\(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^7} \, dx\) [744]

Optimal result

Integrand size = 25, antiderivative size = 539 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {B c^3 x}{e^7}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{6 e^8 (d+e x)^6}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{5 e^8 (d+e x)^5}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{4 e^8 (d+e x)^4}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{3 e^8 (d+e x)^3}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^8 (d+e x)^2}+\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {c^2 (7 B c d-3 b B e-A c e) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 868, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {A e \left (-c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+e^3 \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+3 c e^2 \left (a^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+6 c^2 e \left (a e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 b \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )\right )+B \left (c^3 \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )+e^3 \left (2 a^3 e^3 (d+6 e x)+3 a^2 b e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+3 a b^2 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b^3 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+3 c e^2 \left (a^2 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+4 a b e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+10 b^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+3 c^2 e \left (10 a e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )-b d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )\right )+60 c^2 (7 B c d-3 b B e-A c e) (d+e x)^6 \log (d+e x)}{60 e^8 (d+e x)^6} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x]
 

Output:

-1/60*(A*e*(-(c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e 
^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) + e^3*(10*a^3*e^3 + 6*a^2*b*e^2*(d 
 + 6*e*x) + 3*a*b^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x 
+ 15*d*e^2*x^2 + 20*e^3*x^3)) + 3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x 
^2) + 2*a*b*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^2*(d^4 + 
 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 6*c^2*e*(a*e*( 
d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*b*(d^5 + 
 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5))) 
 + B*(c^3*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x^3 + 
4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7) + e^3*(2* 
a^3*e^3*(d + 6*e*x) + 3*a^2*b*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 3*a*b^2*e 
*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^3*(d^4 + 6*d^3*e*x + 
15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 3*c*e^2*(a^2*e^2*(d^3 + 6*d 
^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 4*a*b*e*(d^4 + 6*d^3*e*x + 15*d^2*e^ 
2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 10*b^2*(d^5 + 6*d^4*e*x + 15*d^3*e^2* 
x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) + 3*c^2*e*(10*a*e*(d^5 + 
 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - 
 b*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d 
*e^4*x^4 + 360*e^5*x^5))) + 60*c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6 
*Log[d + e*x])/(e^8*(d + e*x)^6)
 

Rubi [A] (verified)

Time = 2.13 (sec) , antiderivative size = 539, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x]
 

Output:

(B*c^3*x)/e^7 + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(6*e^8*(d + e*x)^6 
) - ((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)))/(5*e^8*(d + e*x)^5) + (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))))/(4*e^8*(d + e*x)^4) + (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)))/(3*e^8*(d + e*x)^3) + (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)))/(2*e^8*(d + e*x)^2) + (3*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)) - (c^2*(7*B*c*d - 3* 
b*B*e - A*c*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 = 1.56 (sec) , antiderivative size = 1047, normalized size of antiderivative = 1.94

method	result	size
norman	\(\text {Expression too large to display}\)	\(1047\)
default	\(\text {Expression too large to display}\)	\(1058\)
risch	\(\text {Expression too large to display}\)	\(1067\)
parallelrisch	\(\text {Expression too large to display}\)	\(1587\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

(B*c^3*x^7/e-1/60*(10*A*a^3*e^7+6*A*a^2*b*d*e^6+3*A*a^2*c*d^2*e^5+3*A*a*b^ 
2*d^2*e^5+6*A*a*b*c*d^3*e^4+6*A*a*c^2*d^4*e^3+A*b^3*d^3*e^4+6*A*b^2*c*d^4* 
e^3+30*A*b*c^2*d^5*e^2-147*A*c^3*d^6*e+2*B*a^3*d*e^6+3*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+12*B*a*b*c*d^4*e^3+30*B*a*c^2*d^5*e^2+2*B 
*b^3*d^4*e^3+30*B*b^2*c*d^5*e^2-441*B*b*c^2*d^6*e+1029*B*c^3*d^7)/e^8-3*(A 
*b*c^2*e^2-2*A*c^3*d*e+B*a*c^2*e^2+B*b^2*c*e^2-6*B*b*c^2*d*e+14*B*c^3*d^2) 
/e^3*x^5-1/2*(3*A*a*c^2*e^3+3*A*b^2*c*e^3+15*A*b*c^2*d*e^2-45*A*c^3*d^2*e+ 
6*B*a*b*c*e^3+15*B*a*c^2*d*e^2+B*b^3*e^3+15*B*b^2*c*d*e^2-135*B*b*c^2*d^2* 
e+315*B*c^3*d^3)/e^4*x^4-1/3*(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+30*A*b*c^2*d^2*e^2-110*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+30*B*a*c^2*d^2*e^2+2*B*b^3*d*e^3+30*B*b^2*c*d^2*e^2-330* 
B*b*c^2*d^3*e+770*B*c^3*d^4)/e^5*x^3-1/4*(3*A*a^2*c*e^5+3*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+30*A*b*c^2*d^3 
*e^2-125*A*c^3*d^4*e+3*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+30*B*a*c^2*d^3*e^2+2*B*b^3*d^2*e^3+30*B*b^2*c*d^3*e^2-375*B*b* 
c^2*d^4*e+875*B*c^3*d^5)/e^6*x^2-1/10*(6*A*a^2*b*e^6+3*A*a^2*c*d*e^5+3*A*a 
*b^2*d*e^5+6*A*a*b*c*d^2*e^4+6*A*a*c^2*d^3*e^3+A*b^3*d^2*e^4+6*A*b^2*c*d^3 
*e^3+30*A*b*c^2*d^4*e^2-137*A*c^3*d^5*e+2*B*a^3*e^6+3*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+12*B*a*b*c*d^3*e^3+30*B*a*c^2*d^4*e^2+2*B*b^ 
3*d^3*e^3+30*B*b^2*c*d^4*e^2-411*B*b*c^2*d^5*e+959*B*c^3*d^6)/e^7*x)/(e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (530) = 1060\).

Time = 0.10 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="fricas")
 

Output:

1/60*(60*B*c^3*e^7*x^7 + 360*B*c^3*d*e^6*x^6 - 669*B*c^3*d^7 - 10*A*a^3*e^ 
7 + 147*(3*B*b*c^2 + A*c^3)*d^6*e - 30*(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*a*b)*c)*d^3*e^4 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2 
*e^5 - 2*(B*a^3 + 3*A*a^2*b)*d*e^6 - 180*(2*B*c^3*d^2*e^5 - 2*(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 - 30*(135*B*c^3*d^3*e 
^4 - 45*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15*(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 - 20*(410*B*c^3*d^ 
4*e^3 - 110*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 30*(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 - 15*(515*B*c^3*d^5*e^2 - 125*(3 
*B*b*c^2 + A*c^3)*d^4*e^3 + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 2*(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 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 - 6* 
(599*B*c^3*d^6*e - 137*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 30*(B*b^2*c + (B*a + 
A*b)*c^2)*d^4*e^3 + 2*(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 + 3*(B*a^2*b + A*a*b 
^2 + A*a^2*c)*d*e^6 + 2*(B*a^3 + 3*A*a^2*b)*e^7)*x - 60*(7*B*c^3*d^7 - (3* 
B*b*c^2 + A*c^3)*d^6*e + (7*B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 6 
*(7*B*c^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d*e^6)*x^5 + 15*(7*B*c^3*d^3*e^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/(e*x+d)**7,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 905, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="maxima")
 

Output:

-1/60*(669*B*c^3*d^7 + 10*A*a^3*e^7 - 147*(3*B*b*c^2 + A*c^3)*d^6*e + 30*( 
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*a*b)*c)*d^3*e^4 + 3 
*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 2*(B*a^3 + 3*A*a^2*b)*d*e^6 + 180 
*(7*B*c^3*d^2*e^5 - 2*(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 + 30*(175*B*c^3*d^3*e^4 - 45*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15 
*(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 + 20*(455*B*c^3*d^4*e^3 - 110*(3*B*b*c^2 + A*c^3)*d^3*e^4 
+ 30*(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 + 15*(539*B*c^3*d^5*e^2 - 125*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 30*(B*b^2*c 
+ (B*a + A*b)*c^2)*d^3*e^4 + 2*(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 + 3*(B*a^2*b 
+ A*a*b^2 + A*a^2*c)*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 137*(3*B*b*c^2 + A*c^ 
3)*d^5*e^2 + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + 2*(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 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + 2*(B*a^3 + 3*A*a^2 
*b)*e^7)*x)/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 
 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) + B*c^3*x/e^7 - (7*B*c^3*d - (3* 
B*b*c^2 + A*c^3)*e)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (530) = 1060\).

Time = 0.19 (sec) , antiderivative size = 1067, normalized size of antiderivative = 1.98 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="giac")
 

Output:

B*c^3*x/e^7 - (7*B*c^3*d - 3*B*b*c^2*e - A*c^3*e)*log(abs(e*x + d))/e^8 - 
1/60*(669*B*c^3*d^7 - 441*B*b*c^2*d^6*e - 147*A*c^3*d^6*e + 30*B*b^2*c*d^5 
*e^2 + 30*B*a*c^2*d^5*e^2 + 30*A*b*c^2*d^5*e^2 + 2*B*b^3*d^4*e^3 + 12*B*a* 
b*c*d^4*e^3 + 6*A*b^2*c*d^4*e^3 + 6*A*a*c^2*d^4*e^3 + 3*B*a*b^2*d^3*e^4 + 
A*b^3*d^3*e^4 + 3*B*a^2*c*d^3*e^4 + 6*A*a*b*c*d^3*e^4 + 3*B*a^2*b*d^2*e^5 
+ 3*A*a*b^2*d^2*e^5 + 3*A*a^2*c*d^2*e^5 + 2*B*a^3*d*e^6 + 6*A*a^2*b*d*e^6 
+ 10*A*a^3*e^7 + 180*(7*B*c^3*d^2*e^5 - 6*B*b*c^2*d*e^6 - 2*A*c^3*d*e^6 + 
B*b^2*c*e^7 + B*a*c^2*e^7 + A*b*c^2*e^7)*x^5 + 30*(175*B*c^3*d^3*e^4 - 135 
*B*b*c^2*d^2*e^5 - 45*A*c^3*d^2*e^5 + 15*B*b^2*c*d*e^6 + 15*B*a*c^2*d*e^6 
+ 15*A*b*c^2*d*e^6 + B*b^3*e^7 + 6*B*a*b*c*e^7 + 3*A*b^2*c*e^7 + 3*A*a*c^2 
*e^7)*x^4 + 20*(455*B*c^3*d^4*e^3 - 330*B*b*c^2*d^3*e^4 - 110*A*c^3*d^3*e^ 
4 + 30*B*b^2*c*d^2*e^5 + 30*B*a*c^2*d^2*e^5 + 30*A*b*c^2*d^2*e^5 + 2*B*b^3 
*d*e^6 + 12*B*a*b*c*d*e^6 + 6*A*b^2*c*d*e^6 + 6*A*a*c^2*d*e^6 + 3*B*a*b^2* 
e^7 + A*b^3*e^7 + 3*B*a^2*c*e^7 + 6*A*a*b*c*e^7)*x^3 + 15*(539*B*c^3*d^5*e 
^2 - 375*B*b*c^2*d^4*e^3 - 125*A*c^3*d^4*e^3 + 30*B*b^2*c*d^3*e^4 + 30*B*a 
*c^2*d^3*e^4 + 30*A*b*c^2*d^3*e^4 + 2*B*b^3*d^2*e^5 + 12*B*a*b*c*d^2*e^5 + 
 6*A*b^2*c*d^2*e^5 + 6*A*a*c^2*d^2*e^5 + 3*B*a*b^2*d*e^6 + A*b^3*d*e^6 + 3 
*B*a^2*c*d*e^6 + 6*A*a*b*c*d*e^6 + 3*B*a^2*b*e^7 + 3*A*a*b^2*e^7 + 3*A*a^2 
*c*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 411*B*b*c^2*d^5*e^2 - 137*A*c^3*d^5*e^2 
 + 30*B*b^2*c*d^4*e^3 + 30*B*a*c^2*d^4*e^3 + 30*A*b*c^2*d^4*e^3 + 2*B*b...
 

Mupad [B] (verification not implemented)

Time = 11.57 (sec) , antiderivative size = 1598, normalized size of antiderivative = 2.96 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x)
 

Output:

-(10*A*a^3*e^7 + 669*B*c^3*d^7 + 2*B*a^3*d*e^6 - 147*A*c^3*d^6*e + 420*B*c 
^3*d^7*log(d + e*x) + 12*B*a^3*e^7*x + A*b^3*d^3*e^4 + 2*B*b^3*d^4*e^3 + 2 
0*A*b^3*e^7*x^3 + 30*B*b^3*e^7*x^4 - 60*B*c^3*e^7*x^7 + 3594*B*c^3*d^6*e*x 
 + 3*A*a*b^2*d^2*e^5 + 6*A*a*c^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 + 3*B*a*b^2*d 
^3*e^4 + 3*B*a^2*b*d^2*e^5 + 30*A*b*c^2*d^5*e^2 + 6*A*b^2*c*d^4*e^3 + 30*B 
*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 + 30*B*b^2*c*d^5*e^2 + 45*A*a*b^2*e^7*x 
^2 + 45*A*a^2*c*e^7*x^2 + 45*B*a^2*b*e^7*x^2 + 60*B*a*b^2*e^7*x^3 + 90*A*a 
*c^2*e^7*x^4 + 60*B*a^2*c*e^7*x^3 + 90*A*b^2*c*e^7*x^4 + 180*A*b*c^2*e^7*x 
^5 + 180*B*a*c^2*e^7*x^5 + 6*A*b^3*d^2*e^5*x + 15*A*b^3*d*e^6*x^2 + 180*B* 
b^2*c*e^7*x^5 - 822*A*c^3*d^5*e^2*x + 12*B*b^3*d^3*e^4*x + 40*B*b^3*d*e^6* 
x^3 - 360*A*c^3*d*e^6*x^5 - 360*B*c^3*d*e^6*x^6 - 60*A*c^3*e^7*x^6*log(d + 
 e*x) - 1875*A*c^3*d^4*e^3*x^2 + 30*B*b^3*d^2*e^5*x^2 - 2200*A*c^3*d^3*e^4 
*x^3 - 1350*A*c^3*d^2*e^5*x^4 + 7725*B*c^3*d^5*e^2*x^2 + 8200*B*c^3*d^4*e^ 
3*x^3 + 4050*B*c^3*d^3*e^4*x^4 + 360*B*c^3*d^2*e^5*x^5 + 6*A*a^2*b*d*e^6 - 
 441*B*b*c^2*d^6*e - 60*A*c^3*d^6*e*log(d + e*x) + 36*A*a^2*b*e^7*x + 90*A 
*a*c^2*d^2*e^5*x^2 + 450*A*b*c^2*d^3*e^4*x^2 + 90*A*b^2*c*d^2*e^5*x^2 + 45 
0*B*a*c^2*d^3*e^4*x^2 + 600*A*b*c^2*d^2*e^5*x^3 + 600*B*a*c^2*d^2*e^5*x^3 
- 5625*B*b*c^2*d^4*e^3*x^2 + 450*B*b^2*c*d^3*e^4*x^2 - 6600*B*b*c^2*d^3*e^ 
4*x^3 + 600*B*b^2*c*d^2*e^5*x^3 - 4050*B*b*c^2*d^2*e^5*x^4 - 900*A*c^3*d^4 
*e^3*x^2*log(d + e*x) - 1200*A*c^3*d^3*e^4*x^3*log(d + e*x) - 900*A*c^3...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1228, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^7,x)
 

Output:

(60*log(d + e*x)*a*c**3*d**7*e + 360*log(d + e*x)*a*c**3*d**6*e**2*x + 900 
*log(d + e*x)*a*c**3*d**5*e**3*x**2 + 1200*log(d + e*x)*a*c**3*d**4*e**4*x 
**3 + 900*log(d + e*x)*a*c**3*d**3*e**5*x**4 + 360*log(d + e*x)*a*c**3*d** 
2*e**6*x**5 + 60*log(d + e*x)*a*c**3*d*e**7*x**6 + 180*log(d + e*x)*b**2*c 
**2*d**7*e + 1080*log(d + e*x)*b**2*c**2*d**6*e**2*x + 2700*log(d + e*x)*b 
**2*c**2*d**5*e**3*x**2 + 3600*log(d + e*x)*b**2*c**2*d**4*e**4*x**3 + 270 
0*log(d + e*x)*b**2*c**2*d**3*e**5*x**4 + 1080*log(d + e*x)*b**2*c**2*d**2 
*e**6*x**5 + 180*log(d + e*x)*b**2*c**2*d*e**7*x**6 - 420*log(d + e*x)*b*c 
**3*d**8 - 2520*log(d + e*x)*b*c**3*d**7*e*x - 6300*log(d + e*x)*b*c**3*d* 
*6*e**2*x**2 - 8400*log(d + e*x)*b*c**3*d**5*e**3*x**3 - 6300*log(d + e*x) 
*b*c**3*d**4*e**4*x**4 - 2520*log(d + e*x)*b*c**3*d**3*e**5*x**5 - 420*log 
(d + e*x)*b*c**3*d**2*e**6*x**6 - 10*a**4*d*e**7 - 8*a**3*b*d**2*e**6 - 48 
*a**3*b*d*e**7*x - 3*a**3*c*d**3*e**5 - 18*a**3*c*d**2*e**6*x - 45*a**3*c* 
d*e**7*x**2 - 6*a**2*b**2*d**3*e**5 - 36*a**2*b**2*d**2*e**6*x - 90*a**2*b 
**2*d*e**7*x**2 - 9*a**2*b*c*d**4*e**4 - 54*a**2*b*c*d**3*e**5*x - 135*a** 
2*b*c*d**2*e**6*x**2 - 180*a**2*b*c*d*e**7*x**3 - 6*a**2*c**2*d**5*e**3 - 
36*a**2*c**2*d**4*e**4*x - 90*a**2*c**2*d**3*e**5*x**2 - 120*a**2*c**2*d** 
2*e**6*x**3 - 90*a**2*c**2*d*e**7*x**4 - 4*a*b**3*d**4*e**4 - 24*a*b**3*d* 
*3*e**5*x - 60*a*b**3*d**2*e**6*x**2 - 80*a*b**3*d*e**7*x**3 - 18*a*b**2*c 
*d**5*e**3 - 108*a*b**2*c*d**4*e**4*x - 270*a*b**2*c*d**3*e**5*x**2 - 3...