e^2 - 2*A*b^2*c*d^2*e + 6*B* 
a^2*c*d*e^2 + C*a*b^2*d^2*e + C*a^2*b*d*e^2 - 8*C*a^2*c*d^2*e - 3*A*a*b*c* 
d*e^2 + 2*B*a*b*c*d^2*e)/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^ 
2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 
8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) - (x^2*(6*A*a*c^2*e 
^3 - 2*A*b^2*c*e^3 - 2*C*a^2*c*e^3 - 2*A*c^3*d^2*e + B*a*b*c*e^3 + 2*A*b*c 
^2*d*e^2 - 8*B*a*c^2*d*e^2 + B*b*c^2*d^2*e + B*b^2*c*d*e^2 + 6*C*a*c^2*d^2 
*e - 2*C*b^2*c*d^2*e + 2*C*a*b*c*d*e^2))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2* 
b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 
2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) + ( 
x*(2*A*b^3*e^3 + 2*A*c^3*d^3 - B*a*b^2*e^3 - B*b*c^2*d^3 + 2*B*a^2*c*e^3 - 
 2*C*a*c^2*d^3 + C*a^2*b*e^3 + C*b^2*c*d^3 - B*b^3*d*e^2 + C*b^3*d^2*e - 7 
*A*a*b*c*e^3 + 2*A*a*c^2*d*e^2 - A*b*c^2*d^2*e - A*b^2*c*d*e^2 + 2*B*a*c^2 
*d^2*e - 2*C*a^2*c*d*e^2 + 5*B*a*b*c*d*e^2 - 5*C*a*b*c*d^2*e))/(4*a*c^3*d^ 
4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2* 
e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 
2*a*b^2*c*d^2*e^2))/(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3) + sy 
msum(log((x*(36*A^2*a^2*c^5*e^7 + 4*A^2*b^4*c^3*e^7 + 4*C^2*a^4*c^3*e^7 + 
4*A^2*c^7*d^4*e^3 + B^2*a^2*b^2*c^3*e^7 + 12*A^2*b^2*c^5*d^2*e^5 + 64*B...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d**2*e** 
2 - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d*e**3*x 
 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*d**2*e** 
2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*d*e**3* 
x - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*d**3*e - 
 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*d**2*e**2*x 
 + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c**2*d**4 + 4 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c**2*d**3*e*x + 4 
*log(a + b*x + c*x**2)*a*b*c*d**2*e**2 + 4*log(a + b*x + c*x**2)*a*b*c*d*e 
**3*x - 8*log(a + b*x + c*x**2)*a*c**2*d**3*e - 8*log(a + b*x + c*x**2)*a* 
c**2*d**2*e**2*x - log(a + b*x + c*x**2)*b**3*d**2*e**2 - log(a + b*x + c* 
x**2)*b**3*d*e**3*x + 2*log(a + b*x + c*x**2)*b**2*c*d**3*e + 2*log(a + b* 
x + c*x**2)*b**2*c*d**2*e**2*x - 8*log(d + e*x)*a*b*c*d**2*e**2 - 8*log(d 
+ e*x)*a*b*c*d*e**3*x + 16*log(d + e*x)*a*c**2*d**3*e + 16*log(d + e*x)*a* 
c**2*d**2*e**2*x + 2*log(d + e*x)*b**3*d**2*e**2 + 2*log(d + e*x)*b**3*d*e 
**3*x - 4*log(d + e*x)*b**2*c*d**3*e - 4*log(d + e*x)*b**2*c*d**2*e**2*x + 
 8*a**2*c*e**4*x - 2*a*b**2*e**4*x - 8*a*b*c*d*e**3*x + 8*a*c**2*d**2*e**2 
*x + 2*b**3*d*e**3*x - 2*b**2*c*d**2*e**2*x)/(2*d*(4*a**3*c*d*e**4 + 4*a** 
3*c*e**5*x - a**2*b**2*d*e**4 - a**2*b**2*e**5*x - 8*a**2*b*c*d**2*e**3 - 
8*a**2*b*c*d*e**4*x + 8*a**2*c**2*d**3*e**2 + 8*a**2*c**2*d**2*e**3*x +...