\(\int \frac {(d+e x)^4 (f+g x)}{(a+b x+c x^2)^2} \, dx\) [756]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [F(-2)]
Giac [A] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 25, antiderivative size = 478 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^2 \left (12 c^3 d^2 f-3 b^3 e^2 g+b c e (2 b e f+8 b d g+11 a e g)-2 c^2 (b d (2 e f+3 d g)+3 a e (e f+4 d g))\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {e^3 \left (16 c^2 d f+3 b^2 e g-2 c (b e f+4 b d g+4 a e g)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {e^4 (2 c f-b g) x^3}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (b f-2 a g+(2 c f-b g) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (4 c^5 d^4 f+3 b^5 e^4 g-2 b^3 c e^3 (b e f+4 b d g+10 a e g)-2 c^4 d^2 (b d (4 e f+d g)-4 a e (3 e f+2 d g))-12 a c^3 e^2 (b d (2 e f+3 d g)+a e (e f+4 d g))+2 b c^2 e^2 \left (15 a^2 e^2 g+b^2 d (2 e f+3 d g)+6 a b e (e f+4 d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {e^2 \left (3 b^2 e^2 g+2 c^2 d (2 e f+3 d g)-2 c e (b e f+4 b d g+a e g)\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \] Output:

e^2*(12*c^3*d^2*f-3*b^3*e^2*g+b*c*e*(11*a*e*g+8*b*d*g+2*b*e*f)-2*c^2*(b*d* 
(3*d*g+2*e*f)+3*a*e*(4*d*g+e*f)))*x/c^3/(-4*a*c+b^2)+1/2*e^3*(16*c^2*d*f+3 
*b^2*e*g-2*c*(4*a*e*g+4*b*d*g+b*e*f))*x^2/c^2/(-4*a*c+b^2)+e^4*(-b*g+2*c*f 
)*x^3/c/(-4*a*c+b^2)-(e*x+d)^4*(b*f-2*a*g+(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(c* 
x^2+b*x+a)+(4*c^5*d^4*f+3*b^5*e^4*g-2*b^3*c*e^3*(10*a*e*g+4*b*d*g+b*e*f)-2 
*c^4*d^2*(b*d*(d*g+4*e*f)-4*a*e*(2*d*g+3*e*f))-12*a*c^3*e^2*(b*d*(3*d*g+2* 
e*f)+a*e*(4*d*g+e*f))+2*b*c^2*e^2*(15*a^2*e^2*g+b^2*d*(3*d*g+2*e*f)+6*a*b* 
e*(4*d*g+e*f)))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*c+b^2)^(3/ 
2)+1/2*e^2*(3*b^2*e^2*g+2*c^2*d*(3*d*g+2*e*f)-2*c*e*(a*e*g+4*b*d*g+b*e*f)) 
*ln(c*x^2+b*x+a)/c^4
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2 c e^3 (c e f+4 c d g-2 b e g) x+c^2 e^4 g x^2+\frac {2 \left (b^5 e^4 g x+b^4 e^3 (a e g-c (e f+4 d g) x)-b^3 c e^2 (-2 c d (2 e f+3 d g) x+a e (e f+4 d g+5 e g x))+2 c^2 \left (a^3 e^4 g-c^3 d^4 f x-a^2 c e^2 \left (6 d^2 g+e^2 f x+4 d e (f+g x)\right )+a c^2 d^2 \left (d^2 g+6 e^2 f x+4 d e (f+g x)\right )\right )+b c^2 \left (c^2 d^3 (-d f+4 e f x+d g x)+a^2 e^3 (3 e f+12 d g+5 e g x)-2 a c d e \left (2 d^2 g+6 e^2 f x+3 d e (f+3 g x)\right )\right )+2 b^2 c e \left (-2 a^2 e^3 g-c^2 d^2 (3 e f+2 d g) x+a c e \left (3 d^2 g+2 e^2 f x+2 d e (f+4 g x)\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \left (4 c^5 d^4 f+3 b^5 e^4 g-2 b^3 c e^3 (b e f+4 b d g+10 a e g)-2 c^4 d^2 (b d (4 e f+d g)-4 a e (3 e f+2 d g))-12 a c^3 e^2 (b d (2 e f+3 d g)+a e (e f+4 d g))+2 b c^2 e^2 \left (15 a^2 e^2 g+b^2 d (2 e f+3 d g)+6 a b e (e f+4 d g)\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+e^2 \left (3 b^2 e^2 g+2 c^2 d (2 e f+3 d g)-2 c e (b e f+4 b d g+a e g)\right ) \log (a+x (b+c x))}{2 c^4} \] Input:

Integrate[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^2,x]
 

Output:

(2*c*e^3*(c*e*f + 4*c*d*g - 2*b*e*g)*x + c^2*e^4*g*x^2 + (2*(b^5*e^4*g*x + 
 b^4*e^3*(a*e*g - c*(e*f + 4*d*g)*x) - b^3*c*e^2*(-2*c*d*(2*e*f + 3*d*g)*x 
 + a*e*(e*f + 4*d*g + 5*e*g*x)) + 2*c^2*(a^3*e^4*g - c^3*d^4*f*x - a^2*c*e 
^2*(6*d^2*g + e^2*f*x + 4*d*e*(f + g*x)) + a*c^2*d^2*(d^2*g + 6*e^2*f*x + 
4*d*e*(f + g*x))) + b*c^2*(c^2*d^3*(-(d*f) + 4*e*f*x + d*g*x) + a^2*e^3*(3 
*e*f + 12*d*g + 5*e*g*x) - 2*a*c*d*e*(2*d^2*g + 6*e^2*f*x + 3*d*e*(f + 3*g 
*x))) + 2*b^2*c*e*(-2*a^2*e^3*g - c^2*d^2*(3*e*f + 2*d*g)*x + a*c*e*(3*d^2 
*g + 2*e^2*f*x + 2*d*e*(f + 4*g*x)))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + 
 (2*(4*c^5*d^4*f + 3*b^5*e^4*g - 2*b^3*c*e^3*(b*e*f + 4*b*d*g + 10*a*e*g) 
- 2*c^4*d^2*(b*d*(4*e*f + d*g) - 4*a*e*(3*e*f + 2*d*g)) - 12*a*c^3*e^2*(b* 
d*(2*e*f + 3*d*g) + a*e*(e*f + 4*d*g)) + 2*b*c^2*e^2*(15*a^2*e^2*g + b^2*d 
*(2*e*f + 3*d*g) + 6*a*b*e*(e*f + 4*d*g)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 
4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^2*(3*b^2*e^2*g + 2*c^2*d*(2*e*f + 3*d*g) 
 - 2*c*e*(b*e*f + 4*b*d*g + a*e*g))*Log[a + x*(b + c*x)])/(2*c^4)
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1233, 25, 1200, 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.

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1200

\(\displaystyle \frac {(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \left (-\frac {\left (3 e g b^2+4 c^2 d f-2 c (b e f+b d g+4 a e g)\right ) x e^3}{c}-\frac {\left (-3 e^2 g b^3+c e (2 b e f+8 b d g+11 a e g) b+6 c^3 d^2 f-c^2 (b d (4 e f+3 d g)+6 a e (e f+4 d g))\right ) e^2}{c^2}+\frac {2 c^4 f d^4-c^3 (b d (4 e f+d g)-4 a e (3 e f+2 d g)) d^2-3 a b^3 e^4 g+a b c e^3 (2 b e f+8 b d g+11 a e g)-2 a c^2 e^2 (b d (2 e f+3 d g)+3 a e (e f+4 d g))-\left (b^2-4 a c\right ) e^2 \left (2 d (2 e f+3 d g) c^2-2 e (b e f+4 b d g+a e g) c+3 b^2 e^2 g\right ) x}{c^2 \left (c x^2+b x+a\right )}\right )dx}{c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 b c^2 e^2 \left (15 a^2 e^2 g+6 a b e (4 d g+e f)+b^2 d (3 d g+2 e f)\right )-2 b^3 c e^3 (10 a e g+4 b d g+b e f)-2 c^4 d^2 (b d (d g+4 e f)-4 a e (2 d g+3 e f))-12 a c^3 e^2 (a e (4 d g+e f)+b d (3 d g+2 e f))+3 b^5 e^4 g+4 c^5 d^4 f\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {e^2 x \left (-c^2 (6 a e (4 d g+e f)+b d (3 d g+4 e f))+b c e (11 a e g+8 b d g+2 b e f)-3 b^3 e^2 g+6 c^3 d^2 f\right )}{c^2}-\frac {e^3 x^2 \left (-2 c (4 a e g+b d g+b e f)+3 b^2 e g+4 c^2 d f\right )}{2 c}-\frac {e^2 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right ) \left (-2 c e (a e g+4 b d g+b e f)+3 b^2 e^2 g+2 c^2 d (3 d g+2 e f)\right )}{2 c^3}}{c \left (b^2-4 a c\right )}\)

Input:

Int[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^2,x]
 

Output:

((d + e*x)^3*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g 
 - c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - 
(-((e^2*(6*c^3*d^2*f - 3*b^3*e^2*g + b*c*e*(2*b*e*f + 8*b*d*g + 11*a*e*g) 
- c^2*(b*d*(4*e*f + 3*d*g) + 6*a*e*(e*f + 4*d*g)))*x)/c^2) - (e^3*(4*c^2*d 
*f + 3*b^2*e*g - 2*c*(b*e*f + b*d*g + 4*a*e*g))*x^2)/(2*c) - ((4*c^5*d^4*f 
 + 3*b^5*e^4*g - 2*b^3*c*e^3*(b*e*f + 4*b*d*g + 10*a*e*g) - 2*c^4*d^2*(b*d 
*(4*e*f + d*g) - 4*a*e*(3*e*f + 2*d*g)) - 12*a*c^3*e^2*(b*d*(2*e*f + 3*d*g 
) + a*e*(e*f + 4*d*g)) + 2*b*c^2*e^2*(15*a^2*e^2*g + b^2*d*(2*e*f + 3*d*g) 
 + 6*a*b*e*(e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sq 
rt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*e^2*(3*b^2*e^2*g + 2*c^2*d*(2*e*f + 3*d* 
g) - 2*c*e*(b*e*f + 4*b*d*g + a*e*g))*Log[a + b*x + c*x^2])/(2*c^3))/(c*(b 
^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1200
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1233
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1009\) vs. \(2(470)=940\).

Time = 1.91 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.11

method result size
default \(\text {Expression too large to display}\) \(1010\)
risch \(\text {Expression too large to display}\) \(30349\)

Input:

int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-e^3/c^3*(-1/2*c*e*g*x^2+2*b*e*g*x-4*c*d*g*x-c*e*f*x)+1/c^3*((-(5*a^2*b*c^ 
2*e^4*g-8*a^2*c^3*d*e^3*g-2*a^2*c^3*e^4*f-5*a*b^3*c*e^4*g+16*a*b^2*c^2*d*e 
^3*g+4*a*b^2*c^2*e^4*f-18*a*b*c^3*d^2*e^2*g-12*a*b*c^3*d*e^3*f+8*a*c^4*d^3 
*e*g+12*a*c^4*d^2*e^2*f+b^5*e^4*g-4*b^4*c*d*e^3*g-b^4*c*e^4*f+6*b^3*c^2*d^ 
2*e^2*g+4*b^3*c^2*d*e^3*f-4*b^2*c^3*d^3*e*g-6*b^2*c^3*d^2*e^2*f+b*c^4*d^4* 
g+4*b*c^4*d^3*e*f-2*c^5*d^4*f)/(4*a*c-b^2)/c*x-(2*a^3*c^2*e^4*g-4*a^2*b^2* 
c*e^4*g+12*a^2*b*c^2*d*e^3*g+3*a^2*b*c^2*e^4*f-12*a^2*c^3*d^2*e^2*g-8*a^2* 
c^3*d*e^3*f+a*b^4*e^4*g-4*a*b^3*c*d*e^3*g-a*b^3*c*e^4*f+6*a*b^2*c^2*d^2*e^ 
2*g+4*a*b^2*c^2*d*e^3*f-4*a*b*c^3*d^3*e*g-6*a*b*c^3*d^2*e^2*f+2*a*c^4*d^4* 
g+8*a*c^4*d^3*e*f-b*c^4*d^4*f)/(4*a*c-b^2)/c)/(c*x^2+b*x+a)+1/(4*a*c-b^2)* 
(1/2*(-8*a^2*c^2*e^4*g+14*a*b^2*c*e^4*g-32*a*b*c^2*d*e^3*g-8*a*b*c^2*e^4*f 
+24*a*c^3*d^2*e^2*g+16*a*c^3*d*e^3*f-3*b^4*e^4*g+8*b^3*c*d*e^3*g+2*b^3*c*e 
^4*f-6*b^2*c^2*d^2*e^2*g-4*b^2*c^2*d*e^3*f)/c*ln(c*x^2+b*x+a)+2*(11*a^2*b* 
e^4*g*c-24*a^2*c^2*d*e^3*g-6*a^2*c^2*e^4*f-3*a*b^3*e^4*g+8*a*b^2*c*d*e^3*g 
+2*a*b^2*c*e^4*f-6*a*b*c^2*d^2*e^2*g-4*a*b*c^2*d*e^3*f+8*a*c^3*d^3*e*g+12* 
a*c^3*d^2*e^2*f-b*c^3*d^4*g-4*b*c^3*d^3*e*f+2*c^4*d^4*f-1/2*(-8*a^2*c^2*e^ 
4*g+14*a*b^2*c*e^4*g-32*a*b*c^2*d*e^3*g-8*a*b*c^2*e^4*f+24*a*c^3*d^2*e^2*g 
+16*a*c^3*d*e^3*f-3*b^4*e^4*g+8*b^3*c*d*e^3*g+2*b^3*c*e^4*f-6*b^2*c^2*d^2* 
e^2*g-4*b^2*c^2*d*e^3*f)*b/c)/(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. 2144 vs. \(2 (470) = 940\).

Time = 0.87 (sec) , antiderivative size = 4309, normalized size of antiderivative = 9.01 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4692 vs. \(2 (500) = 1000\).

Time = 105.98 (sec) , antiderivative size = 4692, normalized size of antiderivative = 9.82 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**4*(g*x+f)/(c*x**2+b*x+a)**2,x)
 

Output:

x*(-2*b*e**4*g/c**3 + 4*d*e**3*g/c**2 + e**4*f/c**2) + (-e**2*(2*a*c*e**2* 
g - 3*b**2*e**2*g + 8*b*c*d*e*g + 2*b*c*e**2*f - 6*c**2*d**2*g - 4*c**2*d* 
e*f)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(30*a**2*b*c**2*e**4*g - 48*a**2* 
c**3*d*e**3*g - 12*a**2*c**3*e**4*f - 20*a*b**3*c*e**4*g + 48*a*b**2*c**2* 
d*e**3*g + 12*a*b**2*c**2*e**4*f - 36*a*b*c**3*d**2*e**2*g - 24*a*b*c**3*d 
*e**3*f + 16*a*c**4*d**3*e*g + 24*a*c**4*d**2*e**2*f + 3*b**5*e**4*g - 8*b 
**4*c*d*e**3*g - 2*b**4*c*e**4*f + 6*b**3*c**2*d**2*e**2*g + 4*b**3*c**2*d 
*e**3*f - 2*b*c**4*d**4*g - 8*b*c**4*d**3*e*f + 4*c**5*d**4*f)/(2*c**4*(64 
*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (16*a**3*c* 
*2*e**4*g - 17*a**2*b**2*c*e**4*g + 40*a**2*b*c**2*d*e**3*g + 10*a**2*b*c* 
*2*e**4*f + 16*a**2*c**5*(-e**2*(2*a*c*e**2*g - 3*b**2*e**2*g + 8*b*c*d*e* 
g + 2*b*c*e**2*f - 6*c**2*d**2*g - 4*c**2*d*e*f)/(2*c**4) - sqrt(-(4*a*c - 
 b**2)**3)*(30*a**2*b*c**2*e**4*g - 48*a**2*c**3*d*e**3*g - 12*a**2*c**3*e 
**4*f - 20*a*b**3*c*e**4*g + 48*a*b**2*c**2*d*e**3*g + 12*a*b**2*c**2*e**4 
*f - 36*a*b*c**3*d**2*e**2*g - 24*a*b*c**3*d*e**3*f + 16*a*c**4*d**3*e*g + 
 24*a*c**4*d**2*e**2*f + 3*b**5*e**4*g - 8*b**4*c*d*e**3*g - 2*b**4*c*e**4 
*f + 6*b**3*c**2*d**2*e**2*g + 4*b**3*c**2*d*e**3*f - 2*b*c**4*d**4*g - 8* 
b*c**4*d**3*e*f + 4*c**5*d**4*f)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 
 + 12*a*b**4*c - b**6))) - 48*a**2*c**3*d**2*e**2*g - 32*a**2*c**3*d*e**3* 
f + 3*a*b**4*e**4*g - 8*a*b**3*c*d*e**3*g - 2*a*b**3*c*e**4*f - 8*a*b**...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^4*(g*x+f)/(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 = 0.22 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.79 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^2,x, algorithm="giac")
 

Output:

-(4*c^5*d^4*f - 8*b*c^4*d^3*e*f + 24*a*c^4*d^2*e^2*f + 4*b^3*c^2*d*e^3*f - 
 24*a*b*c^3*d*e^3*f - 2*b^4*c*e^4*f + 12*a*b^2*c^2*e^4*f - 12*a^2*c^3*e^4* 
f - 2*b*c^4*d^4*g + 16*a*c^4*d^3*e*g + 6*b^3*c^2*d^2*e^2*g - 36*a*b*c^3*d^ 
2*e^2*g - 8*b^4*c*d*e^3*g + 48*a*b^2*c^2*d*e^3*g - 48*a^2*c^3*d*e^3*g + 3* 
b^5*e^4*g - 20*a*b^3*c*e^4*g + 30*a^2*b*c^2*e^4*g)*arctan((2*c*x + b)/sqrt 
(-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4*a*c)) + 1/2*(4*c^2*d*e^ 
3*f - 2*b*c*e^4*f + 6*c^2*d^2*e^2*g - 8*b*c*d*e^3*g + 3*b^2*e^4*g - 2*a*c* 
e^4*g)*log(c*x^2 + b*x + a)/c^4 + 1/2*(c^2*e^4*g*x^2 + 2*c^2*e^4*f*x + 8*c 
^2*d*e^3*g*x - 4*b*c*e^4*g*x)/c^4 - (b*c^4*d^4*f - 8*a*c^4*d^3*e*f + 6*a*b 
*c^3*d^2*e^2*f - 4*a*b^2*c^2*d*e^3*f + 8*a^2*c^3*d*e^3*f + a*b^3*c*e^4*f - 
 3*a^2*b*c^2*e^4*f - 2*a*c^4*d^4*g + 4*a*b*c^3*d^3*e*g - 6*a*b^2*c^2*d^2*e 
^2*g + 12*a^2*c^3*d^2*e^2*g + 4*a*b^3*c*d*e^3*g - 12*a^2*b*c^2*d*e^3*g - a 
*b^4*e^4*g + 4*a^2*b^2*c*e^4*g - 2*a^3*c^2*e^4*g + (2*c^5*d^4*f - 4*b*c^4* 
d^3*e*f + 6*b^2*c^3*d^2*e^2*f - 12*a*c^4*d^2*e^2*f - 4*b^3*c^2*d*e^3*f + 1 
2*a*b*c^3*d*e^3*f + b^4*c*e^4*f - 4*a*b^2*c^2*e^4*f + 2*a^2*c^3*e^4*f - b* 
c^4*d^4*g + 4*b^2*c^3*d^3*e*g - 8*a*c^4*d^3*e*g - 6*b^3*c^2*d^2*e^2*g + 18 
*a*b*c^3*d^2*e^2*g + 4*b^4*c*d*e^3*g - 16*a*b^2*c^2*d*e^3*g + 8*a^2*c^3*d* 
e^3*g - b^5*e^4*g + 5*a*b^3*c*e^4*g - 5*a^2*b*c^2*e^4*g)*x)/((c*x^2 + b*x 
+ a)*(b^2 - 4*a*c)*c^4)
 

Mupad [B] (verification not implemented)

Time = 15.03 (sec) , antiderivative size = 2222, normalized size of antiderivative = 4.65 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(((f + g*x)*(d + e*x)^4)/(a + b*x + c*x^2)^2,x)
 

Output:

x*((e^4*f + 4*d*e^3*g)/c^2 - (2*b*e^4*g)/c^3) - ((2*a^3*c^2*e^4*g + a*b^4* 
e^4*g + 2*a*c^4*d^4*g - b*c^4*d^4*f - a*b^3*c*e^4*f + 8*a*c^4*d^3*e*f + 3* 
a^2*b*c^2*e^4*f - 4*a^2*b^2*c*e^4*g - 8*a^2*c^3*d*e^3*f - 12*a^2*c^3*d^2*e 
^2*g + 6*a*b^2*c^2*d^2*e^2*g - 4*a*b*c^3*d^3*e*g - 4*a*b^3*c*d*e^3*g - 6*a 
*b*c^3*d^2*e^2*f + 4*a*b^2*c^2*d*e^3*f + 12*a^2*b*c^2*d*e^3*g)/(c*(4*a*c - 
 b^2)) - (x*(2*c^5*d^4*f - b^5*e^4*g + 2*a^2*c^3*e^4*f - b*c^4*d^4*g + b^4 
*c*e^4*f + 5*a*b^3*c*e^4*g - 8*a*c^4*d^3*e*g - 4*b*c^4*d^3*e*f + 4*b^4*c*d 
*e^3*g - 4*a*b^2*c^2*e^4*f - 5*a^2*b*c^2*e^4*g - 12*a*c^4*d^2*e^2*f + 8*a^ 
2*c^3*d*e^3*g - 4*b^3*c^2*d*e^3*f + 4*b^2*c^3*d^3*e*g + 6*b^2*c^3*d^2*e^2* 
f - 6*b^3*c^2*d^2*e^2*g + 12*a*b*c^3*d*e^3*f + 18*a*b*c^3*d^2*e^2*g - 16*a 
*b^2*c^2*d*e^3*g))/(c*(4*a*c - b^2)))/(a*c^3 + c^4*x^2 + b*c^3*x) - (log(( 
(6*a^2*c^2*e^4*f - 2*c^4*d^4*f + 3*a*b^3*e^4*g + b*c^3*d^4*g - 2*a*b^2*c*e 
^4*f - 11*a^2*b*c*e^4*g - 8*a*c^3*d^3*e*g + 4*b*c^3*d^3*e*f - 12*a*c^3*d^2 
*e^2*f + 24*a^2*c^2*d*e^3*g + 4*a*b*c^2*d*e^3*f - 8*a*b^2*c*d*e^3*g + 6*a* 
b*c^2*d^2*e^2*g)/(c^3*(4*a*c - b^2)) + ((b + 2*c*x)*(c^4*(-(4*c^5*d^4*f + 
3*b^5*e^4*g - 12*a^2*c^3*e^4*f - 2*b*c^4*d^4*g - 2*b^4*c*e^4*f - 20*a*b^3* 
c*e^4*g + 16*a*c^4*d^3*e*g - 8*b*c^4*d^3*e*f - 8*b^4*c*d*e^3*g + 12*a*b^2* 
c^2*e^4*f + 30*a^2*b*c^2*e^4*g + 24*a*c^4*d^2*e^2*f - 48*a^2*c^3*d*e^3*g + 
 4*b^3*c^2*d*e^3*f + 6*b^3*c^2*d^2*e^2*g - 24*a*b*c^3*d*e^3*f - 36*a*b*c^3 
*d^2*e^2*g + 48*a*b^2*c^2*d*e^3*g)^2/(c^8*(4*a*c - b^2)^3))^(1/2) + 3*b...
 

Reduce [B] (verification not implemented)

Time = 4.45 (sec) , antiderivative size = 4909, normalized size of antiderivative = 10.27 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^2,x)
 

Output:

(60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**2 
*e**4*g - 96*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3* 
b*c**3*d*e**3*g - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a**3*b*c**3*e**4*f - 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
- b**2))*a**2*b**4*c*e**4*g + 96*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*a**2*b**3*c**2*d*e**3*g + 24*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**2*e**4*f + 60*sqrt(4*a*c - b**2)*at 
an((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**2*e**4*g*x - 72*sqrt(4*a*c 
 - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**3*d**2*e**2*g - 
 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**3 
*d*e**3*f - 96*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a** 
2*b**2*c**3*d*e**3*g*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
 - b**2))*a**2*b**2*c**3*e**4*f*x + 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*a**2*b**2*c**3*e**4*g*x**2 + 32*sqrt(4*a*c - b**2)*at 
an((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**4*d**3*e*g + 48*sqrt(4*a*c - 
b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**4*d**2*e**2*f - 96*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**4*d*e**3*g 
*x**2 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* 
c**4*e**4*f*x**2 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a*b**6*e**4*g - 16*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - ...