\(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^3} \, dx\) [740]

Optimal result

Integrand size = 25, antiderivative size = 530 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {\left (A e \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 B \left (5 c^3 d^4-2 c^2 d^2 e (5 b d-3 a e)-b^2 e^3 (b d-a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )\right ) x}{e^7}-\frac {\left (B \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 A c e \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c \left (A c e (c d-b e)-B \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^3}{e^5}-\frac {c^2 (3 B c d-3 b B e-A c e) x^4}{4 e^4}+\frac {B c^3 x^5}{5 e^3}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{2 e^8 (d+e x)^2}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^8 (d+e x)}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \log (d+e x)}{e^8} \] Output:

-(A*e*(10*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a*e+3*b*d)-9*c^2*d*e*(-a*e+2*b*d)) 
-3*B*(5*c^3*d^4-2*c^2*d^2*e*(-3*a*e+5*b*d)-b^2*e^3*(-a*e+b*d)+c*e^2*(a^2*e 
^2-6*a*b*d*e+6*b^2*d^2)))*x/e^7-1/2*(B*(10*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a 
*e+3*b*d)-9*c^2*d*e*(-a*e+2*b*d))-3*A*c*e*(2*c^2*d^2+b^2*e^2-c*e*(-a*e+3*b 
*d)))*x^2/e^6-c*(A*c*e*(-b*e+c*d)-B*(2*c^2*d^2+b^2*e^2-c*e*(-a*e+3*b*d)))* 
x^3/e^5-1/4*c^2*(-A*c*e-3*B*b*e+3*B*c*d)*x^4/e^4+1/5*B*c^3*x^5/e^3+1/2*(-A 
*e+B*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^2-(a*e^2-b*d*e+c*d^2)^2*(7*B*c*d 
^2-B*e*(-a*e+4*b*d)-3*A*e*(-b*e+2*c*d))/e^8/(e*x+d)-3*(a*e^2-b*d*e+c*d^2)* 
(B*(7*c^2*d^3-c*d*e*(-3*a*e+8*b*d)+b*e^2*(-a*e+2*b*d))-A*e*(5*c^2*d^2+b^2* 
e^2-c*e*(-a*e+5*b*d)))*ln(e*x+d)/e^8
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 503, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {20 e \left (A e \left (-10 c^3 d^3+b^3 e^3+9 c^2 d e (2 b d-a e)+3 b c e^2 (-3 b d+2 a e)\right )+3 B \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )\right ) x+10 e^2 \left (3 A c e \left (2 c^2 d^2+b^2 e^2+c e (-3 b d+a e)\right )+B \left (-10 c^3 d^3+b^3 e^3+9 c^2 d e (2 b d-a e)+3 b c e^2 (-3 b d+2 a e)\right )\right ) x^2+20 c e^3 \left (A c e (-c d+b e)+B \left (2 c^2 d^2+b^2 e^2+c e (-3 b d+a e)\right )\right ) x^3+5 c^2 e^4 (-3 B c d+3 b B e+A c e) x^4+4 B c^3 e^5 x^5+\frac {10 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}-\frac {20 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{d+e x}-60 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right ) \log (d+e x)}{20 e^8} \] Input:

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

Output:

(20*e*(A*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(- 
3*b*d + 2*a*e)) + 3*B*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(- 
5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x + 10*e^2*(3*A 
*c*e*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)) + B*(-10*c^3*d^3 + b^3*e^3 
 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e)))*x^2 + 20*c*e^3*( 
A*c*e*(-(c*d) + b*e) + B*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)))*x^3 + 
 5*c^2*e^4*(-3*B*c*d + 3*b*B*e + A*c*e)*x^4 + 4*B*c^3*e^5*x^5 + (10*(B*d - 
 A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 - (20*(c*d^2 + e*(-(b*d) + 
 a*e))^2*(7*B*c*d^2 + B*e*(-4*b*d + a*e) + 3*A*e*(-2*c*d + b*e)))/(d + e*x 
) - 60*(c*d^2 + e*(-(b*d) + a*e))*(-(A*e*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b* 
d + a*e))) + B*(7*c^2*d^3 + b*e^2*(2*b*d - a*e) + c*d*e*(-8*b*d + 3*a*e))) 
*Log[d + e*x])/(20*e^8)
 

Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.

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

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )-A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^7}+\frac {3 c x^2 \left (B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )-A c e (c d-b e)\right )}{e^5}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)}+\frac {x \left (3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )-B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )\right )}{e^6}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^2}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^3}+\frac {c^2 x^3 (A c e+3 b B e-3 B c d)}{e^4}+\frac {B c^3 x^4}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x \left (A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac {c x^3 \left (A c e (c d-b e)-B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{e^5}-\frac {3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}-\frac {x^2 \left (B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{2 e^6}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^8 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac {B c^3 x^5}{5 e^3}\)

Input:

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

Output:

-(((A*e*(10*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b 
*d - a*e)) - 3*B*(5*c^3*d^4 - 2*c^2*d^2*e*(5*b*d - 3*a*e) - b^2*e^3*(b*d - 
 a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x)/e^7) - ((B*(10*c^3*d^ 
3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*A*c 
*e*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*e)))*x^2)/(2*e^6) - (c*(A*c*e*(c* 
d - b*e) - B*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*e)))*x^3)/e^5 - (c^2*(3 
*B*c*d - 3*b*B*e - A*c*e)*x^4)/(4*e^4) + (B*c^3*x^5)/(5*e^3) + ((B*d - A*e 
)*(c*d^2 - b*d*e + a*e^2)^3)/(2*e^8*(d + e*x)^2) - ((c*d^2 - b*d*e + a*e^2 
)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e)))/(e^8*(d + e*x)) 
 - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^ 
2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*Log[d + 
e*x])/e^8
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 1044, normalized size of antiderivative = 1.97

method	result	size
norman	\(\text {Expression too large to display}\)	\(1044\)
default	\(\text {Expression too large to display}\)	\(1116\)
risch	\(\text {Expression too large to display}\)	\(1215\)
parallelrisch	\(\text {Expression too large to display}\)	\(1951\)

Input:

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

Output:

((6*A*a*b*c*e^4-6*A*a*c^2*d*e^3+A*b^3*e^4-6*A*b^2*c*d*e^3+10*A*b*c^2*d^2*e 
^2-5*A*c^3*d^3*e+3*B*a^2*c*e^4+3*B*a*b^2*e^4-12*B*a*b*c*d*e^3+10*B*a*c^2*d 
^2*e^2-2*B*b^3*d*e^3+10*B*b^2*c*d^2*e^2-15*B*b*c^2*d^3*e+7*B*c^3*d^4)/e^5* 
x^3-1/2*(A*a^3*e^7+3*A*a^2*b*d*e^6-9*A*a^2*c*d^2*e^5-9*A*a*b^2*d^2*e^5+54* 
A*a*b*c*d^3*e^4-54*A*a*c^2*d^4*e^3+9*A*b^3*d^3*e^4-54*A*b^2*c*d^4*e^3+90*A 
*b*c^2*d^5*e^2-45*A*c^3*d^6*e+B*a^3*d*e^6-9*B*a^2*b*d^2*e^5+27*B*a^2*c*d^3 
*e^4+27*B*a*b^2*d^3*e^4-108*B*a*b*c*d^4*e^3+90*B*a*c^2*d^5*e^2-18*B*b^3*d^ 
4*e^3+90*B*b^2*c*d^5*e^2-135*B*b*c^2*d^6*e+63*B*c^3*d^7)/e^8+1/4*(6*A*a*c^ 
2*e^3+6*A*b^2*c*e^3-10*A*b*c^2*d*e^2+5*A*c^3*d^2*e+12*B*a*b*c*e^3-10*B*a*c 
^2*d*e^2+2*B*b^3*e^3-10*B*b^2*c*d*e^2+15*B*b*c^2*d^2*e-7*B*c^3*d^3)/e^4*x^ 
4-(3*A*a^2*b*e^6-6*A*a^2*c*d*e^5-6*A*a*b^2*d*e^5+36*A*a*b*c*d^2*e^4-36*A*a 
*c^2*d^3*e^3+6*A*b^3*d^2*e^4-36*A*b^2*c*d^3*e^3+60*A*b*c^2*d^4*e^2-30*A*c^ 
3*d^5*e+B*a^3*e^6-6*B*a^2*b*d*e^5+18*B*a^2*c*d^2*e^4+18*B*a*b^2*d^2*e^4-72 
*B*a*b*c*d^3*e^3+60*B*a*c^2*d^4*e^2-12*B*b^3*d^3*e^3+60*B*b^2*c*d^4*e^2-90 
*B*b*c^2*d^5*e+42*B*c^3*d^6)/e^7*x+1/5*B*c^3*x^7/e+1/10*c*(10*A*b*c*e^2-5* 
A*c^2*d*e+10*B*a*c*e^2+10*B*b^2*e^2-15*B*b*c*d*e+7*B*c^2*d^2)/e^3*x^5+1/20 
*c^2*(5*A*c*e+15*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^2+3*(A*a^2*c*e^5+A*a*b^2* 
e^5-6*A*a*b*c*d*e^4+6*A*a*c^2*d^2*e^3-A*b^3*d*e^4+6*A*b^2*c*d^2*e^3-10*A*b 
*c^2*d^3*e^2+5*A*c^3*d^4*e+B*a^2*b*e^5-3*B*a^2*c*d*e^4-3*B*a*b^2*d*e^4+12* 
B*a*b*c*d^2*e^3-10*B*a*c^2*d^3*e^2+2*B*b^3*d^2*e^3-10*B*b^2*c*d^3*e^2+1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1311 vs. \(2 (523) = 1046\).

Time = 0.09 (sec) , antiderivative size = 1311, normalized size of antiderivative = 2.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/20*(4*B*c^3*e^7*x^7 - 130*B*c^3*d^7 - 10*A*a^3*e^7 + 110*(3*B*b*c^2 + A* 
c^3)*d^6*e - 270*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*a*c 
^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 - 50*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2 
*A*a*b)*c)*d^3*e^4 + 90*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - 10*(B*a^3 
+ 3*A*a^2*b)*d*e^6 - (7*B*c^3*d*e^6 - 5*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 2*( 
7*B*c^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d*e^6 + 10*(B*b^2*c + (B*a + A*b)* 
c^2)*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 10*(B 
*b^2*c + (B*a + A*b)*c^2)*d*e^6 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^ 
2)*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 - 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 10* 
(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + 
A*b^2)*c)*d*e^6 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 1 
0*(50*B*c^3*d^5*e^2 - 34*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c + (B*a 
+ A*b)*c^2)*d^3*e^4 - 11*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e 
^5 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6)*x^2 + 20*(8*B*c^ 
3*d^6*e - 4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^ 
4*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 - 2*(3*B*a*b^2 
 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 6*(B*a^2*b + A*a*b^2 + A*a^2*c 
)*d*e^6 - (B*a^3 + 3*A*a^2*b)*e^7)*x - 60*(7*B*c^3*d^7 - 5*(3*B*b*c^2 + A* 
c^3)*d^6*e + 10*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 2*(B*b^3 + 3*A*a*c^2 
 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1149 vs. \(2 (520) = 1040\).

Time = 35.74 (sec) , antiderivative size = 1149, normalized size of antiderivative = 2.17 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

B*c**3*x**5/(5*e**3) + x**4*(A*c**3/(4*e**3) + 3*B*b*c**2/(4*e**3) - 3*B*c 
**3*d/(4*e**4)) + x**3*(A*b*c**2/e**3 - A*c**3*d/e**4 + B*a*c**2/e**3 + B* 
b**2*c/e**3 - 3*B*b*c**2*d/e**4 + 2*B*c**3*d**2/e**5) + x**2*(3*A*a*c**2/( 
2*e**3) + 3*A*b**2*c/(2*e**3) - 9*A*b*c**2*d/(2*e**4) + 3*A*c**3*d**2/e**5 
 + 3*B*a*b*c/e**3 - 9*B*a*c**2*d/(2*e**4) + B*b**3/(2*e**3) - 9*B*b**2*c*d 
/(2*e**4) + 9*B*b*c**2*d**2/e**5 - 5*B*c**3*d**3/e**6) + x*(6*A*a*b*c/e**3 
 - 9*A*a*c**2*d/e**4 + A*b**3/e**3 - 9*A*b**2*c*d/e**4 + 18*A*b*c**2*d**2/ 
e**5 - 10*A*c**3*d**3/e**6 + 3*B*a**2*c/e**3 + 3*B*a*b**2/e**3 - 18*B*a*b* 
c*d/e**4 + 18*B*a*c**2*d**2/e**5 - 3*B*b**3*d/e**4 + 18*B*b**2*c*d**2/e**5 
 - 30*B*b*c**2*d**3/e**6 + 15*B*c**3*d**4/e**7) + (-A*a**3*e**7 - 3*A*a**2 
*b*d*e**6 + 9*A*a**2*c*d**2*e**5 + 9*A*a*b**2*d**2*e**5 - 30*A*a*b*c*d**3* 
e**4 + 21*A*a*c**2*d**4*e**3 - 5*A*b**3*d**3*e**4 + 21*A*b**2*c*d**4*e**3 
- 27*A*b*c**2*d**5*e**2 + 11*A*c**3*d**6*e - B*a**3*d*e**6 + 9*B*a**2*b*d* 
*2*e**5 - 15*B*a**2*c*d**3*e**4 - 15*B*a*b**2*d**3*e**4 + 42*B*a*b*c*d**4* 
e**3 - 27*B*a*c**2*d**5*e**2 + 7*B*b**3*d**4*e**3 - 27*B*b**2*c*d**5*e**2 
+ 33*B*b*c**2*d**6*e - 13*B*c**3*d**7 + x*(-6*A*a**2*b*e**7 + 12*A*a**2*c* 
d*e**6 + 12*A*a*b**2*d*e**6 - 36*A*a*b*c*d**2*e**5 + 24*A*a*c**2*d**3*e**4 
 - 6*A*b**3*d**2*e**5 + 24*A*b**2*c*d**3*e**4 - 30*A*b*c**2*d**4*e**3 + 12 
*A*c**3*d**5*e**2 - 2*B*a**3*e**7 + 12*B*a**2*b*d*e**6 - 18*B*a**2*c*d**2* 
e**5 - 18*B*a*b**2*d**2*e**5 + 48*B*a*b*c*d**3*e**4 - 30*B*a*c**2*d**4*...
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^3,x, algorithm="maxima")
 

Output:

-1/2*(13*B*c^3*d^7 + A*a^3*e^7 - 11*(3*B*b*c^2 + A*c^3)*d^6*e + 27*(B*b^2* 
c + (B*a + A*b)*c^2)*d^5*e^2 - 7*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)* 
c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 9*(B* 
a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (B*a^3 + 3*A*a^2*b)*d*e^6 + 2*(7*B*c^ 
3*d^6*e - 6*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d 
^4*e^3 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 3*(3*B*a* 
b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 6*(B*a^2*b + A*a*b^2 + A*a^ 
2*c)*d*e^6 + (B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8) 
+ 1/20*(4*B*c^3*e^4*x^5 - 5*(3*B*c^3*d*e^3 - (3*B*b*c^2 + A*c^3)*e^4)*x^4 
+ 20*(2*B*c^3*d^2*e^2 - (3*B*b*c^2 + A*c^3)*d*e^3 + (B*b^2*c + (B*a + A*b) 
*c^2)*e^4)*x^3 - 10*(10*B*c^3*d^3*e - 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 9*(B 
*b^2*c + (B*a + A*b)*c^2)*d*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2) 
*c)*e^4)*x^2 + 20*(15*B*c^3*d^4 - 10*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2 
*c + (B*a + A*b)*c^2)*d^2*e^2 - 3*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2) 
*c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^4)*x)/e^7 - 3*(7 
*B*c^3*d^5 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 10*(B*b^2*c + (B*a + A*b)*c^2)* 
d^3*e^2 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^3 + (3*B*a*b 
^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^4 - (B*a^2*b + A*a*b^2 + A*a^2*c)* 
e^5)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (523) = 1046\).

Time = 0.20 (sec) , antiderivative size = 1127, normalized size of antiderivative = 2.13 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-3*(7*B*c^3*d^5 - 15*B*b*c^2*d^4*e - 5*A*c^3*d^4*e + 10*B*b^2*c*d^3*e^2 + 
10*B*a*c^2*d^3*e^2 + 10*A*b*c^2*d^3*e^2 - 2*B*b^3*d^2*e^3 - 12*B*a*b*c*d^2 
*e^3 - 6*A*b^2*c*d^2*e^3 - 6*A*a*c^2*d^2*e^3 + 3*B*a*b^2*d*e^4 + A*b^3*d*e 
^4 + 3*B*a^2*c*d*e^4 + 6*A*a*b*c*d*e^4 - B*a^2*b*e^5 - A*a*b^2*e^5 - A*a^2 
*c*e^5)*log(abs(e*x + d))/e^8 - 1/2*(13*B*c^3*d^7 - 33*B*b*c^2*d^6*e - 11* 
A*c^3*d^6*e + 27*B*b^2*c*d^5*e^2 + 27*B*a*c^2*d^5*e^2 + 27*A*b*c^2*d^5*e^2 
 - 7*B*b^3*d^4*e^3 - 42*B*a*b*c*d^4*e^3 - 21*A*b^2*c*d^4*e^3 - 21*A*a*c^2* 
d^4*e^3 + 15*B*a*b^2*d^3*e^4 + 5*A*b^3*d^3*e^4 + 15*B*a^2*c*d^3*e^4 + 30*A 
*a*b*c*d^3*e^4 - 9*B*a^2*b*d^2*e^5 - 9*A*a*b^2*d^2*e^5 - 9*A*a^2*c*d^2*e^5 
 + B*a^3*d*e^6 + 3*A*a^2*b*d*e^6 + A*a^3*e^7 + 2*(7*B*c^3*d^6*e - 18*B*b*c 
^2*d^5*e^2 - 6*A*c^3*d^5*e^2 + 15*B*b^2*c*d^4*e^3 + 15*B*a*c^2*d^4*e^3 + 1 
5*A*b*c^2*d^4*e^3 - 4*B*b^3*d^3*e^4 - 24*B*a*b*c*d^3*e^4 - 12*A*b^2*c*d^3* 
e^4 - 12*A*a*c^2*d^3*e^4 + 9*B*a*b^2*d^2*e^5 + 3*A*b^3*d^2*e^5 + 9*B*a^2*c 
*d^2*e^5 + 18*A*a*b*c*d^2*e^5 - 6*B*a^2*b*d*e^6 - 6*A*a*b^2*d*e^6 - 6*A*a^ 
2*c*d*e^6 + B*a^3*e^7 + 3*A*a^2*b*e^7)*x)/((e*x + d)^2*e^8) + 1/20*(4*B*c^ 
3*e^12*x^5 - 15*B*c^3*d*e^11*x^4 + 15*B*b*c^2*e^12*x^4 + 5*A*c^3*e^12*x^4 
+ 40*B*c^3*d^2*e^10*x^3 - 60*B*b*c^2*d*e^11*x^3 - 20*A*c^3*d*e^11*x^3 + 20 
*B*b^2*c*e^12*x^3 + 20*B*a*c^2*e^12*x^3 + 20*A*b*c^2*e^12*x^3 - 100*B*c^3* 
d^3*e^9*x^2 + 180*B*b*c^2*d^2*e^10*x^2 + 60*A*c^3*d^2*e^10*x^2 - 90*B*b^2* 
c*d*e^11*x^2 - 90*B*a*c^2*d*e^11*x^2 - 90*A*b*c^2*d*e^11*x^2 + 10*B*b^3...
 

Mupad [B] (verification not implemented)

Time = 11.45 (sec) , antiderivative size = 1297, normalized size of antiderivative = 2.45 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

x*((A*b^3 + 3*B*a*b^2 + 3*B*a^2*c + 6*A*a*b*c)/e^3 + (3*d^2*((3*d*((A*c^3 
+ 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2* 
c)/e^3 + (3*B*c^3*d^2)/e^5))/e^2 - (3*d*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 
6*B*a*b*c)/e^3 - (3*d^2*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^2 + 
 (3*d*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*A*b*c^2 + 
3*B*a*c^2 + 3*B*b^2*c)/e^3 + (3*B*c^3*d^2)/e^5))/e - (B*c^3*d^3)/e^6))/e - 
 (d^3*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^3) - ((A*a^3*e^7 + 13 
*B*c^3*d^7 + B*a^3*d*e^6 - 11*A*c^3*d^6*e + 5*A*b^3*d^3*e^4 - 7*B*b^3*d^4* 
e^3 - 9*A*a*b^2*d^2*e^5 - 21*A*a*c^2*d^4*e^3 - 9*A*a^2*c*d^2*e^5 + 15*B*a* 
b^2*d^3*e^4 - 9*B*a^2*b*d^2*e^5 + 27*A*b*c^2*d^5*e^2 - 21*A*b^2*c*d^4*e^3 
+ 27*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^4 + 27*B*b^2*c*d^5*e^2 + 3*A*a^2*b 
*d*e^6 - 33*B*b*c^2*d^6*e + 30*A*a*b*c*d^3*e^4 - 42*B*a*b*c*d^4*e^3)/(2*e) 
 + x*(B*a^3*e^6 + 7*B*c^3*d^6 + 3*A*a^2*b*e^6 - 6*A*c^3*d^5*e + 3*A*b^3*d^ 
2*e^4 - 4*B*b^3*d^3*e^3 - 12*A*a*c^2*d^3*e^3 + 9*B*a*b^2*d^2*e^4 + 15*A*b* 
c^2*d^4*e^2 - 12*A*b^2*c*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 9*B*a^2*c*d^2*e^4 
+ 15*B*b^2*c*d^4*e^2 - 6*A*a*b^2*d*e^5 - 6*A*a^2*c*d*e^5 - 6*B*a^2*b*d*e^5 
 - 18*B*b*c^2*d^5*e + 18*A*a*b*c*d^2*e^4 - 24*B*a*b*c*d^3*e^3))/(d^2*e^7 + 
 e^9*x^2 + 2*d*e^8*x) - x^3*((d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4 
))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/(3*e^3) + (B*c^3*d^2)/e^5) + x^ 
4*((A*c^3 + 3*B*b*c^2)/(4*e^3) - (3*B*c^3*d)/(4*e^4)) + x^2*((B*b^3 + 3...
 

Reduce [B] (verification not implemented)

Time = 5.02 (sec) , antiderivative size = 1485, normalized size of antiderivative = 2.80 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(60*log(d + e*x)*a**3*c*d**3*e**5 + 120*log(d + e*x)*a**3*c*d**2*e**6*x + 
60*log(d + e*x)*a**3*c*d*e**7*x**2 + 120*log(d + e*x)*a**2*b**2*d**3*e**5 
+ 240*log(d + e*x)*a**2*b**2*d**2*e**6*x + 120*log(d + e*x)*a**2*b**2*d*e* 
*7*x**2 - 540*log(d + e*x)*a**2*b*c*d**4*e**4 - 1080*log(d + e*x)*a**2*b*c 
*d**3*e**5*x - 540*log(d + e*x)*a**2*b*c*d**2*e**6*x**2 + 360*log(d + e*x) 
*a**2*c**2*d**5*e**3 + 720*log(d + e*x)*a**2*c**2*d**4*e**4*x + 360*log(d 
+ e*x)*a**2*c**2*d**3*e**5*x**2 - 240*log(d + e*x)*a*b**3*d**4*e**4 - 480* 
log(d + e*x)*a*b**3*d**3*e**5*x - 240*log(d + e*x)*a*b**3*d**2*e**6*x**2 + 
 1080*log(d + e*x)*a*b**2*c*d**5*e**3 + 2160*log(d + e*x)*a*b**2*c*d**4*e* 
*4*x + 1080*log(d + e*x)*a*b**2*c*d**3*e**5*x**2 - 1200*log(d + e*x)*a*b*c 
**2*d**6*e**2 - 2400*log(d + e*x)*a*b*c**2*d**5*e**3*x - 1200*log(d + e*x) 
*a*b*c**2*d**4*e**4*x**2 + 300*log(d + e*x)*a*c**3*d**7*e + 600*log(d + e* 
x)*a*c**3*d**6*e**2*x + 300*log(d + e*x)*a*c**3*d**5*e**3*x**2 + 120*log(d 
 + e*x)*b**4*d**5*e**3 + 240*log(d + e*x)*b**4*d**4*e**4*x + 120*log(d + e 
*x)*b**4*d**3*e**5*x**2 - 600*log(d + e*x)*b**3*c*d**6*e**2 - 1200*log(d + 
 e*x)*b**3*c*d**5*e**3*x - 600*log(d + e*x)*b**3*c*d**4*e**4*x**2 + 900*lo 
g(d + e*x)*b**2*c**2*d**7*e + 1800*log(d + e*x)*b**2*c**2*d**6*e**2*x + 90 
0*log(d + e*x)*b**2*c**2*d**5*e**3*x**2 - 420*log(d + e*x)*b*c**3*d**8 - 8 
40*log(d + e*x)*b*c**3*d**7*e*x - 420*log(d + e*x)*b*c**3*d**6*e**2*x**2 - 
 10*a**4*d*e**7 + 40*a**3*b*e**8*x**2 + 30*a**3*c*d**3*e**5 - 60*a**3*c...