\(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^6} \, dx\) [743]

Optimal result

Integrand size = 25, antiderivative size = 532 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {c^2 (6 B c d-3 b B e-A c e) x}{e^7}+\frac {B c^3 x^2}{2 e^6}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{5 e^8 (d+e x)^5}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{4 e^8 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)^3}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{2 e^8 (d+e x)^2}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^8 (d+e x)}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 885, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {A e \left (-2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )-e^3 \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 e \left (-12 a e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+b d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )\right )+B \left (c^3 \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )-e^3 \left (a^3 e^3 (d+5 e x)+2 a^2 b e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b^2 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 b^3 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c e^2 \left (-3 a^2 e^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )-24 a b e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+b^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+c^2 e \left (a d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-6 b \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )\right )+60 c \left (A c e (-2 c d+b e)+B \left (7 c^2 d^2+b^2 e^2+c e (-6 b d+a e)\right )\right ) (d+e x)^5 \log (d+e x)}{20 e^8 (d+e x)^5} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x]
 

Output:

(A*e*(-2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 5 
0*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) - e^3*(4*a^3*e^3 + 3*a^2*b*e^2* 
(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e* 
x + 10*d*e^2*x^2 + 10*e^3*x^3)) - 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2 
*x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 
 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(-12*a* 
e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + b*d*(137 
*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4))) + B 
*(c^3*(459*d^7 + 1875*d^6*e*x + 2700*d^5*e^2*x^2 + 1300*d^4*e^3*x^3 - 400* 
d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*x^7) - e^3*(a^3*e^3* 
(d + 5*e*x) + 2*a^2*b*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b^2*e*(d^3 + 
5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*b^3*(d^4 + 5*d^3*e*x + 10*d^2*e 
^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c*e^2*(-3*a^2*e^2*(d^3 + 5*d^2*e*x + 
 10*d*e^2*x^2 + 10*e^3*x^3) - 24*a*b*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 
 10*d*e^3*x^3 + 5*e^4*x^4) + b^2*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x 
^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + c^2*e*(a*d*e*(137*d^4 + 625*d^3*e*x + 
 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) - 6*b*(87*d^6 + 375*d^5*e 
*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 1 
0*e^6*x^6))) + 60*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)))*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 532, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x]
 

Output:

-((c^2*(6*B*c*d - 3*b*B*e - A*c*e)*x)/e^7) + (B*c^3*x^2)/(2*e^6) + ((B*d - 
 A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^8*(d + e*x)^5) - ((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)))/(4*e^8*(d + 
 e*x)^4) + ((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))))/( 
e^8*(d + e*x)^3) + (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)))/(2*e^8*(d + e*x)^2) 
 + (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)))/(e^8*(d + e 
*x)) - (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)))*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 = 1042, normalized size of antiderivative = 1.96

method	result	size
norman	\(\text {Expression too large to display}\)	\(1042\)
default	\(\text {Expression too large to display}\)	\(1053\)
risch	\(\text {Expression too large to display}\)	\(1089\)
parallelrisch	\(\text {Expression too large to display}\)	\(1854\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

(-1/20*(4*A*a^3*e^7+3*A*a^2*b*d*e^6+2*A*a^2*c*d^2*e^5+2*A*a*b^2*d^2*e^5+6* 
A*a*b*c*d^3*e^4+12*A*a*c^2*d^4*e^3+A*b^3*d^3*e^4+12*A*b^2*c*d^4*e^3-137*A* 
b*c^2*d^5*e^2+274*A*c^3*d^6*e+B*a^3*d*e^6+2*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+24*B*a*b*c*d^4*e^3-137*B*a*c^2*d^5*e^2+4*B*b^3*d^4*e 
^3-137*B*b^2*c*d^5*e^2+822*B*b*c^2*d^6*e-959*B*c^3*d^7)/e^8-(3*A*a*c^2*e^3 
+3*A*b^2*c*e^3-15*A*b*c^2*d*e^2+30*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+90*B*b*c^2*d^2*e-105*B*c^3*d^3)/e^4*x^4-1/2 
*(6*A*a*b*c*e^4+12*A*a*c^2*d*e^3+A*b^3*e^4+12*A*b^2*c*d*e^3-90*A*b*c^2*d^2 
*e^2+180*A*c^3*d^3*e+3*B*a^2*c*e^4+3*B*a*b^2*e^4+24*B*a*b*c*d*e^3-90*B*a*c 
^2*d^2*e^2+4*B*b^3*d*e^3-90*B*b^2*c*d^2*e^2+540*B*b*c^2*d^3*e-630*B*c^3*d^ 
4)/e^5*x^3-1/2*(2*A*a^2*c*e^5+2*A*a*b^2*e^5+6*A*a*b*c*d*e^4+12*A*a*c^2*d^2 
*e^3+A*b^3*d*e^4+12*A*b^2*c*d^2*e^3-110*A*b*c^2*d^3*e^2+220*A*c^3*d^4*e+2* 
B*a^2*b*e^5+3*B*a^2*c*d*e^4+3*B*a*b^2*d*e^4+24*B*a*b*c*d^2*e^3-110*B*a*c^2 
*d^3*e^2+4*B*b^3*d^2*e^3-110*B*b^2*c*d^3*e^2+660*B*b*c^2*d^4*e-770*B*c^3*d 
^5)/e^6*x^2-1/4*(3*A*a^2*b*e^6+2*A*a^2*c*d*e^5+2*A*a*b^2*d*e^5+6*A*a*b*c*d 
^2*e^4+12*A*a*c^2*d^3*e^3+A*b^3*d^2*e^4+12*A*b^2*c*d^3*e^3-125*A*b*c^2*d^4 
*e^2+250*A*c^3*d^5*e+B*a^3*e^6+2*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+24*B*a*b*c*d^3*e^3-125*B*a*c^2*d^4*e^2+4*B*b^3*d^3*e^3-125*B*b^2* 
c*d^4*e^2+750*B*b*c^2*d^5*e-875*B*c^3*d^6)/e^7*x+1/2*B*c^3*x^7/e+1/2*c^2*( 
2*A*c*e+6*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^5+3/e^8*c*(A*b*c*e^2-2*A*c^2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (527) = 1054\).

Time = 0.10 (sec) , antiderivative size = 1247, normalized size of antiderivative = 2.34 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="fricas")
 

Output:

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A* 
c^3)*d^6*e + 137*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 4*(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 - 2*(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 - 10*(7*B*c^3*d*e^6 - 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 - 100*(5*B 
*c^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*( 
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 + 10*(130*B*c^3*d^4*e^3 - 8 
0*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 4 
*(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 + 10*(270*B*c^3*d^5*e^2 - 120*(3*B*b*c^2 + 
 A*c^3)*d^4*e^3 + 110*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 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 - 2*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 5*(375*B*c^ 
3*d^6*e - 150*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(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*B*a 
*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 2*(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 - 2*(3*B*b*c^2 
+ A*c^3)*d^6*e + (B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + (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 + 5*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/(e*x+d)**6,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 898, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="maxima")
 

Output:

1/20*(459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A*c^3)*d^6*e + 137*(B 
*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 4*(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 - 2* 
(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 + 20*(35 
*B*c^3*d^3*e^4 - 15*(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 + 10*( 
245*B*c^3*d^4*e^3 - 100*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(B*b^2*c + (B*a + 
 A*b)*c^2)*d^2*e^5 - 4*(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 + 10*(329*B*c^3*d^5* 
e^2 - 130*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(B*b^2*c + (B*a + A*b)*c^2)*d^ 
3*e^4 - 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 - 2*(B*a^2*b + A*a*b^2 + A*a^2*c)* 
e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(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*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 2*(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^13*x^5 
+ 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8 
) + 1/2*(B*c^3*e*x^2 - 2*(6*B*c^3*d - (3*B*b*c^2 + A*c^3)*e)*x)/e^7 + 3*(7 
*B*c^3*d^2 - 2*(3*B*b*c^2 + A*c^3)*d*e + (B*b^2*c + (B*a + A*b)*c^2)*e^2)* 
log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1077 vs. \(2 (527) = 1054\).

Time = 0.16 (sec) , antiderivative size = 1077, normalized size of antiderivative = 2.02 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="giac")
 

Output:

3*(7*B*c^3*d^2 - 6*B*b*c^2*d*e - 2*A*c^3*d*e + B*b^2*c*e^2 + B*a*c^2*e^2 + 
 A*b*c^2*e^2)*log(abs(e*x + d))/e^8 + 1/2*(B*c^3*e^6*x^2 - 12*B*c^3*d*e^5* 
x + 6*B*b*c^2*e^6*x + 2*A*c^3*e^6*x)/e^12 + 1/20*(459*B*c^3*d^7 - 522*B*b* 
c^2*d^6*e - 174*A*c^3*d^6*e + 137*B*b^2*c*d^5*e^2 + 137*B*a*c^2*d^5*e^2 + 
137*A*b*c^2*d^5*e^2 - 4*B*b^3*d^4*e^3 - 24*B*a*b*c*d^4*e^3 - 12*A*b^2*c*d^ 
4*e^3 - 12*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 - 2*B*a^2*b*d^2*e^5 - 2*A*a*b^2*d^2*e^5 - 2*A 
*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^2*b*d*e^6 - 4*A*a^3*e^7 + 20*(35*B*c^ 
3*d^3*e^4 - 45*B*b*c^2*d^2*e^5 - 15*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 + 10*(245*B*c^3*d^4*e^3 - 300*B*b*c^2*d^3*e^4 - 10 
0*A*c^3*d^3*e^4 + 90*B*b^2*c*d^2*e^5 + 90*B*a*c^2*d^2*e^5 + 90*A*b*c^2*d^2 
*e^5 - 4*B*b^3*d*e^6 - 24*B*a*b*c*d*e^6 - 12*A*b^2*c*d*e^6 - 12*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 + 10* 
(329*B*c^3*d^5*e^2 - 390*B*b*c^2*d^4*e^3 - 130*A*c^3*d^4*e^3 + 110*B*b^2*c 
*d^3*e^4 + 110*B*a*c^2*d^3*e^4 + 110*A*b*c^2*d^3*e^4 - 4*B*b^3*d^2*e^5 - 2 
4*B*a*b*c*d^2*e^5 - 12*A*b^2*c*d^2*e^5 - 12*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 - 2*B*a^2*b*e^7 - 2* 
A*a*b^2*e^7 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 462*B*b*c^2*d^5*e^ 
2 - 154*A*c^3*d^5*e^2 + 125*B*b^2*c*d^4*e^3 + 125*B*a*c^2*d^4*e^3 + 125...
 

Mupad [B] (verification not implemented)

Time = 11.50 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x)
 

Output:

x*((A*c^3 + 3*B*b*c^2)/e^6 - (6*B*c^3*d)/e^7) - ((4*A*a^3*e^7 - 459*B*c^3* 
d^7 + B*a^3*d*e^6 + 174*A*c^3*d^6*e + A*b^3*d^3*e^4 + 4*B*b^3*d^4*e^3 + 2* 
A*a*b^2*d^2*e^5 + 12*A*a*c^2*d^4*e^3 + 2*A*a^2*c*d^2*e^5 + 3*B*a*b^2*d^3*e 
^4 + 2*B*a^2*b*d^2*e^5 - 137*A*b*c^2*d^5*e^2 + 12*A*b^2*c*d^4*e^3 - 137*B* 
a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 - 137*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 
+ 522*B*b*c^2*d^6*e + 6*A*a*b*c*d^3*e^4 + 24*B*a*b*c*d^4*e^3)/(20*e) + x^4 
*(B*b^3*e^6 + 3*A*a*c^2*e^6 + 3*A*b^2*c*e^6 + 15*A*c^3*d^2*e^4 - 35*B*c^3* 
d^3*e^3 + 45*B*b*c^2*d^2*e^4 + 6*B*a*b*c*e^6 - 15*A*b*c^2*d*e^5 - 15*B*a*c 
^2*d*e^5 - 15*B*b^2*c*d*e^5) + x^3*((A*b^3*e^6)/2 + (3*B*a*b^2*e^6)/2 + (3 
*B*a^2*c*e^6)/2 + 2*B*b^3*d*e^5 + 50*A*c^3*d^3*e^3 - (245*B*c^3*d^4*e^2)/2 
 - 45*A*b*c^2*d^2*e^4 - 45*B*a*c^2*d^2*e^4 + 150*B*b*c^2*d^3*e^3 - 45*B*b^ 
2*c*d^2*e^4 + 3*A*a*b*c*e^6 + 6*A*a*c^2*d*e^5 + 6*A*b^2*c*d*e^5 + 12*B*a*b 
*c*d*e^5) + x*((B*a^3*e^6)/4 - (399*B*c^3*d^6)/4 + (3*A*a^2*b*e^6)/4 + (77 
*A*c^3*d^5*e)/2 + (A*b^3*d^2*e^4)/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)/4 - (125*A*b*c^2*d^4*e^2)/4 + 3*A*b^2*c*d^3*e^3 - (125 
*B*a*c^2*d^4*e^2)/4 + (3*B*a^2*c*d^2*e^4)/4 - (125*B*b^2*c*d^4*e^2)/4 + (A 
*a*b^2*d*e^5)/2 + (A*a^2*c*d*e^5)/2 + (B*a^2*b*d*e^5)/2 + (231*B*b*c^2*d^5 
*e)/2 + (3*A*a*b*c*d^2*e^4)/2 + 6*B*a*b*c*d^3*e^3) + x^2*(A*a*b^2*e^6 + A* 
a^2*c*e^6 + B*a^2*b*e^6 + (A*b^3*d*e^5)/2 - (329*B*c^3*d^5*e)/2 + 65*A*c^3 
*d^4*e^2 + 2*B*b^3*d^2*e^4 + 6*A*a*c^2*d^2*e^4 - 55*A*b*c^2*d^3*e^3 + 6...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1340, normalized size of antiderivative = 2.52 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^6,x)
 

Output:

(120*log(d + e*x)*a*b*c**2*d**6*e**2 + 600*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 + 1200*log(d + e*x)*a*b*c** 
2*d**3*e**5*x**3 + 600*log(d + e*x)*a*b*c**2*d**2*e**6*x**4 + 120*log(d + 
e*x)*a*b*c**2*d*e**7*x**5 - 120*log(d + e*x)*a*c**3*d**7*e - 600*log(d + e 
*x)*a*c**3*d**6*e**2*x - 1200*log(d + e*x)*a*c**3*d**5*e**3*x**2 - 1200*lo 
g(d + e*x)*a*c**3*d**4*e**4*x**3 - 600*log(d + e*x)*a*c**3*d**3*e**5*x**4 
- 120*log(d + e*x)*a*c**3*d**2*e**6*x**5 + 60*log(d + e*x)*b**3*c*d**6*e** 
2 + 300*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 + 600*log(d + e*x)*b**3*c*d**3*e**5*x**3 + 300*log(d + e*x)*b**3*c* 
d**2*e**6*x**4 + 60*log(d + e*x)*b**3*c*d*e**7*x**5 - 360*log(d + e*x)*b** 
2*c**2*d**7*e - 1800*log(d + e*x)*b**2*c**2*d**6*e**2*x - 3600*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 - 
1800*log(d + e*x)*b**2*c**2*d**3*e**5*x**4 - 360*log(d + e*x)*b**2*c**2*d* 
*2*e**6*x**5 + 420*log(d + e*x)*b*c**3*d**8 + 2100*log(d + e*x)*b*c**3*d** 
7*e*x + 4200*log(d + e*x)*b*c**3*d**6*e**2*x**2 + 4200*log(d + e*x)*b*c**3 
*d**5*e**3*x**3 + 2100*log(d + e*x)*b*c**3*d**4*e**4*x**4 + 420*log(d + e* 
x)*b*c**3*d**3*e**5*x**5 - 4*a**4*d*e**7 - 4*a**3*b*d**2*e**6 - 20*a**3*b* 
d*e**7*x - 2*a**3*c*d**3*e**5 - 10*a**3*c*d**2*e**6*x - 20*a**3*c*d*e**7*x 
**2 - 4*a**2*b**2*d**3*e**5 - 20*a**2*b**2*d**2*e**6*x - 40*a**2*b**2*d*e* 
*7*x**2 - 9*a**2*b*c*d**4*e**4 - 45*a**2*b*c*d**3*e**5*x - 90*a**2*b*c*...