\(\int \frac {(d+e x)^5}{(a+b x+c x^2)^3} \, dx\) [485]

Optimal result

Integrand size = 20, antiderivative size = 405 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {e^2 \left (18 c^3 d^3+b^3 e^3-7 a b c e^3-3 c^2 d e (7 b d-10 a e)\right ) x}{c^2 \left (b^2-4 a c\right )^2}-\frac {e^3 \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x^2}{2 c \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^3 \left (7 b^2 d e-4 a c d e-6 b \left (c d^2+a e^2\right )-\left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {e^5 \log \left (a+b x+c x^2\right )}{2 c^3} \] Output:

-e^2*(18*c^3*d^3+b^3*e^3-7*a*b*c*e^3-3*c^2*d*e*(-10*a*e+7*b*d))*x/c^2/(-4* 
a*c+b^2)^2-1/2*e^3*(12*c^2*d^2-b^2*e^2-4*c*e*(-4*a*e+3*b*d))*x^2/c/(-4*a*c 
+b^2)^2-1/2*(e*x+d)^4*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a 
)^2-1/2*(e*x+d)^3*(7*b^2*d*e-4*a*c*d*e-6*b*(a*e^2+c*d^2)-(12*c^2*d^2-b^2*e 
^2-4*c*e*(-4*a*e+3*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-(12*c^5*d^5-b^5*e 
^5+10*a*b^3*c*e^5-30*a^2*b*c^2*e^5-10*c^4*d^3*e*(-4*a*e+3*b*d)+20*c^3*d*e^ 
2*(3*a^2*e^2-3*a*b*d*e+b^2*d^2))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3 
/(-4*a*c+b^2)^(5/2)+1/2*e^5*ln(c*x^2+b*x+a)/c^3
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {b^5 e^5 x+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (-2 c d^2 x+a e (d+e x)\right )-2 b^2 c e^2 \left (2 a^2 e^3+5 c^2 d^3 x-5 a c d e (d+2 e x)\right )+2 c^2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )+b c^2 \left (-c^2 d^4 (d-5 e x)+5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {-b^6 e^5+b^5 c e^4 (5 d+4 e x)+b^4 c e^3 \left (11 a e^2-10 c d (d+e x)\right )+10 b^3 c^2 e^2 \left (c d^3-a e^2 (4 d+3 e x)\right )+4 c^3 \left (8 a^3 e^5+3 c^3 d^5 x+10 a c^2 d^3 e^2 x-5 a^2 c d e^3 (8 d+5 e x)\right )+2 b c^3 \left (3 c^2 d^4 (d-5 e x)+10 a c d^2 e^2 (d-3 e x)+5 a^2 e^4 (11 d+5 e x)\right )+b^2 c^2 e \left (-39 a^2 e^4-5 c^2 d^3 (3 d-4 e x)+10 a c d e^2 (5 d+8 e x)\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {2 c (2 c d-b e) \left (6 c^4 d^4+b^4 e^4+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+2 c^2 e^2 \left (2 b^2 d^2-10 a b d e+15 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+c e^5 \log (a+x (b+c x))}{2 c^4} \] Input:

Integrate[(d + e*x)^5/(a + b*x + c*x^2)^3,x]
 

Output:

((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) + (-(b^6*e^5) + 
b^5*c*e^4*(5*d + 4*e*x) + b^4*c*e^3*(11*a*e^2 - 10*c*d*(d + e*x)) + 10*b^3 
*c^2*e^2*(c*d^3 - a*e^2*(4*d + 3*e*x)) + 4*c^3*(8*a^3*e^5 + 3*c^3*d^5*x + 
10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(8*d + 5*e*x)) + 2*b*c^3*(3*c^2*d^4*(d 
- 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(11*d + 5*e*x)) + b^2*c^ 
2*e*(-39*a^2*e^4 - 5*c^2*d^3*(3*d - 4*e*x) + 10*a*c*d*e^2*(5*d + 8*e*x)))/ 
((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*c*(2*c*d - b*e)*(6*c^4*d^4 + b^4* 
e^4 + 2*b^2*c*e^3*(b*d - 5*a*e) - 4*c^3*d^2*e*(3*b*d - 5*a*e) + 2*c^2*e^2* 
(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)^(5/2) + c*e^5*Log[a + x*(b + c*x)])/(2*c^4)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1164, 1233, 27, 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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(d+e x)^3 \left (6 c d^2-e (7 b d-8 a e)-e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {\frac {\int -\frac {2 (d+e x) \left (6 c^3 d^4-c^2 e (15 b d-14 a e) d^2-a b^2 e^4+c e^2 \left (10 b^2 d^2-21 a b e d+16 a^2 e^2\right )-e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}+\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \frac {(d+e x) \left (6 c^3 d^4-c^2 e (15 b d-14 a e) d^2-a b^2 e^4+c e^2 \left (10 b^2 d^2-21 a b e d+16 a^2 e^2\right )-e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1200

\(\displaystyle -\frac {\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \left (\frac {6 c^4 d^5-5 c^3 e (3 b d-4 a e) d^3+10 c^2 e^2 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right ) d+a b^3 e^5-7 a^2 b c e^5+\left (b^2-4 a c\right )^2 e^5 x}{c \left (c x^2+b x+a\right )}-\frac {e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right )}{c}\right )dx}{c \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {e^2 x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )}{c}+\frac {e^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )}{2 c^2}\right )}{c \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

Int[(d + e*x)^5/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*((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)^2) - (((d + e*x)^2*(8*a*c*e*(c*d^2 + 2*a*e^2) - 6*b*c*d*(c*d^2 + 
 3*a*e^2) + b^2*(7*c*d^2*e - a*e^3) - (2*c*d - b*e)*(6*c^2*d^2 - b^2*e^2 - 
 2*c*e*(3*b*d - 5*a*e))*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*(-((e 
^2*(2*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/c) - ((12* 
c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b* 
d - 4*a*e) + 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 
2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)^2*e^5* 
Log[a + b*x + c*x^2])/(2*c^2)))/(c*(b^2 - 4*a*c)))/(2*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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. \(973\) vs. \(2(393)=786\).

Time = 1.10 (sec) , antiderivative size = 974, normalized size of antiderivative = 2.40

method	result	size
default	\(\frac {\frac {\left (25 a^{2} b \,c^{2} e^{5}-50 a^{2} c^{3} d \,e^{4}-15 a \,b^{3} c \,e^{5}+40 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}+2 b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{2} c^{3} d^{3} e^{2}-15 b \,c^{4} d^{4} e +6 d^{5} c^{5}\right ) x^{3}}{c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {\left (32 a^{3} c^{3} e^{5}+11 a^{2} b^{2} c^{2} e^{5}+10 a^{2} b \,c^{3} d \,e^{4}-160 a^{2} c^{4} d^{2} e^{3}-19 a \,b^{4} c \,e^{5}+40 a \,b^{3} c^{2} d \,e^{4}-10 a \,b^{2} c^{3} d^{2} e^{3}+60 a b \,c^{4} d^{3} e^{2}+3 b^{6} e^{5}-5 b^{5} c d \,e^{4}-10 b^{4} c^{2} d^{2} e^{3}+30 b^{3} c^{3} d^{3} e^{2}-45 b^{2} c^{4} d^{4} e +18 b \,c^{5} d^{5}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c^{3}}+\frac {\left (31 a^{3} b \,c^{2} e^{5}-30 a^{3} c^{3} d \,e^{4}-22 a^{2} b^{3} c \,e^{5}+50 a^{2} b^{2} c^{2} d \,e^{4}-50 a^{2} b \,c^{3} d^{2} e^{3}-20 a^{2} c^{4} d^{3} e^{2}+3 a \,b^{5} e^{5}-5 a \,b^{4} c d \,e^{4}-10 a \,b^{3} c^{2} d^{2} e^{3}+50 a \,b^{2} c^{3} d^{3} e^{2}-25 a b \,c^{4} d^{4} e +10 a \,c^{5} d^{5}-5 b^{3} c^{3} d^{4} e +2 b^{2} c^{4} d^{5}\right ) x}{\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) c^{3}}+\frac {24 a^{4} c^{2} e^{5}-21 a^{3} b^{2} c \,e^{5}+50 a^{3} b \,c^{2} d \,e^{4}-80 a^{3} c^{3} d^{2} e^{3}+3 a^{2} b^{4} e^{5}-5 a^{2} b^{3} c d \,e^{4}-10 a^{2} b^{2} c^{2} d^{2} e^{3}+60 a^{2} b \,c^{3} d^{3} e^{2}-40 a^{2} c^{4} d^{4} e -5 a \,b^{2} c^{3} d^{4} e +10 a b \,c^{4} d^{5}-b^{3} c^{3} d^{5}}{2 c^{3} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{2} e^{5}-8 a \,b^{2} e^{5} c +b^{4} e^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-7 a^{2} b \,e^{5} c +30 a^{2} c^{2} d \,e^{4}+a \,b^{3} e^{5}-30 a b \,c^{2} d^{2} e^{3}+20 a \,c^{3} d^{3} e^{2}+10 b^{2} c^{2} d^{3} e^{2}-15 b \,c^{3} d^{4} e +6 c^{4} d^{5}-\frac {\left (16 a^{2} c^{2} e^{5}-8 a \,b^{2} e^{5} c +b^{4} e^{5}\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}}}}{c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\)	\(974\)
risch	\(\text {Expression too large to display}\)	\(5551\)

Input:

int((e*x+d)^5/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

((25*a^2*b*c^2*e^5-50*a^2*c^3*d*e^4-15*a*b^3*c*e^5+40*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+2*b^5*e^5-5*b^4*c*d*e^4+10*b^2*c^3*d^3*e^2 
-15*b*c^4*d^4*e+6*c^5*d^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(32*a^3* 
c^3*e^5+11*a^2*b^2*c^2*e^5+10*a^2*b*c^3*d*e^4-160*a^2*c^4*d^2*e^3-19*a*b^4 
*c*e^5+40*a*b^3*c^2*d*e^4-10*a*b^2*c^3*d^2*e^3+60*a*b*c^4*d^3*e^2+3*b^6*e^ 
5-5*b^5*c*d*e^4-10*b^4*c^2*d^2*e^3+30*b^3*c^3*d^3*e^2-45*b^2*c^4*d^4*e+18* 
b*c^5*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+(31*a^3*b*c^2*e^5-30*a^3*c^3 
*d*e^4-22*a^2*b^3*c*e^5+50*a^2*b^2*c^2*d*e^4-50*a^2*b*c^3*d^2*e^3-20*a^2*c 
^4*d^3*e^2+3*a*b^5*e^5-5*a*b^4*c*d*e^4-10*a*b^3*c^2*d^2*e^3+50*a*b^2*c^3*d 
^3*e^2-25*a*b*c^4*d^4*e+10*a*c^5*d^5-5*b^3*c^3*d^4*e+2*b^2*c^4*d^5)/(16*a^ 
2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2/c^3*(24*a^4*c^2*e^5-21*a^3*b^2*c*e^5+50*a^3 
*b*c^2*d*e^4-80*a^3*c^3*d^2*e^3+3*a^2*b^4*e^5-5*a^2*b^3*c*d*e^4-10*a^2*b^2 
*c^2*d^2*e^3+60*a^2*b*c^3*d^3*e^2-40*a^2*c^4*d^4*e-5*a*b^2*c^3*d^4*e+10*a* 
b*c^4*d^5-b^3*c^3*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+1/c^2/( 
16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^2*c^2*e^5-8*a*b^2*c*e^5+b^4*e^5)/c*ln 
(c*x^2+b*x+a)+2*(-7*a^2*b*e^5*c+30*a^2*c^2*d*e^4+a*b^3*e^5-30*a*b*c^2*d^2* 
e^3+20*a*c^3*d^3*e^2+10*b^2*c^2*d^3*e^2-15*b*c^3*d^4*e+6*c^4*d^5-1/2*(16*a 
^2*c^2*e^5-8*a*b^2*c*e^5+b^4*e^5)*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. 1923 vs. \(2 (393) = 786\).

Time = 0.18 (sec) , antiderivative size = 3865, normalized size of antiderivative = 9.54 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^5/(c*x^2+b*x+a)^3,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. 3403 vs. \(2 (403) = 806\).

Time = 67.77 (sec) , antiderivative size = 3403, normalized size of antiderivative = 8.40 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**5/(c*x**2+b*x+a)**3,x)
 

Output:

(e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 
 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 
 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d** 
4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 16 
0*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x + (-64*a**3*c**5*(e**5/(2* 
c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b 
**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3 
*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c** 
3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b* 
*6*c**2 + 20*a*b**8*c - b**10))) + 32*a**3*c**2*e**5 + 48*a**2*b**2*c**4*( 
e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 
- 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 
+ 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4 
)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160 
*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 9*a**2*b**2*c*e**5 - 30*a**2*b* 
c**2*d*e**4 - 12*a*b**4*c**3*(e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b* 
e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20 
*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 
12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c 
**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) ...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^5/(c*x^2+b*x+a)^3,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (393) = 786\).

Time = 0.37 (sec) , antiderivative size = 845, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^5/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

1/2*e^5*log(c*x^2 + b*x + a)/c^3 + (12*c^5*d^5 - 30*b*c^4*d^4*e + 20*b^2*c 
^3*d^3*e^2 + 40*a*c^4*d^3*e^2 - 60*a*b*c^3*d^2*e^3 + 60*a^2*c^3*d*e^4 - b^ 
5*e^5 + 10*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^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) - 1/2*(b 
^3*c^3*d^5 - 10*a*b*c^4*d^5 + 5*a*b^2*c^3*d^4*e + 40*a^2*c^4*d^4*e - 60*a^ 
2*b*c^3*d^3*e^2 + 10*a^2*b^2*c^2*d^2*e^3 + 80*a^3*c^3*d^2*e^3 + 5*a^2*b^3* 
c*d*e^4 - 50*a^3*b*c^2*d*e^4 - 3*a^2*b^4*e^5 + 21*a^3*b^2*c*e^5 - 24*a^4*c 
^2*e^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 20*a*c^5*d^3 
*e^2 - 30*a*b*c^4*d^2*e^3 - 5*b^4*c^2*d*e^4 + 40*a*b^2*c^3*d*e^4 - 50*a^2* 
c^4*d*e^4 + 2*b^5*c*e^5 - 15*a*b^3*c^2*e^5 + 25*a^2*b*c^3*e^5)*x^3 - (18*b 
*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 + 60*a*b*c^4*d^3*e^2 - 10 
*b^4*c^2*d^2*e^3 - 10*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d^2*e^3 - 5*b^5*c*d* 
e^4 + 40*a*b^3*c^2*d*e^4 + 10*a^2*b*c^3*d*e^4 + 3*b^6*e^5 - 19*a*b^4*c*e^5 
 + 11*a^2*b^2*c^2*e^5 + 32*a^3*c^3*e^5)*x^2 - 2*(2*b^2*c^4*d^5 + 10*a*c^5* 
d^5 - 5*b^3*c^3*d^4*e - 25*a*b*c^4*d^4*e + 50*a*b^2*c^3*d^3*e^2 - 20*a^2*c 
^4*d^3*e^2 - 10*a*b^3*c^2*d^2*e^3 - 50*a^2*b*c^3*d^2*e^3 - 5*a*b^4*c*d*e^4 
 + 50*a^2*b^2*c^2*d*e^4 - 30*a^3*c^3*d*e^4 + 3*a*b^5*e^5 - 22*a^2*b^3*c*e^ 
5 + 31*a^3*b*c^2*e^5)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^3)
 

Mupad [B] (verification not implemented)

Time = 7.34 (sec) , antiderivative size = 1486, normalized size of antiderivative = 3.67 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d + e*x)^5/(a + b*x + c*x^2)^3,x)
 

Output:

(atan((((x*(b*e - 2*c*d)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*a*c^3* 
d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d* 
e^3 - 20*a*b*c^2*d*e^3))/(c^2*(4*a*c - b^2)^5) + ((b*e - 2*c*d)*(b^5*c^2 - 
 8*a*b^3*c^3 + 16*a^2*b*c^4)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*a* 
c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3* 
c*d*e^3 - 20*a*b*c^2*d*e^3))/(2*c^5*(4*a*c - b^2)^5*(b^4 + 16*a^2*c^2 - 8* 
a*b^2*c)))*(32*a^2*c^5*(4*a*c - b^2)^(5/2) + 2*b^4*c^3*(4*a*c - b^2)^(5/2) 
 - 16*a*b^2*c^4*(4*a*c - b^2)^(5/2)))/(12*c^5*d^5 - b^5*e^5 - 30*a^2*b*c^2 
*e^5 + 40*a*c^4*d^3*e^2 + 60*a^2*c^3*d*e^4 + 20*b^2*c^3*d^3*e^2 + 10*a*b^3 
*c*e^5 - 30*b*c^4*d^4*e - 60*a*b*c^3*d^2*e^3))*(b*e - 2*c*d)*(b^4*e^4 + 6* 
c^4*d^4 + 30*a^2*c^2*e^4 + 20*a*c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2 
*c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d*e^3 - 20*a*b*c^2*d*e^3))/(c^3*(4*a*c - 
 b^2)^(5/2)) - (log(a + b*x + c*x^2)*(b^10*e^5 - 1024*a^5*c^5*e^5 + 160*a^ 
2*b^6*c^2*e^5 - 640*a^3*b^4*c^3*e^5 + 1280*a^4*b^2*c^4*e^5 - 20*a*b^8*c*e^ 
5))/(2*(1024*a^5*c^8 - b^10*c^3 + 20*a*b^8*c^4 - 160*a^2*b^6*c^5 + 640*a^3 
*b^4*c^6 - 1280*a^4*b^2*c^7)) - ((b^3*c^3*d^5 - 24*a^4*c^2*e^5 - 3*a^2*b^4 
*e^5 + 21*a^3*b^2*c*e^5 + 40*a^2*c^4*d^4*e + 80*a^3*c^3*d^2*e^3 - 10*a*b*c 
^4*d^5 + 10*a^2*b^2*c^2*d^2*e^3 + 5*a*b^2*c^3*d^4*e + 5*a^2*b^3*c*d*e^4 - 
50*a^3*b*c^2*d*e^4 - 60*a^2*b*c^3*d^3*e^2)/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a* 
b^2*c)) + (x*(22*a^2*b^3*c*e^5 - 10*a*c^5*d^5 - 2*b^2*c^4*d^5 - 3*a*b^5...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 4384, normalized size of antiderivative = 10.82 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^5/(c*x^2+b*x+a)^3,x)
 

Output:

( - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*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** 
4*b*c**3*d*e**4 + 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a**3*b**4*c*e**5 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a**3*b**3*c**2*e**5*x - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq 
rt(4*a*c - b**2))*a**3*b**2*c**3*d**2*e**3 + 240*sqrt(4*a*c - b**2)*atan(( 
b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3*d*e**4*x - 120*sqrt(4*a*c - 
b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3*e**5*x**2 + 80*s 
qrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**4*d**3*e* 
*2 + 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c* 
*4*d*e**4*x**2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a**2*b**6*e**5 + 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a**2*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**2*b**4*c**2*e**5*x**2 + 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*a**2*b**3*c**3*d**3*e**2 - 240*sqrt(4*a*c - b**2)*ata 
n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*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**2*b**3*c**3*d*e**4*x** 
2 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3* 
c**3*e**5*x**3 - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a**2*b**2*c**4*d**4*e + 160*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(...