\(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^4} \, dx\) [741]

Optimal result

Integrand size = 25, antiderivative size = 519 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {\left (B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 488, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {12 e \left (A c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )-B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )\right ) x+6 c e^2 \left (A c e (-4 c d+3 b e)+B \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )\right ) x^2+4 c^2 e^3 (-4 B c d+3 b B e+A c e) x^3+3 B c^3 e^4 x^4+\frac {4 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {6 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{(d+e x)^2}+\frac {36 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right )}{d+e x}+12 \left (A e (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )+B \left (35 c^3 d^4+30 c^2 d^2 e (-2 b d+a e)+b^2 e^3 (-4 b d+3 a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{12 e^8} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]
 

Output:

(12*e*(A*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)) - B*(2*c*d - 
b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e)))*x + 6*c*e^2*(A*c*e*( 
-4*c*d + 3*b*e) + B*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)))*x^2 + 
 4*c^2*e^3*(-4*B*c*d + 3*b*B*e + A*c*e)*x^3 + 3*B*c^3*e^4*x^4 + (4*(B*d - 
A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (6*(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)^ 
2 + (36*(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)) 
))/(d + e*x) + 12*(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 + 30*c^2*d^2*e*(-2*b*d + a*e) + b^2*e^3*(-4*b* 
d + 3*a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*Log[d + e*x])/(1 
2*e^8)
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 519, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]
 

Output:

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

Leaf count of result is larger than twice the leaf count of optimal. \(1044\) vs. \(2(509)=1018\).

Time = 1.73 (sec) , antiderivative size = 1045, normalized size of antiderivative = 2.01

method	result	size
norman	\(\text {Expression too large to display}\)	\(1045\)
default	\(\text {Expression too large to display}\)	\(1089\)
risch	\(\text {Expression too large to display}\)	\(1168\)
parallelrisch	\(\text {Expression too large to display}\)	\(2116\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

(-1/6*(2*A*a^3*e^7+3*A*a^2*b*d*e^6+6*A*a^2*c*d^2*e^5+6*A*a*b^2*d^2*e^5-66* 
A*a*b*c*d^3*e^4+132*A*a*c^2*d^4*e^3-11*A*b^3*d^3*e^4+132*A*b^2*c*d^4*e^3-3 
30*A*b*c^2*d^5*e^2+220*A*c^3*d^6*e+B*a^3*d*e^6+6*B*a^2*b*d^2*e^5-33*B*a^2* 
c*d^3*e^4-33*B*a*b^2*d^3*e^4+264*B*a*b*c*d^4*e^3-330*B*a*c^2*d^5*e^2+44*B* 
b^3*d^4*e^3-330*B*b^2*c*d^5*e^2+660*B*b*c^2*d^6*e-385*B*c^3*d^7)/e^8+1/4*( 
12*A*a*c^2*e^3+12*A*b^2*c*e^3-30*A*b*c^2*d*e^2+20*A*c^3*d^2*e+24*B*a*b*c*e 
^3-30*B*a*c^2*d*e^2+4*B*b^3*e^3-30*B*b^2*c*d*e^2+60*B*b*c^2*d^2*e-35*B*c^3 
*d^3)/e^4*x^4-3*(A*a^2*c*e^5+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^3-30*A*b*c^2*d^3*e^2+20*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+24*B*a*b*c*d^2*e^3-30*B*a*c^2*d^3*e^2 
+4*B*b^3*d^2*e^3-30*B*b^2*c*d^3*e^2+60*B*b*c^2*d^4*e-35*B*c^3*d^5)/e^6*x^2 
-1/2*(3*A*a^2*b*e^6+6*A*a^2*c*d*e^5+6*A*a*b^2*d*e^5-54*A*a*b*c*d^2*e^4+108 
*A*a*c^2*d^3*e^3-9*A*b^3*d^2*e^4+108*A*b^2*c*d^3*e^3-270*A*b*c^2*d^4*e^2+1 
80*A*c^3*d^5*e+B*a^3*e^6+6*B*a^2*b*d*e^5-27*B*a^2*c*d^2*e^4-27*B*a*b^2*d^2 
*e^4+216*B*a*b*c*d^3*e^3-270*B*a*c^2*d^4*e^2+36*B*b^3*d^3*e^3-270*B*b^2*c* 
d^4*e^2+540*B*b*c^2*d^5*e-315*B*c^3*d^6)/e^7*x+1/4*B*c^3*x^7/e+1/4*c*(6*A* 
b*c*e^2-4*A*c^2*d*e+6*B*a*c*e^2+6*B*b^2*e^2-12*B*b*c*d*e+7*B*c^2*d^2)/e^3* 
x^5+1/12*c^2*(4*A*c*e+12*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^3+1/e^8*(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+30*A*b*c^2*d^2*e^2-20*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+30*B*a*c^2*d^2*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1369 vs. \(2 (508) = 1016\).

Time = 0.10 (sec) , antiderivative size = 1369, normalized size of antiderivative = 2.64 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="fricas")
 

Output:

1/12*(3*B*c^3*e^7*x^7 + 214*B*c^3*d^7 - 4*A*a^3*e^7 - 148*(3*B*b*c^2 + A*c 
^3)*d^6*e + 282*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 52*(B*b^3 + 3*A*a*c^ 
2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 22*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2* 
A*a*b)*c)*d^3*e^4 - 12*(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 - (7*B*c^3*d*e^6 - 4*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7* 
B*c^3*d^2*e^5 - 4*(3*B*b*c^2 + A*c^3)*d*e^6 + 6*(B*b^2*c + (B*a + A*b)*c^2 
)*e^7)*x^5 - 3*(35*B*c^3*d^3*e^4 - 20*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 30*(B* 
b^2*c + (B*a + A*b)*c^2)*d*e^6 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2 
)*c)*e^7)*x^4 - 2*(278*B*c^3*d^4*e^3 - 146*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 1 
89*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 18*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a* 
b + A*b^2)*c)*d*e^6)*x^3 - 6*(68*B*c^3*d^5*e^2 - 26*(3*B*b*c^2 + A*c^3)*d^ 
4*e^3 + 9*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 6*(B*b^3 + 3*A*a*c^2 + 3*( 
2*B*a*b + A*b^2)*c)*d^2*e^5 - 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c 
)*d*e^6 + 6*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 6*(37*B*c^3*d^6*e - 3 
4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 81*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 1 
8*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 9*(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 + 12*(35*B*c^3*d^7 - 20*(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 - 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...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**3/(e*x+d)**4,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 875, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/6*(107*B*c^3*d^7 - 2*A*a^3*e^7 - 74*(3*B*b*c^2 + A*c^3)*d^6*e + 141*(B*b 
^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 26*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b 
^2)*c)*d^4*e^3 + 11*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 
6*(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 + 18*( 
7*B*c^3*d^5*e^2 - 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 10*(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 - (B*a^2*b + A*a*b^2 + A* 
a^2*c)*e^7)*x^2 + 3*(77*B*c^3*d^6*e - 54*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 105 
*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 20*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b 
+ A*b^2)*c)*d^3*e^4 + 9*(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 
^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*B*c^3*e^3*x^4 - 
4*(4*B*c^3*d*e^2 - (3*B*b*c^2 + A*c^3)*e^3)*x^3 + 6*(10*B*c^3*d^2*e - 4*(3 
*B*b*c^2 + A*c^3)*d*e^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^3)*x^2 - 12*(20* 
B*c^3*d^3 - 10*(3*B*b*c^2 + A*c^3)*d^2*e + 12*(B*b^2*c + (B*a + A*b)*c^2)* 
d*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^3)*x)/e^7 + (35*B*c^ 
3*d^4 - 20*(3*B*b*c^2 + A*c^3)*d^3*e + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^2* 
e^2 - 4*(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)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1100 vs. \(2 (508) = 1016\).

Time = 0.17 (sec) , antiderivative size = 1100, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="giac")
 

Output:

(35*B*c^3*d^4 - 60*B*b*c^2*d^3*e - 20*A*c^3*d^3*e + 30*B*b^2*c*d^2*e^2 + 3 
0*B*a*c^2*d^2*e^2 + 30*A*b*c^2*d^2*e^2 - 4*B*b^3*d*e^3 - 24*B*a*b*c*d*e^3 
- 12*A*b^2*c*d*e^3 - 12*A*a*c^2*d*e^3 + 3*B*a*b^2*e^4 + A*b^3*e^4 + 3*B*a^ 
2*c*e^4 + 6*A*a*b*c*e^4)*log(abs(e*x + d))/e^8 + 1/6*(107*B*c^3*d^7 - 222* 
B*b*c^2*d^6*e - 74*A*c^3*d^6*e + 141*B*b^2*c*d^5*e^2 + 141*B*a*c^2*d^5*e^2 
 + 141*A*b*c^2*d^5*e^2 - 26*B*b^3*d^4*e^3 - 156*B*a*b*c*d^4*e^3 - 78*A*b^2 
*c*d^4*e^3 - 78*A*a*c^2*d^4*e^3 + 33*B*a*b^2*d^3*e^4 + 11*A*b^3*d^3*e^4 + 
33*B*a^2*c*d^3*e^4 + 66*A*a*b*c*d^3*e^4 - 6*B*a^2*b*d^2*e^5 - 6*A*a*b^2*d^ 
2*e^5 - 6*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^2*b*d*e^6 - 2*A*a^3*e^7 + 
18*(7*B*c^3*d^5*e^2 - 15*B*b*c^2*d^4*e^3 - 5*A*c^3*d^4*e^3 + 10*B*b^2*c*d^ 
3*e^4 + 10*B*a*c^2*d^3*e^4 + 10*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 - B*a^2*b*e^7 - A*a*b^2*e^7 
 - A*a^2*c*e^7)*x^2 + 3*(77*B*c^3*d^6*e - 162*B*b*c^2*d^5*e^2 - 54*A*c^3*d 
^5*e^2 + 105*B*b^2*c*d^4*e^3 + 105*B*a*c^2*d^4*e^3 + 105*A*b*c^2*d^4*e^3 - 
 20*B*b^3*d^3*e^4 - 120*B*a*b*c*d^3*e^4 - 60*A*b^2*c*d^3*e^4 - 60*A*a*c^2* 
d^3*e^4 + 27*B*a*b^2*d^2*e^5 + 9*A*b^3*d^2*e^5 + 27*B*a^2*c*d^2*e^5 + 54*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)^3*e^8) + 1/12*(3*B*c^3*e^12*x^4 - 16 
*B*c^3*d*e^11*x^3 + 12*B*b*c^2*e^12*x^3 + 4*A*c^3*e^12*x^3 + 60*B*c^3*d...
 

Mupad [B] (verification not implemented)

Time = 11.54 (sec) , antiderivative size = 1151, normalized size of antiderivative = 2.22 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x)
 

Output:

x*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*B*a*b*c)/e^4 - (6*d^2*((A*c^3 + 3*B* 
b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e^2 + (4*d*((4*d*((A*c^3 + 3*B*b*c^2)/e^4 - 
 (4*B*c^3*d)/e^5))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^4 + (6*B*c^3* 
d^2)/e^6))/e - (4*B*c^3*d^3)/e^7) - x^2*((2*d*((A*c^3 + 3*B*b*c^2)/e^4 - ( 
4*B*c^3*d)/e^5))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/(2*e^4) + (3*B*c^ 
3*d^2)/e^6) - ((2*A*a^3*e^7 - 107*B*c^3*d^7 + B*a^3*d*e^6 + 74*A*c^3*d^6*e 
 - 11*A*b^3*d^3*e^4 + 26*B*b^3*d^4*e^3 + 6*A*a*b^2*d^2*e^5 + 78*A*a*c^2*d^ 
4*e^3 + 6*A*a^2*c*d^2*e^5 - 33*B*a*b^2*d^3*e^4 + 6*B*a^2*b*d^2*e^5 - 141*A 
*b*c^2*d^5*e^2 + 78*A*b^2*c*d^4*e^3 - 141*B*a*c^2*d^5*e^2 - 33*B*a^2*c*d^3 
*e^4 - 141*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 + 222*B*b*c^2*d^6*e - 66*A*a* 
b*c*d^3*e^4 + 156*B*a*b*c*d^4*e^3)/(6*e) + x*((B*a^3*e^6)/2 - (77*B*c^3*d^ 
6)/2 + (3*A*a^2*b*e^6)/2 + 27*A*c^3*d^5*e - (9*A*b^3*d^2*e^4)/2 + 10*B*b^3 
*d^3*e^3 + 30*A*a*c^2*d^3*e^3 - (27*B*a*b^2*d^2*e^4)/2 - (105*A*b*c^2*d^4* 
e^2)/2 + 30*A*b^2*c*d^3*e^3 - (105*B*a*c^2*d^4*e^2)/2 - (27*B*a^2*c*d^2*e^ 
4)/2 - (105*B*b^2*c*d^4*e^2)/2 + 3*A*a*b^2*d*e^5 + 3*A*a^2*c*d*e^5 + 3*B*a 
^2*b*d*e^5 + 81*B*b*c^2*d^5*e - 27*A*a*b*c*d^2*e^4 + 60*B*a*b*c*d^3*e^3) + 
 x^2*(3*A*a*b^2*e^6 + 3*A*a^2*c*e^6 + 3*B*a^2*b*e^6 - 3*A*b^3*d*e^5 - 21*B 
*c^3*d^5*e + 15*A*c^3*d^4*e^2 + 6*B*b^3*d^2*e^4 + 18*A*a*c^2*d^2*e^4 - 30* 
A*b*c^2*d^3*e^3 + 18*A*b^2*c*d^2*e^4 - 30*B*a*c^2*d^3*e^3 + 45*B*b*c^2*d^4 
*e^2 - 30*B*b^2*c*d^3*e^3 - 9*B*a*b^2*d*e^5 - 9*B*a^2*c*d*e^5 + 36*B*a*...
 

Reduce [B] (verification not implemented)

Time = 14.24 (sec) , antiderivative size = 1551, normalized size of antiderivative = 2.99 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^3/(e*x+d)^4,x)
 

Output:

(108*log(d + e*x)*a**2*b*c*d**4*e**4 + 324*log(d + e*x)*a**2*b*c*d**3*e**5 
*x + 324*log(d + e*x)*a**2*b*c*d**2*e**6*x**2 + 108*log(d + e*x)*a**2*b*c* 
d*e**7*x**3 - 144*log(d + e*x)*a**2*c**2*d**5*e**3 - 432*log(d + e*x)*a**2 
*c**2*d**4*e**4*x - 432*log(d + e*x)*a**2*c**2*d**3*e**5*x**2 - 144*log(d 
+ e*x)*a**2*c**2*d**2*e**6*x**3 + 48*log(d + e*x)*a*b**3*d**4*e**4 + 144*l 
og(d + e*x)*a*b**3*d**3*e**5*x + 144*log(d + e*x)*a*b**3*d**2*e**6*x**2 + 
48*log(d + e*x)*a*b**3*d*e**7*x**3 - 432*log(d + e*x)*a*b**2*c*d**5*e**3 - 
 1296*log(d + e*x)*a*b**2*c*d**4*e**4*x - 1296*log(d + e*x)*a*b**2*c*d**3* 
e**5*x**2 - 432*log(d + e*x)*a*b**2*c*d**2*e**6*x**3 + 720*log(d + e*x)*a* 
b*c**2*d**6*e**2 + 2160*log(d + e*x)*a*b*c**2*d**5*e**3*x + 2160*log(d + e 
*x)*a*b*c**2*d**4*e**4*x**2 + 720*log(d + e*x)*a*b*c**2*d**3*e**5*x**3 - 2 
40*log(d + e*x)*a*c**3*d**7*e - 720*log(d + e*x)*a*c**3*d**6*e**2*x - 720* 
log(d + e*x)*a*c**3*d**5*e**3*x**2 - 240*log(d + e*x)*a*c**3*d**4*e**4*x** 
3 - 48*log(d + e*x)*b**4*d**5*e**3 - 144*log(d + e*x)*b**4*d**4*e**4*x - 1 
44*log(d + e*x)*b**4*d**3*e**5*x**2 - 48*log(d + e*x)*b**4*d**2*e**6*x**3 
+ 360*log(d + e*x)*b**3*c*d**6*e**2 + 1080*log(d + e*x)*b**3*c*d**5*e**3*x 
 + 1080*log(d + e*x)*b**3*c*d**4*e**4*x**2 + 360*log(d + e*x)*b**3*c*d**3* 
e**5*x**3 - 720*log(d + e*x)*b**2*c**2*d**7*e - 2160*log(d + e*x)*b**2*c** 
2*d**6*e**2*x - 2160*log(d + e*x)*b**2*c**2*d**5*e**3*x**2 - 720*log(d + e 
*x)*b**2*c**2*d**4*e**4*x**3 + 420*log(d + e*x)*b*c**3*d**8 + 1260*log(...