\(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^9} \, dx\) [746]

Optimal result

Integrand size = 25, antiderivative size = 548 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{8 e^8 (d+e x)^8}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{7 e^8 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{2 e^8 (d+e x)^6}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{5 e^8 (d+e x)^5}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^8 (d+e x)^4}+\frac {c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^8 (d+e x)^3}+\frac {c^2 (7 B c d-3 b B e-A c e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \] Output:

1/8*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^8-1/7*(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)^7+1/2*(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)))/e^8/(e*x+d)^6+1/5*(A*e*(-b*e+2*c*d)* 
(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))-B*(35*c^3*d^4-b^2*e^3*(-3*a*e+4* 
b*d)-30*c^2*d^2*e*(-a*e+2*b*d)+3*c*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2)))/e^ 
8/(e*x+d)^5+1/4*(B*(35*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a*e+5*b*d)-15*c^2*d*e 
*(-a*e+3*b*d))-3*A*c*e*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))/e^8/(e*x+d)^4 
+c*(A*c*e*(-b*e+2*c*d)-B*(7*c^2*d^2+b^2*e^2-c*e*(-a*e+6*b*d)))/e^8/(e*x+d) 
^3+1/2*c^2*(-A*c*e-3*B*b*e+7*B*c*d)/e^8/(e*x+d)^2-B*c^3/e^8/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 847, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {A e \left (5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+e^3 \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+c e^2 \left (5 a^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a b e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+c^2 e \left (3 a e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )\right )+B \left (35 c^3 \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )+e^3 \left (5 a^3 e^3 (d+8 e x)+5 a^2 b e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a b^2 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^3 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+c e^2 \left (3 a^2 e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+6 a b e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+5 c^2 e \left (a e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 b \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )\right )}{280 e^8 (d+e x)^8} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^9,x]
 

Output:

-1/280*(A*e*(5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70 
*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6) + e^3*(35*a^3*e^3 + 15*a^2*b*e^2 
*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2*e 
*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + c*e^2*(5*a^2*e^2*(d^2 + 8*d*e*x + 28*e^ 
2*x^2) + 6*a*b*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*(d^ 
4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c^2*e*(3*a* 
e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*(d^ 
5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^ 
5))) + B*(35*c^3*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70*d 
^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7) + e^3*(5*a^3*e^3*( 
d + 8*e*x) + 5*a^2*b*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*b^2*e*(d^3 + 8 
*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^3*(d^4 + 8*d^3*e*x + 28*d^2*e^2* 
x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c*e^2*(3*a^2*e^2*(d^3 + 8*d^2*e*x + 28 
*d*e^2*x^2 + 56*e^3*x^3) + 6*a*b*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56* 
d*e^3*x^3 + 70*e^4*x^4) + 5*b^2*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2 
*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) + 5*c^2*e*(a*e*(d^5 + 8*d^4*e*x + 2 
8*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 3*b*(d^6 + 8 
*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 
 + 28*e^6*x^6))))/(e^8*(d + e*x)^8)
 

Rubi [A] (verified)

Time = 2.15 (sec) , antiderivative size = 548, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)^6}+\frac {3 c \left (B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )-A c e (2 c d-b e)\right )}{e^7 (d+e x)^4}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)^7}+\frac {3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )}{e^7 (d+e x)^5}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^8}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^9}+\frac {c^2 (A c e+3 b B e-7 B c d)}{e^7 (d+e x)^3}+\frac {B c^3}{e^7 (d+e x)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{5 e^8 (d+e x)^5}+\frac {c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8 (d+e x)^6}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^8 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{8 e^8 (d+e x)^8}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^9,x]
 

Output:

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(8*e^8*(d + e*x)^8) - ((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)))/(7*e 
^8*(d + e*x)^7) + ((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))))/(2*e^8*(d + e*x)^6) + (A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c 
*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2 
*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))/(5*e^8*(d 
+ e*x)^5) + (B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2* 
d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(4 
*e^8*(d + e*x)^4) + (c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e 
*(6*b*d - a*e))))/(e^8*(d + e*x)^3) + (c^2*(7*B*c*d - 3*b*B*e - A*c*e))/(2 
*e^8*(d + e*x)^2) - (B*c^3)/(e^8*(d + e*x))
 

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 = 1.49 (sec) , antiderivative size = 1034, normalized size of antiderivative = 1.89

method	result	size
risch	\(\text {Expression too large to display}\)	\(1034\)
norman	\(\text {Expression too large to display}\)	\(1044\)
default	\(\text {Expression too large to display}\)	\(1067\)
gosper	\(\text {Expression too large to display}\)	\(1211\)
parallelrisch	\(\text {Expression too large to display}\)	\(1211\)
orering	\(\text {Expression too large to display}\)	\(1211\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^9,x,method=_RETURNVERBOSE)
 

Output:

(-B*c^3*x^7/e-1/2*c^2*(A*c*e+3*B*b*e+7*B*c*d)/e^2*x^6-c*(A*b*c*e^2+A*c^2*d 
*e+B*a*c*e^2+B*b^2*e^2+3*B*b*c*d*e+7*B*c^2*d^2)/e^3*x^5-1/4*(3*A*a*c^2*e^3 
+3*A*b^2*c*e^3+5*A*b*c^2*d*e^2+5*A*c^3*d^2*e+6*B*a*b*c*e^3+5*B*a*c^2*d*e^2 
+B*b^3*e^3+5*B*b^2*c*d*e^2+15*B*b*c^2*d^2*e+35*B*c^3*d^3)/e^4*x^4-1/5*(6*A 
*a*b*c*e^4+3*A*a*c^2*d*e^3+A*b^3*e^4+3*A*b^2*c*d*e^3+5*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+6*B*a*b*c*d*e^3+5*B*a*c^2*d^2*e^2+B 
*b^3*d*e^3+5*B*b^2*c*d^2*e^2+15*B*b*c^2*d^3*e+35*B*c^3*d^4)/e^5*x^3-1/10*( 
5*A*a^2*c*e^5+5*A*a*b^2*e^5+6*A*a*b*c*d*e^4+3*A*a*c^2*d^2*e^3+A*b^3*d*e^4+ 
3*A*b^2*c*d^2*e^3+5*A*b*c^2*d^3*e^2+5*A*c^3*d^4*e+5*B*a^2*b*e^5+3*B*a^2*c* 
d*e^4+3*B*a*b^2*d*e^4+6*B*a*b*c*d^2*e^3+5*B*a*c^2*d^3*e^2+B*b^3*d^2*e^3+5* 
B*b^2*c*d^3*e^2+15*B*b*c^2*d^4*e+35*B*c^3*d^5)/e^6*x^2-1/35*(15*A*a^2*b*e^ 
6+5*A*a^2*c*d*e^5+5*A*a*b^2*d*e^5+6*A*a*b*c*d^2*e^4+3*A*a*c^2*d^3*e^3+A*b^ 
3*d^2*e^4+3*A*b^2*c*d^3*e^3+5*A*b*c^2*d^4*e^2+5*A*c^3*d^5*e+5*B*a^3*e^6+5* 
B*a^2*b*d*e^5+3*B*a^2*c*d^2*e^4+3*B*a*b^2*d^2*e^4+6*B*a*b*c*d^3*e^3+5*B*a* 
c^2*d^4*e^2+B*b^3*d^3*e^3+5*B*b^2*c*d^4*e^2+15*B*b*c^2*d^5*e+35*B*c^3*d^6) 
/e^7*x-1/280*(35*A*a^3*e^7+15*A*a^2*b*d*e^6+5*A*a^2*c*d^2*e^5+5*A*a*b^2*d^ 
2*e^5+6*A*a*b*c*d^3*e^4+3*A*a*c^2*d^4*e^3+A*b^3*d^3*e^4+3*A*b^2*c*d^4*e^3+ 
5*A*b*c^2*d^5*e^2+5*A*c^3*d^6*e+5*B*a^3*d*e^6+5*B*a^2*b*d^2*e^5+3*B*a^2*c* 
d^3*e^4+3*B*a*b^2*d^3*e^4+6*B*a*b*c*d^4*e^3+5*B*a*c^2*d^5*e^2+B*b^3*d^4*e^ 
3+5*B*b^2*c*d^5*e^2+15*B*b*c^2*d^6*e+35*B*c^3*d^7)/e^8)/(e*x+d)^8
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^9,x, algorithm="fricas")
 

Output:

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 35*A*a^3*e^7 + 5*(3*B*b*c^2 + A 
*c^3)*d^6*e + 5*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^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*a*b) 
*c)*d^3*e^4 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 5*(B*a^3 + 3*A*a^2 
*b)*d*e^6 + 140*(7*B*c^3*d*e^6 + (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 280*(7*B*c 
^3*d^2*e^5 + (3*B*b*c^2 + A*c^3)*d*e^6 + (B*b^2*c + (B*a + A*b)*c^2)*e^7)* 
x^5 + 70*(35*B*c^3*d^3*e^4 + 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 5*(B*b^2*c + 
(B*a + A*b)*c^2)*d*e^6 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)* 
x^4 + 56*(35*B*c^3*d^4*e^3 + 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 5*(B*b^2*c + 
(B*a + A*b)*c^2)*d^2*e^5 + (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 + 28*(35*B*c^3*d 
^5*e^2 + 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 5*(B*b^2*c + (B*a + A*b)*c^2)*d^3 
*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + (3*B*a*b^2 + 
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7 
)*x^2 + 8*(35*B*c^3*d^6*e + 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 5*(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 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 5*(B*a^2*b + A 
*a*b^2 + A*a^2*c)*d*e^6 + 5*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^16*x^8 + 8*d*e^ 
15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11 
*x^3 + 28*d^6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/(e*x+d)**9,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^9,x, algorithm="maxima")
 

Output:

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 35*A*a^3*e^7 + 5*(3*B*b*c^2 + A 
*c^3)*d^6*e + 5*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^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*a*b) 
*c)*d^3*e^4 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 5*(B*a^3 + 3*A*a^2 
*b)*d*e^6 + 140*(7*B*c^3*d*e^6 + (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 280*(7*B*c 
^3*d^2*e^5 + (3*B*b*c^2 + A*c^3)*d*e^6 + (B*b^2*c + (B*a + A*b)*c^2)*e^7)* 
x^5 + 70*(35*B*c^3*d^3*e^4 + 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 5*(B*b^2*c + 
(B*a + A*b)*c^2)*d*e^6 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)* 
x^4 + 56*(35*B*c^3*d^4*e^3 + 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 5*(B*b^2*c + 
(B*a + A*b)*c^2)*d^2*e^5 + (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 + 28*(35*B*c^3*d 
^5*e^2 + 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 5*(B*b^2*c + (B*a + A*b)*c^2)*d^3 
*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + (3*B*a*b^2 + 
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 + 5*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7 
)*x^2 + 8*(35*B*c^3*d^6*e + 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 5*(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 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 5*(B*a^2*b + A 
*a*b^2 + A*a^2*c)*d*e^6 + 5*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^16*x^8 + 8*d*e^ 
15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11 
*x^3 + 28*d^6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1210 vs. \(2 (537) = 1074\).

Time = 0.21 (sec) , antiderivative size = 1210, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^9,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/280*(280*B*c^3*e^7*x^7 + 980*B*c^3*d*e^6*x^6 + 420*B*b*c^2*e^7*x^6 + 14 
0*A*c^3*e^7*x^6 + 1960*B*c^3*d^2*e^5*x^5 + 840*B*b*c^2*d*e^6*x^5 + 280*A*c 
^3*d*e^6*x^5 + 280*B*b^2*c*e^7*x^5 + 280*B*a*c^2*e^7*x^5 + 280*A*b*c^2*e^7 
*x^5 + 2450*B*c^3*d^3*e^4*x^4 + 1050*B*b*c^2*d^2*e^5*x^4 + 350*A*c^3*d^2*e 
^5*x^4 + 350*B*b^2*c*d*e^6*x^4 + 350*B*a*c^2*d*e^6*x^4 + 350*A*b*c^2*d*e^6 
*x^4 + 70*B*b^3*e^7*x^4 + 420*B*a*b*c*e^7*x^4 + 210*A*b^2*c*e^7*x^4 + 210* 
A*a*c^2*e^7*x^4 + 1960*B*c^3*d^4*e^3*x^3 + 840*B*b*c^2*d^3*e^4*x^3 + 280*A 
*c^3*d^3*e^4*x^3 + 280*B*b^2*c*d^2*e^5*x^3 + 280*B*a*c^2*d^2*e^5*x^3 + 280 
*A*b*c^2*d^2*e^5*x^3 + 56*B*b^3*d*e^6*x^3 + 336*B*a*b*c*d*e^6*x^3 + 168*A* 
b^2*c*d*e^6*x^3 + 168*A*a*c^2*d*e^6*x^3 + 168*B*a*b^2*e^7*x^3 + 56*A*b^3*e 
^7*x^3 + 168*B*a^2*c*e^7*x^3 + 336*A*a*b*c*e^7*x^3 + 980*B*c^3*d^5*e^2*x^2 
 + 420*B*b*c^2*d^4*e^3*x^2 + 140*A*c^3*d^4*e^3*x^2 + 140*B*b^2*c*d^3*e^4*x 
^2 + 140*B*a*c^2*d^3*e^4*x^2 + 140*A*b*c^2*d^3*e^4*x^2 + 28*B*b^3*d^2*e^5* 
x^2 + 168*B*a*b*c*d^2*e^5*x^2 + 84*A*b^2*c*d^2*e^5*x^2 + 84*A*a*c^2*d^2*e^ 
5*x^2 + 84*B*a*b^2*d*e^6*x^2 + 28*A*b^3*d*e^6*x^2 + 84*B*a^2*c*d*e^6*x^2 + 
 168*A*a*b*c*d*e^6*x^2 + 140*B*a^2*b*e^7*x^2 + 140*A*a*b^2*e^7*x^2 + 140*A 
*a^2*c*e^7*x^2 + 280*B*c^3*d^6*e*x + 120*B*b*c^2*d^5*e^2*x + 40*A*c^3*d^5* 
e^2*x + 40*B*b^2*c*d^4*e^3*x + 40*B*a*c^2*d^4*e^3*x + 40*A*b*c^2*d^4*e^3*x 
 + 8*B*b^3*d^3*e^4*x + 48*B*a*b*c*d^3*e^4*x + 24*A*b^2*c*d^3*e^4*x + 24*A* 
a*c^2*d^3*e^4*x + 24*B*a*b^2*d^2*e^5*x + 8*A*b^3*d^2*e^5*x + 24*B*a^2*c...
 

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^9,x)
 

Output:

-((35*A*a^3*e^7 + 35*B*c^3*d^7 + 5*B*a^3*d*e^6 + 5*A*c^3*d^6*e + A*b^3*d^3 
*e^4 + B*b^3*d^4*e^3 + 5*A*a*b^2*d^2*e^5 + 3*A*a*c^2*d^4*e^3 + 5*A*a^2*c*d 
^2*e^5 + 3*B*a*b^2*d^3*e^4 + 5*B*a^2*b*d^2*e^5 + 5*A*b*c^2*d^5*e^2 + 3*A*b 
^2*c*d^4*e^3 + 5*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 + 5*B*b^2*c*d^5*e^2 + 
 15*A*a^2*b*d*e^6 + 15*B*b*c^2*d^6*e + 6*A*a*b*c*d^3*e^4 + 6*B*a*b*c*d^4*e 
^3)/(280*e^8) + (x^4*(B*b^3*e^3 + 35*B*c^3*d^3 + 3*A*a*c^2*e^3 + 3*A*b^2*c 
*e^3 + 5*A*c^3*d^2*e + 6*B*a*b*c*e^3 + 5*A*b*c^2*d*e^2 + 5*B*a*c^2*d*e^2 + 
 15*B*b*c^2*d^2*e + 5*B*b^2*c*d*e^2))/(4*e^4) + (x*(5*B*a^3*e^6 + 35*B*c^3 
*d^6 + 15*A*a^2*b*e^6 + 5*A*c^3*d^5*e + A*b^3*d^2*e^4 + B*b^3*d^3*e^3 + 3* 
A*a*c^2*d^3*e^3 + 3*B*a*b^2*d^2*e^4 + 5*A*b*c^2*d^4*e^2 + 3*A*b^2*c*d^3*e^ 
3 + 5*B*a*c^2*d^4*e^2 + 3*B*a^2*c*d^2*e^4 + 5*B*b^2*c*d^4*e^2 + 5*A*a*b^2* 
d*e^5 + 5*A*a^2*c*d*e^5 + 5*B*a^2*b*d*e^5 + 15*B*b*c^2*d^5*e + 6*A*a*b*c*d 
^2*e^4 + 6*B*a*b*c*d^3*e^3))/(35*e^7) + (x^2*(35*B*c^3*d^5 + 5*A*a*b^2*e^5 
 + 5*A*a^2*c*e^5 + 5*B*a^2*b*e^5 + A*b^3*d*e^4 + 5*A*c^3*d^4*e + B*b^3*d^2 
*e^3 + 3*A*a*c^2*d^2*e^3 + 5*A*b*c^2*d^3*e^2 + 3*A*b^2*c*d^2*e^3 + 5*B*a*c 
^2*d^3*e^2 + 5*B*b^2*c*d^3*e^2 + 3*B*a*b^2*d*e^4 + 3*B*a^2*c*d*e^4 + 15*B* 
b*c^2*d^4*e + 6*B*a*b*c*d^2*e^3 + 6*A*a*b*c*d*e^4))/(10*e^6) + (x^5*(7*B*c 
^3*d^2 + A*c^3*d*e + A*b*c^2*e^2 + B*a*c^2*e^2 + B*b^2*c*e^2 + 3*B*b*c^2*d 
*e))/e^3 + (x^3*(A*b^3*e^4 + 35*B*c^3*d^4 + 3*B*a*b^2*e^4 + 3*B*a^2*c*e^4 
+ 5*A*c^3*d^3*e + B*b^3*d*e^3 + 5*A*b*c^2*d^2*e^2 + 5*B*a*c^2*d^2*e^2 +...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 911, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=\frac {35 b \,c^{3} e^{7} x^{8}-140 a \,c^{3} d \,e^{6} x^{6}-420 b^{2} c^{2} d \,e^{6} x^{6}-560 a b \,c^{2} d \,e^{6} x^{5}-280 a \,c^{3} d^{2} e^{5} x^{5}-280 b^{3} c d \,e^{6} x^{5}-840 b^{2} c^{2} d^{2} e^{5} x^{5}-210 a^{2} c^{2} d \,e^{6} x^{4}-630 a \,b^{2} c d \,e^{6} x^{4}-700 a b \,c^{2} d^{2} e^{5} x^{4}-350 a \,c^{3} d^{3} e^{4} x^{4}-70 b^{4} d \,e^{6} x^{4}-350 b^{3} c \,d^{2} e^{5} x^{4}-1050 b^{2} c^{2} d^{3} e^{4} x^{4}-504 a^{2} b c d \,e^{6} x^{3}-168 a^{2} c^{2} d^{2} e^{5} x^{3}-224 a \,b^{3} d \,e^{6} x^{3}-504 a \,b^{2} c \,d^{2} e^{5} x^{3}-560 a b \,c^{2} d^{3} e^{4} x^{3}-280 a \,c^{3} d^{4} e^{3} x^{3}-56 b^{4} d^{2} e^{5} x^{3}-280 b^{3} c \,d^{3} e^{4} x^{3}-840 b^{2} c^{2} d^{4} e^{3} x^{3}-140 a^{3} c d \,e^{6} x^{2}-280 a^{2} b^{2} d \,e^{6} x^{2}-252 a^{2} b c \,d^{2} e^{5} x^{2}-84 a^{2} c^{2} d^{3} e^{4} x^{2}-112 a \,b^{3} d^{2} e^{5} x^{2}-252 a \,b^{2} c \,d^{3} e^{4} x^{2}-280 a b \,c^{2} d^{4} e^{3} x^{2}-140 a \,c^{3} d^{5} e^{2} x^{2}-28 b^{4} d^{3} e^{4} x^{2}-140 b^{3} c \,d^{4} e^{3} x^{2}-420 b^{2} c^{2} d^{5} e^{2} x^{2}-160 a^{3} b d \,e^{6} x -40 a^{3} c \,d^{2} e^{5} x -80 a^{2} b^{2} d^{2} e^{5} x -72 a^{2} b c \,d^{3} e^{4} x -24 a^{2} c^{2} d^{4} e^{3} x -32 a \,b^{3} d^{3} e^{4} x -72 a \,b^{2} c \,d^{4} e^{3} x -80 a b \,c^{2} d^{5} e^{2} x -40 a \,c^{3} d^{6} e x -8 b^{4} d^{4} e^{3} x -40 b^{3} c \,d^{5} e^{2} x -120 b^{2} c^{2} d^{6} e x -35 a^{4} d \,e^{6}-20 a^{3} b \,d^{2} e^{5}-5 a^{3} c \,d^{3} e^{4}-10 a^{2} b^{2} d^{3} e^{4}-9 a^{2} b c \,d^{4} e^{3}-3 a^{2} c^{2} d^{5} e^{2}-4 a \,b^{3} d^{4} e^{3}-9 a \,b^{2} c \,d^{5} e^{2}-10 a b \,c^{2} d^{6} e -5 a \,c^{3} d^{7}-b^{4} d^{5} e^{2}-5 b^{3} c \,d^{6} e -15 b^{2} c^{2} d^{7}}{280 d \,e^{7} \left (e^{8} x^{8}+8 d \,e^{7} x^{7}+28 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} e^{4} x^{4}+56 d^{5} e^{3} x^{3}+28 d^{6} e^{2} x^{2}+8 d^{7} e x +d^{8}\right )} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^9,x)
 

Output:

( - 35*a**4*d*e**6 - 20*a**3*b*d**2*e**5 - 160*a**3*b*d*e**6*x - 5*a**3*c* 
d**3*e**4 - 40*a**3*c*d**2*e**5*x - 140*a**3*c*d*e**6*x**2 - 10*a**2*b**2* 
d**3*e**4 - 80*a**2*b**2*d**2*e**5*x - 280*a**2*b**2*d*e**6*x**2 - 9*a**2* 
b*c*d**4*e**3 - 72*a**2*b*c*d**3*e**4*x - 252*a**2*b*c*d**2*e**5*x**2 - 50 
4*a**2*b*c*d*e**6*x**3 - 3*a**2*c**2*d**5*e**2 - 24*a**2*c**2*d**4*e**3*x 
- 84*a**2*c**2*d**3*e**4*x**2 - 168*a**2*c**2*d**2*e**5*x**3 - 210*a**2*c* 
*2*d*e**6*x**4 - 4*a*b**3*d**4*e**3 - 32*a*b**3*d**3*e**4*x - 112*a*b**3*d 
**2*e**5*x**2 - 224*a*b**3*d*e**6*x**3 - 9*a*b**2*c*d**5*e**2 - 72*a*b**2* 
c*d**4*e**3*x - 252*a*b**2*c*d**3*e**4*x**2 - 504*a*b**2*c*d**2*e**5*x**3 
- 630*a*b**2*c*d*e**6*x**4 - 10*a*b*c**2*d**6*e - 80*a*b*c**2*d**5*e**2*x 
- 280*a*b*c**2*d**4*e**3*x**2 - 560*a*b*c**2*d**3*e**4*x**3 - 700*a*b*c**2 
*d**2*e**5*x**4 - 560*a*b*c**2*d*e**6*x**5 - 5*a*c**3*d**7 - 40*a*c**3*d** 
6*e*x - 140*a*c**3*d**5*e**2*x**2 - 280*a*c**3*d**4*e**3*x**3 - 350*a*c**3 
*d**3*e**4*x**4 - 280*a*c**3*d**2*e**5*x**5 - 140*a*c**3*d*e**6*x**6 - b** 
4*d**5*e**2 - 8*b**4*d**4*e**3*x - 28*b**4*d**3*e**4*x**2 - 56*b**4*d**2*e 
**5*x**3 - 70*b**4*d*e**6*x**4 - 5*b**3*c*d**6*e - 40*b**3*c*d**5*e**2*x - 
 140*b**3*c*d**4*e**3*x**2 - 280*b**3*c*d**3*e**4*x**3 - 350*b**3*c*d**2*e 
**5*x**4 - 280*b**3*c*d*e**6*x**5 - 15*b**2*c**2*d**7 - 120*b**2*c**2*d**6 
*e*x - 420*b**2*c**2*d**5*e**2*x**2 - 840*b**2*c**2*d**4*e**3*x**3 - 1050* 
b**2*c**2*d**3*e**4*x**4 - 840*b**2*c**2*d**2*e**5*x**5 - 420*b**2*c**2...