\(\int \frac {(d+e x)^5}{(a+b x+c x^2)^2} \, dx\) [476]

Optimal result

Integrand size = 20, antiderivative size = 434 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^4 (5 c d-2 b e) x}{c^3}+\frac {e^5 x^2}{2 c^2}-\frac {c \left (5 a b^3 d e^4-\frac {a b^4 e^5}{c}-2 a b^2 e^3 \left (5 c d^2-2 a e^2\right )+b c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \] Output:

e^4*(-2*b*e+5*c*d)*x/c^3+1/2*e^5*x^2/c^2-(c*(5*a*b^3*d*e^4-a*b^4*e^5/c-2*a 
*b^2*e^3*(-2*a*e^2+5*c*d^2)+b*c*d*(-15*a^2*e^4+10*a*c*d^2*e^2+c^2*d^4)-2*a 
*c*e*(a^2*e^4-10*a*c*d^2*e^2+5*c^2*d^4))+(-b*e+2*c*d)*(c^4*d^4+b^4*e^4-2*c 
^3*d^2*e*(5*a*e+b*d)-b^2*c*e^3*(5*a*e+3*b*d)+c^2*e^2*(5*a^2*e^2+10*a*b*d*e 
+4*b^2*d^2))*x)/c^4/(-4*a*c+b^2)/(c*x^2+b*x+a)+(-b*e+2*c*d)*(2*c^4*d^4-3*b 
^4*e^4-4*c^3*d^2*e*(-5*a*e+b*d)+4*b^2*c*e^3*(5*a*e+b*d)-2*c^2*e^2*(15*a^2* 
e^2+10*a*b*d*e+b^2*d^2))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*c 
+b^2)^(3/2)+1/2*e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(a*e+5*b*d))*ln(c*x^2+b*x+ 
a)/c^4
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2 c e^4 (5 c d-2 b e) x+c^2 e^5 x^2+\frac {2 \left (b^5 e^5 x+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (-2 c d^2 x+a e (d+e x)\right )-2 b^2 c e^2 \left (2 a^2 e^3+5 c^2 d^3 x-5 a c d e (d+2 e x)\right )+2 c^2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )+b c^2 \left (-c^2 d^4 (d-5 e x)+5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 (-2 c d+b e) \left (-2 c^4 d^4+3 b^4 e^4+4 c^3 d^2 e (b d-5 a e)-4 b^2 c e^3 (b d+5 a e)+2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (a+x (b+c x))}{2 c^4} \] Input:

Integrate[(d + e*x)^5/(a + b*x + c*x^2)^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1164, 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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(d+e x)^3 \left (2 c d^2-e (5 b d-8 a e)-3 e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle -\frac {\int \left (-\frac {3 (2 c d-b e) x^2 e^4}{c}-\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x e^3}{c^2}-\frac {\left (12 c^3 d^3-10 c^2 e (b d+3 a e) d-3 b^3 e^3+b c e^2 (10 b d+11 a e)\right ) e^2}{c^3}+\frac {2 c^4 d^5-5 c^3 e (b d-4 a e) d^3-10 a c^2 e^3 (b d+3 a e) d-3 a b^3 e^5+a b c e^4 (10 b d+11 a e)-\left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{c^3 \left (c x^2+b x+a\right )}\right )dx}{b^2-4 a c}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right )}{c^4 \sqrt {b^2-4 a c}}-\frac {e^2 x \left (-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right )}{c^3}-\frac {e^3 x^2 \left (-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right )}{2 c^2}-\frac {e^3 \left (b^2-4 a c\right ) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {e^4 x^3 (2 c d-b e)}{c}}{b^2-4 a c}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

Int[(d + e*x)^5/(a + b*x + c*x^2)^2,x]
 

Output:

-(((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + 
c*x^2))) - (-((e^2*(12*c^3*d^3 - 3*b^3*e^3 - 10*c^2*d*e*(b*d + 3*a*e) + b* 
c*e^2*(10*b*d + 11*a*e))*x)/c^3) - (e^3*(16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5 
*b*d + 4*a*e))*x^2)/(2*c^2) - (e^4*(2*c*d - b*e)*x^3)/c - ((2*c*d - b*e)*( 
2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a 
*e) - 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x)/S 
qrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*e^3*(10*c^2*d^ 
2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4))/(b^2 - 
 4*a*c)
 

Defintions of rubi rules used

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, 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 2009

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

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.49

method	result	size
default	\(-\frac {e^{4} \left (-\frac {1}{2} c e \,x^{2}+2 b e x -5 c d x \right )}{c^{3}}+\frac {\frac {-\frac {\left (5 a^{2} b \,c^{2} e^{5}-10 a^{2} c^{3} d \,e^{4}-5 a \,b^{3} c \,e^{5}+20 a \,b^{2} c^{2} d \,e^{4}-30 a b \,c^{3} d^{2} e^{3}+20 a \,c^{4} d^{3} e^{2}+b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{3} d^{2} e^{3} c^{2}-10 b^{2} c^{3} d^{3} e^{2}+5 b \,c^{4} d^{4} e -2 d^{5} c^{5}\right ) x}{\left (4 a c -b^{2}\right ) c}-\frac {2 a^{3} c^{2} e^{5}-4 a^{2} b^{2} c \,e^{5}+15 a^{2} b \,c^{2} d \,e^{4}-20 a^{2} c^{3} d^{2} e^{3}+a \,b^{4} e^{5}-5 a \,b^{3} c d \,e^{4}+10 a \,b^{2} c^{2} d^{2} e^{3}-10 a b \,c^{3} d^{3} e^{2}+10 a \,c^{4} d^{4} e -b \,c^{4} d^{5}}{\left (4 a c -b^{2}\right ) c}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 a^{2} c^{2} e^{5}+14 a \,b^{2} e^{5} c -40 a b \,c^{2} d \,e^{4}+40 e^{3} a \,c^{3} d^{2}-3 b^{4} e^{5}+10 b^{3} c d \,e^{4}-10 b^{2} c^{2} d^{2} e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (11 a^{2} b \,e^{5} c -30 a^{2} c^{2} d \,e^{4}-3 a \,b^{3} e^{5}+10 a \,b^{2} c d \,e^{4}-10 a b \,c^{2} d^{2} e^{3}+20 a \,c^{3} d^{3} e^{2}-5 b \,c^{3} d^{4} e +2 c^{4} d^{5}-\frac {\left (-8 a^{2} c^{2} e^{5}+14 a \,b^{2} e^{5} c -40 a b \,c^{2} d \,e^{4}+40 e^{3} a \,c^{3} d^{2}-3 b^{4} e^{5}+10 b^{3} c d \,e^{4}-10 b^{2} c^{2} d^{2} e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{3}}\)	\(645\)
risch	\(\text {Expression too large to display}\)	\(11237\)

Input:

int((e*x+d)^5/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-e^4/c^3*(-1/2*c*e*x^2+2*b*e*x-5*c*d*x)+1/c^3*((-(5*a^2*b*c^2*e^5-10*a^2*c 
^3*d*e^4-5*a*b^3*c*e^5+20*a*b^2*c^2*d*e^4-30*a*b*c^3*d^2*e^3+20*a*c^4*d^3* 
e^2+b^5*e^5-5*b^4*c*d*e^4+10*b^3*c^2*d^2*e^3-10*b^2*c^3*d^3*e^2+5*b*c^4*d^ 
4*e-2*c^5*d^5)/(4*a*c-b^2)/c*x-(2*a^3*c^2*e^5-4*a^2*b^2*c*e^5+15*a^2*b*c^2 
*d*e^4-20*a^2*c^3*d^2*e^3+a*b^4*e^5-5*a*b^3*c*d*e^4+10*a*b^2*c^2*d^2*e^3-1 
0*a*b*c^3*d^3*e^2+10*a*c^4*d^4*e-b*c^4*d^5)/(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^5+14*a*b^2*c*e^5-40*a*b*c^2*d*e^4+40*a*c^3 
*d^2*e^3-3*b^4*e^5+10*b^3*c*d*e^4-10*b^2*c^2*d^2*e^3)/c*ln(c*x^2+b*x+a)+2* 
(11*a^2*b*e^5*c-30*a^2*c^2*d*e^4-3*a*b^3*e^5+10*a*b^2*c*d*e^4-10*a*b*c^2*d 
^2*e^3+20*a*c^3*d^3*e^2-5*b*c^3*d^4*e+2*c^4*d^5-1/2*(-8*a^2*c^2*e^5+14*a*b 
^2*c*e^5-40*a*b*c^2*d*e^4+40*a*c^3*d^2*e^3-3*b^4*e^5+10*b^3*c*d*e^4-10*b^2 
*c^2*d^2*e^3)*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. 1371 vs. \(2 (426) = 852\).

Time = 0.17 (sec) , antiderivative size = 2763, normalized size of antiderivative = 6.37 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="fricas")
 

Output:

[1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*c^4 - 4*a*b*c^ 
5)*d^5 + 20*(a*b^2*c^4 - 4*a^2*c^5)*d^4*e - 20*(a*b^3*c^3 - 4*a^2*b*c^4)*d 
^3*e^2 + 20*(a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e^3 - 10*(a*b^5*c 
- 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*d*e^4 + 2*(a*b^6 - 8*a^2*b^4*c + 18*a^3*b^ 
2*c^2 - 8*a^4*c^3)*e^5 + (10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - 
3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^3 + (10*(b^5*c^2 - 8*a*b^3 
*c^3 + 16*a^2*b*c^4)*d*e^4 - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16 
*a^3*c^4)*e^5)*x^2 - (4*a*c^5*d^5 - 10*a*b*c^4*d^4*e + 40*a^2*c^4*d^3*e^2 
+ 10*(a*b^3*c^2 - 6*a^2*b*c^3)*d^2*e^3 - 10*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a 
^3*c^3)*d*e^4 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*e^5 + (4*c^6*d^5 - 
 10*b*c^5*d^4*e + 40*a*c^5*d^3*e^2 + 10*(b^3*c^3 - 6*a*b*c^4)*d^2*e^3 - 10 
*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d*e^4 + (3*b^5*c - 20*a*b^3*c^2 + 30* 
a^2*b*c^3)*e^5)*x^2 + (4*b*c^5*d^5 - 10*b^2*c^4*d^4*e + 40*a*b*c^4*d^3*e^2 
 + 10*(b^4*c^2 - 6*a*b^2*c^3)*d^2*e^3 - 10*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b* 
c^3)*d*e^4 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*e^5)*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*(2*(b^2*c^5 - 4*a*c^6)*d^5 - 5*(b^3*c^4 - 4*a*b*c^ 
5)*d^4*e + 10*(b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^2 - 10*(b^5*c^2 - 
7*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e^3 + 5*(b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2* 
c^3 - 24*a^3*c^4)*d*e^4 - (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2671 vs. \(2 (440) = 880\).

Time = 16.41 (sec) , antiderivative size = 2671, normalized size of antiderivative = 6.15 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**5/(c*x**2+b*x+a)**2,x)
 

Output:

x*(-2*b*e**5/c**3 + 5*d*e**4/c**2) + (-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10 
*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d) 
*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d* 
*2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3 
*d**3*e - 2*c**4*d**4)/(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**5 - 17*a**2*b**2*c*e**5 + 50*a**2* 
b*c**2*d*e**4 + 16*a**2*c**5*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e 
 - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a** 
2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 
+ 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e 
- 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b 
**6))) - 80*a**2*c**3*d**2*e**3 + 3*a*b**4*e**5 - 10*a*b**3*c*d*e**4 - 8*a 
*b**2*c**4*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/( 
2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a 
*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4* 
b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2* 
c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 10*a*b**2 
*c**2*d**2*e**3 + 20*a*b*c**3*d**3*e**2 + b**4*c**3*(-e**3*(2*a*c*e**2 - 3 
*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3 
)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^5/(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 = 0.32 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {{\left (4 \, c^{5} d^{5} - 10 \, b c^{4} d^{4} e + 40 \, a c^{4} d^{3} e^{2} + 10 \, b^{3} c^{2} d^{2} e^{3} - 60 \, a b c^{3} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 60 \, a b^{2} c^{2} d e^{4} - 60 \, a^{2} c^{3} d e^{4} + 3 \, b^{5} e^{5} - 20 \, a b^{3} c e^{5} + 30 \, a^{2} b c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (10 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} + 3 \, b^{2} e^{5} - 2 \, a c e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} e^{5} x^{2} + 10 \, c^{2} d e^{4} x - 4 \, b c e^{5} x}{2 \, c^{4}} - \frac {b c^{4} d^{5} - 10 \, a c^{4} d^{4} e + 10 \, a b c^{3} d^{3} e^{2} - 10 \, a b^{2} c^{2} d^{2} e^{3} + 20 \, a^{2} c^{3} d^{2} e^{3} + 5 \, a b^{3} c d e^{4} - 15 \, a^{2} b c^{2} d e^{4} - a b^{4} e^{5} + 4 \, a^{2} b^{2} c e^{5} - 2 \, a^{3} c^{2} e^{5} + {\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 20 \, a c^{4} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 30 \, a b c^{3} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - 20 \, a b^{2} c^{2} d e^{4} + 10 \, a^{2} c^{3} d e^{4} - b^{5} e^{5} + 5 \, a b^{3} c e^{5} - 5 \, a^{2} b c^{2} e^{5}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \] Input:

integrate((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="giac")
 

Output:

-(4*c^5*d^5 - 10*b*c^4*d^4*e + 40*a*c^4*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 60* 
a*b*c^3*d^2*e^3 - 10*b^4*c*d*e^4 + 60*a*b^2*c^2*d*e^4 - 60*a^2*c^3*d*e^4 + 
 3*b^5*e^5 - 20*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5)*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*(10*c^2*d^2*e^ 
3 - 10*b*c*d*e^4 + 3*b^2*e^5 - 2*a*c*e^5)*log(c*x^2 + b*x + a)/c^4 + 1/2*( 
c^2*e^5*x^2 + 10*c^2*d*e^4*x - 4*b*c*e^5*x)/c^4 - (b*c^4*d^5 - 10*a*c^4*d^ 
4*e + 10*a*b*c^3*d^3*e^2 - 10*a*b^2*c^2*d^2*e^3 + 20*a^2*c^3*d^2*e^3 + 5*a 
*b^3*c*d*e^4 - 15*a^2*b*c^2*d*e^4 - a*b^4*e^5 + 4*a^2*b^2*c*e^5 - 2*a^3*c^ 
2*e^5 + (2*c^5*d^5 - 5*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - 20*a*c^4*d^3*e^2 
 - 10*b^3*c^2*d^2*e^3 + 30*a*b*c^3*d^2*e^3 + 5*b^4*c*d*e^4 - 20*a*b^2*c^2* 
d*e^4 + 10*a^2*c^3*d*e^4 - b^5*e^5 + 5*a*b^3*c*e^5 - 5*a^2*b*c^2*e^5)*x)/( 
(c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)
 

Mupad [B] (verification not implemented)

Time = 6.36 (sec) , antiderivative size = 1083, normalized size of antiderivative = 2.50 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((d + e*x)^5/(a + b*x + c*x^2)^2,x)
 

Output:

(e^5*x^2)/(2*c^2) - x*((2*b*e^5)/c^3 - (5*d*e^4)/c^2) - (log(a + b*x + c*x 
^2)*(3*b^8*e^5 + 128*a^4*c^4*e^5 + 168*a^2*b^4*c^2*e^5 - 288*a^3*b^2*c^3*e 
^5 - 640*a^3*c^5*d^2*e^3 + 10*b^6*c^2*d^2*e^3 - 38*a*b^6*c*e^5 - 10*b^7*c* 
d*e^4 + 480*a^2*b^2*c^4*d^2*e^3 + 120*a*b^5*c^2*d*e^4 + 640*a^3*b*c^4*d*e^ 
4 - 120*a*b^4*c^3*d^2*e^3 - 480*a^2*b^3*c^3*d*e^4))/(2*(64*a^3*c^7 - b^6*c 
^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)) - ((a*b^4*e^5 - b*c^4*d^5 + 2*a^3*c^2 
*e^5 - 4*a^2*b^2*c*e^5 - 20*a^2*c^3*d^2*e^3 + 10*a*c^4*d^4*e - 5*a*b^3*c*d 
*e^4 - 10*a*b*c^3*d^3*e^2 + 15*a^2*b*c^2*d*e^4 + 10*a*b^2*c^2*d^2*e^3)/(c* 
(4*a*c - b^2)) + (x*(b^5*e^5 - 2*c^5*d^5 + 5*a^2*b*c^2*e^5 + 20*a*c^4*d^3* 
e^2 - 10*a^2*c^3*d*e^4 - 10*b^2*c^3*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 5*a*b^3 
*c*e^5 + 5*b*c^4*d^4*e - 5*b^4*c*d*e^4 - 30*a*b*c^3*d^2*e^3 + 20*a*b^2*c^2 
*d*e^4))/(c*(4*a*c - b^2)))/(a*c^3 + c^4*x^2 + b*c^3*x) - (atan((c^4*(((b^ 
3*c^3 - 4*a*b*c^4)*(b*e - 2*c*d)*(3*b^4*e^4 - 2*c^4*d^4 + 30*a^2*c^2*e^4 - 
 20*a*c^3*d^2*e^2 + 2*b^2*c^2*d^2*e^2 - 20*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 4 
*b^3*c*d*e^3 + 20*a*b*c^2*d*e^3))/(c^7*(4*a*c - b^2)^4) - (2*x*(b*e - 2*c* 
d)*(3*b^4*e^4 - 2*c^4*d^4 + 30*a^2*c^2*e^4 - 20*a*c^3*d^2*e^2 + 2*b^2*c^2* 
d^2*e^2 - 20*a*b^2*c*e^4 + 4*b*c^3*d^3*e - 4*b^3*c*d*e^3 + 20*a*b*c^2*d*e^ 
3))/(c^3*(4*a*c - b^2)^3))*(4*a*c - b^2)^(5/2))/(3*b^5*e^5 + 4*c^5*d^5 + 3 
0*a^2*b*c^2*e^5 + 40*a*c^4*d^3*e^2 - 60*a^2*c^3*d*e^4 + 10*b^3*c^2*d^2*e^3 
 - 20*a*b^3*c*e^5 - 10*b*c^4*d^4*e - 10*b^4*c*d*e^4 - 60*a*b*c^3*d^2*e^...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2993, normalized size of antiderivative = 6.90 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^5/(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**5 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b 
*c**3*d*e**4 - 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**2*b**4*c*e**5 + 120*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**4 + 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a**2*b**3*c**2*e**5*x - 120*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**3 - 120*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**4*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**5*x**2 + 
 80*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**4*d* 
*3*e**2 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2 
*b*c**4*d*e**4*x**2 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b 
**2))*a*b**6*e**5 - 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*a*b**5*c*d*e**4 - 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a*b**5*c*e**5*x + 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* 
c - b**2))*a*b**4*c**2*d**2*e**3 + 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*a*b**4*c**2*d*e**4*x - 40*sqrt(4*a*c - b**2)*atan((b 
+ 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c**2*e**5*x**2 - 120*sqrt(4*a*c - b**2 
)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**3*d**2*e**3*x + 120*sqrt( 
4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**3*d*e**4*x...