\(\int (d+e x)^m (a+b x+c x^2)^4 \, dx\) [761]

Optimal result

Integrand size = 20, antiderivative size = 485 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^{1+m}}{e^9 (1+m)}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{2+m}}{e^9 (2+m)}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{3+m}}{e^9 (3+m)}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{4+m}}{e^9 (4+m)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{5+m}}{e^9 (5+m)}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{6+m}}{e^9 (6+m)}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7+m}}{e^9 (7+m)}-\frac {4 c^3 (2 c d-b e) (d+e x)^{8+m}}{e^9 (8+m)}+\frac {c^4 (d+e x)^{9+m}}{e^9 (9+m)} \] Output:

(a*e^2-b*d*e+c*d^2)^4*(e*x+d)^(1+m)/e^9/(1+m)-4*(-b*e+2*c*d)*(a*e^2-b*d*e+ 
c*d^2)^3*(e*x+d)^(2+m)/e^9/(2+m)+2*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2 
*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(3+m)/e^9/(3+m)-4*(-b*e+2*c*d)*(a*e^2-b*d 
*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^(4+m)/e^9/(4+m)+( 
70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+ 
6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^(5+m)/e^9/(5+m)-4*c*(-b 
*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^(6+m)/e^9/(6+m)+2 
*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(7+m)/e^9/(7+m)-4*c 
^3*(-b*e+2*c*d)*(e*x+d)^(8+m)/e^9/(8+m)+c^4*(e*x+d)^(9+m)/e^9/(9+m)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1197\) vs. \(2(485)=970\).

Time = 3.42 (sec) , antiderivative size = 1197, normalized size of antiderivative = 2.47 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x)^m*(a + b*x + c*x^2)^4,x]
 

Output:

((d + e*x)^(1 + m)*((a + x*(b + c*x))^4 - (4*(d + e*x)*(a + x*(b + c*x))^3 
*(b*e*(15 + m) + 2*c*(-7*d + e*(8 + m)*x)))/(e^2*(8 + m)*(9 + m)) - (12*(d 
 + e*x)*(-2*((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(3 + m)*(840*c^4*d^4 
 - b^4*e^4*(8 + 14*m + 7*m^2 + m^3) - 4*b^2*c*e^3*(2 + 3*m + m^2)*(5*b*d - 
 a*e*(13 + 2*m)) - 80*c^3*d^2*e*(21*b*d + a*e*(-19 + 3*m + m^2)) + 4*c^2*e 
^2*(20*a*b*d*e*(-19 + 3*m + m^2) + 5*b^2*d^2*(44 + 3*m + m^2) - 2*a^2*e^2* 
(-69 + 58*m + 24*m^2 + 2*m^3))) + (2 + m)*(-1680*c^6*d^6 + b^6*e^6*(12 + 1 
9*m + 8*m^2 + m^3) - b^4*c*e^5*(3 + 4*m + m^2)*(b*d*(-8 + 3*m) + a*e*(56 + 
 9*m)) + 80*c^5*d^4*e*(63*b*d + a*e*(-66 - m + 2*m^2)) + b^2*c^2*e^4*(1 + 
m)*(3*b^2*d^2*(12 - 13*m + m^2) + 12*a^2*e^2*(69 + 24*m + 2*m^2) + 8*a*b*d 
*e*(-39 + 11*m + 3*m^2)) - 4*c^4*d^2*e^2*(40*a*b*d*e*(-66 - m + 2*m^2) + 5 
*b^2*d^2*(249 - m + 2*m^2) - 12*a^2*e^2*(-123 - 11*m + 8*m^2 + m^3)) - 8*c 
^3*e^3*(-5*b^3*d^3*(39 - m + 2*m^2) + 3*a*b^2*d^2*e*(207 - 6*m - 2*m^2 + m 
^3) + 6*a^2*b*d*e^2*(-123 - 11*m + 8*m^2 + m^3) + 2*a^3*e^3*(192 + 104*m + 
 18*m^2 + m^3)))*(d + e*x) + e^2*(2 + m)*(3 + m)*(c*e*(5 + m)*(c*e*(b*d - 
2*a*e)*(7 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*(1 + m) - b^2*d*e*(15 + m 
)) - (b*d*(5*c*d - 2*b*e) + a*e*(2*c*d*(1 + m) - b*e*(2 + m)))*(28*c^2*d^2 
 - b^2*e^2*(1 + m) + 4*c*e*(-7*b*d + a*e*(8 + m)))) - (3*c*d - b*e)*(c*e*( 
2*c*d - b*e)*(7 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*(1 + m) - b^2*d*e*( 
15 + m)) - (10*c^2*d^2 - b^2*e^2*(4 + m) + c*e*(b*d*(-3 + m) + 2*a*e*(6...
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.

\(\displaystyle \int \left (a+b x+c x^2\right )^4 (d+e x)^m \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {(d+e x)^{m+4} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac {2 (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 (2 c d-b e) (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8}+\frac {4 c (2 c d-b e) (d+e x)^{m+5} \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8}+\frac {2 c^2 (d+e x)^{m+6} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {(d+e x)^m \left (a e^2-b d e+c d^2\right )^4}{e^8}+\frac {4 (b e-2 c d) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {4 c^3 (2 c d-b e) (d+e x)^{m+7}}{e^8}+\frac {c^4 (d+e x)^{m+8}}{e^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^{m+5} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9 (m+5)}+\frac {2 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (m+3)}-\frac {4 (2 c d-b e) (d+e x)^{m+4} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (m+4)}-\frac {4 c (2 c d-b e) (d+e x)^{m+6} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (m+6)}+\frac {2 c^2 (d+e x)^{m+7} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (m+7)}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^4}{e^9 (m+1)}-\frac {4 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^3}{e^9 (m+2)}-\frac {4 c^3 (2 c d-b e) (d+e x)^{m+8}}{e^9 (m+8)}+\frac {c^4 (d+e x)^{m+9}}{e^9 (m+9)}\)

Input:

Int[(d + e*x)^m*(a + b*x + c*x^2)^4,x]
 

Output:

((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^(1 + m))/(e^9*(1 + m)) - (4*(2*c*d - 
b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(2 + m))/(e^9*(2 + m)) + (2*(c*d^ 
2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e 
*x)^(3 + m))/(e^9*(3 + m)) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c 
^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(4 + m))/(e^9*(4 + m)) + 
 ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b* 
d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(5 + 
 m))/(e^9*(5 + m)) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d 
- 3*a*e))*(d + e*x)^(6 + m))/(e^9*(6 + m)) + (2*c^2*(14*c^2*d^2 + 3*b^2*e^ 
2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7 + m))/(e^9*(7 + m)) - (4*c^3*(2*c*d 
- b*e)*(d + e*x)^(8 + m))/(e^9*(8 + m)) + (c^4*(d + e*x)^(9 + m))/(e^9*(9 
+ m))
 

Defintions of rubi rules used

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, 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. \(7695\) vs. \(2(485)=970\).

Time = 1.37 (sec) , antiderivative size = 7696, normalized size of antiderivative = 15.87

method	result	size
gosper	\(\text {Expression too large to display}\)	\(7696\)
orering	\(\text {Expression too large to display}\)	\(7699\)
risch	\(\text {Expression too large to display}\)	\(9170\)
parallelrisch	\(\text {Expression too large to display}\)	\(13498\)

Input:

int((e*x+d)^m*(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6186 vs. \(2 (485) = 970\).

Time = 0.17 (sec) , antiderivative size = 6186, normalized size of antiderivative = 12.75 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116059 vs. \(2 (466) = 932\).

Time = 28.10 (sec) , antiderivative size = 116059, normalized size of antiderivative = 239.30 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**m*(c*x**2+b*x+a)**4,x)
 

Output:

Piecewise((d**m*(a**4*x + 2*a**3*b*x**2 + 4*a**3*c*x**3/3 + 2*a**2*b**2*x* 
*3 + 3*a**2*b*c*x**4 + 6*a**2*c**2*x**5/5 + a*b**3*x**4 + 12*a*b**2*c*x**5 
/5 + 2*a*b*c**2*x**6 + 4*a*c**3*x**7/7 + b**4*x**5/5 + 2*b**3*c*x**6/3 + 6 
*b**2*c**2*x**7/7 + b*c**3*x**8/2 + c**4*x**9/9), Eq(e, 0)), (-105*a**4*e* 
*8/(840*d**8*e**9 + 6720*d**7*e**10*x + 23520*d**6*e**11*x**2 + 47040*d**5 
*e**12*x**3 + 58800*d**4*e**13*x**4 + 47040*d**3*e**14*x**5 + 23520*d**2*e 
**15*x**6 + 6720*d*e**16*x**7 + 840*e**17*x**8) - 60*a**3*b*d*e**7/(840*d* 
*8*e**9 + 6720*d**7*e**10*x + 23520*d**6*e**11*x**2 + 47040*d**5*e**12*x** 
3 + 58800*d**4*e**13*x**4 + 47040*d**3*e**14*x**5 + 23520*d**2*e**15*x**6 
+ 6720*d*e**16*x**7 + 840*e**17*x**8) - 480*a**3*b*e**8*x/(840*d**8*e**9 + 
 6720*d**7*e**10*x + 23520*d**6*e**11*x**2 + 47040*d**5*e**12*x**3 + 58800 
*d**4*e**13*x**4 + 47040*d**3*e**14*x**5 + 23520*d**2*e**15*x**6 + 6720*d* 
e**16*x**7 + 840*e**17*x**8) - 20*a**3*c*d**2*e**6/(840*d**8*e**9 + 6720*d 
**7*e**10*x + 23520*d**6*e**11*x**2 + 47040*d**5*e**12*x**3 + 58800*d**4*e 
**13*x**4 + 47040*d**3*e**14*x**5 + 23520*d**2*e**15*x**6 + 6720*d*e**16*x 
**7 + 840*e**17*x**8) - 160*a**3*c*d*e**7*x/(840*d**8*e**9 + 6720*d**7*e** 
10*x + 23520*d**6*e**11*x**2 + 47040*d**5*e**12*x**3 + 58800*d**4*e**13*x* 
*4 + 47040*d**3*e**14*x**5 + 23520*d**2*e**15*x**6 + 6720*d*e**16*x**7 + 8 
40*e**17*x**8) - 560*a**3*c*e**8*x**2/(840*d**8*e**9 + 6720*d**7*e**10*x + 
 23520*d**6*e**11*x**2 + 47040*d**5*e**12*x**3 + 58800*d**4*e**13*x**4 ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2342 vs. \(2 (485) = 970\).

Time = 0.10 (sec) , antiderivative size = 2342, normalized size of antiderivative = 4.83 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)^4,x, algorithm="maxima")
 

Output:

4*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^3*b/((m^2 + 3*m + 2)*e^2 
) + (e*x + d)^(m + 1)*a^4/(e*(m + 1)) + 6*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 
+ m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a^2*b^2/((m^3 + 6*m^2 + 
11*m + 6)*e^3) + 4*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2* 
e*m*x + 2*d^3)*(e*x + d)^m*a^3*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 4*((m^3 
+ 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)* 
d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*a*b^3/((m^4 + 10*m^3 + 35*m 
^2 + 50*m + 24)*e^4) + 12*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 
 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + 
d)^m*a^2*b*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 
35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*( 
m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 
 24*d^5)*(e*x + d)^m*b^4/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)* 
e^5) + 12*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11 
*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d 
^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*a*b^2*c/((m^5 + 15*m^4 + 8 
5*m^3 + 225*m^2 + 274*m + 120)*e^5) + 6*((m^4 + 10*m^3 + 35*m^2 + 50*m + 2 
4)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m 
)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d 
)^m*a^2*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 4*(...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 13146 vs. \(2 (485) = 970\).

Time = 0.21 (sec) , antiderivative size = 13146, normalized size of antiderivative = 27.11 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)^4,x, algorithm="giac")
 

Output:

((e*x + d)^m*c^4*e^9*m^8*x^9 + (e*x + d)^m*c^4*d*e^8*m^8*x^8 + 4*(e*x + d) 
^m*b*c^3*e^9*m^8*x^8 + 36*(e*x + d)^m*c^4*e^9*m^7*x^9 + 4*(e*x + d)^m*b*c^ 
3*d*e^8*m^8*x^7 + 6*(e*x + d)^m*b^2*c^2*e^9*m^8*x^7 + 4*(e*x + d)^m*a*c^3* 
e^9*m^8*x^7 + 28*(e*x + d)^m*c^4*d*e^8*m^7*x^8 + 148*(e*x + d)^m*b*c^3*e^9 
*m^7*x^8 + 546*(e*x + d)^m*c^4*e^9*m^6*x^9 + 6*(e*x + d)^m*b^2*c^2*d*e^8*m 
^8*x^6 + 4*(e*x + d)^m*a*c^3*d*e^8*m^8*x^6 + 4*(e*x + d)^m*b^3*c*e^9*m^8*x 
^6 + 12*(e*x + d)^m*a*b*c^2*e^9*m^8*x^6 - 8*(e*x + d)^m*c^4*d^2*e^7*m^7*x^ 
7 + 120*(e*x + d)^m*b*c^3*d*e^8*m^7*x^7 + 228*(e*x + d)^m*b^2*c^2*e^9*m^7* 
x^7 + 152*(e*x + d)^m*a*c^3*e^9*m^7*x^7 + 322*(e*x + d)^m*c^4*d*e^8*m^6*x^ 
8 + 2296*(e*x + d)^m*b*c^3*e^9*m^6*x^8 + 4536*(e*x + d)^m*c^4*e^9*m^5*x^9 
+ 4*(e*x + d)^m*b^3*c*d*e^8*m^8*x^5 + 12*(e*x + d)^m*a*b*c^2*d*e^8*m^8*x^5 
 + (e*x + d)^m*b^4*e^9*m^8*x^5 + 12*(e*x + d)^m*a*b^2*c*e^9*m^8*x^5 + 6*(e 
*x + d)^m*a^2*c^2*e^9*m^8*x^5 - 28*(e*x + d)^m*b*c^3*d^2*e^7*m^7*x^6 + 192 
*(e*x + d)^m*b^2*c^2*d*e^8*m^7*x^6 + 128*(e*x + d)^m*a*c^3*d*e^8*m^7*x^6 + 
 156*(e*x + d)^m*b^3*c*e^9*m^7*x^6 + 468*(e*x + d)^m*a*b*c^2*e^9*m^7*x^6 - 
 168*(e*x + d)^m*c^4*d^2*e^7*m^6*x^7 + 1456*(e*x + d)^m*b*c^3*d*e^8*m^6*x^ 
7 + 3624*(e*x + d)^m*b^2*c^2*e^9*m^6*x^7 + 2416*(e*x + d)^m*a*c^3*e^9*m^6* 
x^7 + 1960*(e*x + d)^m*c^4*d*e^8*m^5*x^8 + 19432*(e*x + d)^m*b*c^3*e^9*m^5 
*x^8 + 22449*(e*x + d)^m*c^4*e^9*m^4*x^9 + (e*x + d)^m*b^4*d*e^8*m^8*x^4 + 
 12*(e*x + d)^m*a*b^2*c*d*e^8*m^8*x^4 + 6*(e*x + d)^m*a^2*c^2*d*e^8*m^8...
 

Mupad [B] (verification not implemented)

Time = 11.36 (sec) , antiderivative size = 5907, normalized size of antiderivative = 12.18 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx=\text {Too large to display} \] Input:

int((d + e*x)^m*(a + b*x + c*x^2)^4,x)
 

Output:

((d + e*x)^m*(40320*c^4*d^9 + 362880*a^4*d*e^8 + 72576*b^4*d^5*e^4 - 36288 
0*a*b^3*d^4*e^5 - 725760*a^3*b*d^2*e^7 + 207360*a*c^3*d^7*e^2 + 483840*a^3 
*c*d^3*e^6 - 241920*b^3*c*d^6*e^3 + 509004*a^4*d*e^8*m^2 + 214676*a^4*d*e^ 
8*m^3 + 54649*a^4*d*e^8*m^4 + 8624*a^4*d*e^8*m^5 + 826*a^4*d*e^8*m^6 + 44* 
a^4*d*e^8*m^7 + a^4*d*e^8*m^8 + 39600*b^4*d^5*e^4*m + 725760*a^2*b^2*d^3*e 
^6 + 435456*a^2*c^2*d^5*e^4 + 311040*b^2*c^2*d^7*e^2 + 8040*b^4*d^5*e^4*m^ 
2 + 720*b^4*d^5*e^4*m^3 + 24*b^4*d^5*e^4*m^4 - 181440*b*c^3*d^8*e + 663696 
*a^4*d*e^8*m - 20160*b*c^3*d^8*e*m + 294888*a^2*b^2*d^3*e^6*m^2 + 63180*a^ 
2*b^2*d^3*e^6*m^3 + 7500*a^2*b^2*d^3*e^6*m^4 + 468*a^2*b^2*d^3*e^6*m^5 + 1 
2*a^2*b^2*d^3*e^6*m^6 + 48240*a^2*c^2*d^5*e^4*m^2 + 4320*a^2*c^2*d^5*e^4*m 
^3 + 144*a^2*c^2*d^5*e^4*m^4 + 4320*b^2*c^2*d^7*e^2*m^2 - 725760*a*b*c^2*d 
^6*e^3 + 870912*a*b^2*c*d^5*e^4 - 1088640*a^2*b*c*d^4*e^5 - 270576*a*b^3*d 
^4*e^5*m - 964512*a^3*b*d^2*e^7*m + 48960*a*c^3*d^7*e^2*m + 481728*a^3*c*d 
^3*e^6*m - 91680*b^3*c*d^6*e^3*m + 722592*a^2*b^2*d^3*e^6*m - 79800*a*b^3* 
d^4*e^5*m^2 - 535752*a^3*b*d^2*e^7*m^2 - 11640*a*b^3*d^4*e^5*m^3 - 161476* 
a^3*b*d^2*e^7*m^3 - 840*a*b^3*d^4*e^5*m^4 - 28560*a^3*b*d^2*e^7*m^4 - 24*a 
*b^3*d^4*e^5*m^5 - 2968*a^3*b*d^2*e^7*m^5 - 168*a^3*b*d^2*e^7*m^6 - 4*a^3* 
b*d^2*e^7*m^7 + 237600*a^2*c^2*d^5*e^4*m + 2880*a*c^3*d^7*e^2*m^2 + 196592 
*a^3*c*d^3*e^6*m^2 + 42120*a^3*c*d^3*e^6*m^3 + 5000*a^3*c*d^3*e^6*m^4 + 31 
2*a^3*c*d^3*e^6*m^5 + 8*a^3*c*d^3*e^6*m^6 + 73440*b^2*c^2*d^7*e^2*m - 1...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 9169, normalized size of antiderivative = 18.91 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx =\text {Too large to display} \] Input:

int((e*x+d)^m*(c*x^2+b*x+a)^4,x)
 

Output:

((d + e*x)**m*(a**4*d*e**8*m**8 + 44*a**4*d*e**8*m**7 + 826*a**4*d*e**8*m* 
*6 + 8624*a**4*d*e**8*m**5 + 54649*a**4*d*e**8*m**4 + 214676*a**4*d*e**8*m 
**3 + 509004*a**4*d*e**8*m**2 + 663696*a**4*d*e**8*m + 362880*a**4*d*e**8 
+ a**4*e**9*m**8*x + 44*a**4*e**9*m**7*x + 826*a**4*e**9*m**6*x + 8624*a** 
4*e**9*m**5*x + 54649*a**4*e**9*m**4*x + 214676*a**4*e**9*m**3*x + 509004* 
a**4*e**9*m**2*x + 663696*a**4*e**9*m*x + 362880*a**4*e**9*x - 4*a**3*b*d* 
*2*e**7*m**7 - 168*a**3*b*d**2*e**7*m**6 - 2968*a**3*b*d**2*e**7*m**5 - 28 
560*a**3*b*d**2*e**7*m**4 - 161476*a**3*b*d**2*e**7*m**3 - 535752*a**3*b*d 
**2*e**7*m**2 - 964512*a**3*b*d**2*e**7*m - 725760*a**3*b*d**2*e**7 + 4*a* 
*3*b*d*e**8*m**8*x + 168*a**3*b*d*e**8*m**7*x + 2968*a**3*b*d*e**8*m**6*x 
+ 28560*a**3*b*d*e**8*m**5*x + 161476*a**3*b*d*e**8*m**4*x + 535752*a**3*b 
*d*e**8*m**3*x + 964512*a**3*b*d*e**8*m**2*x + 725760*a**3*b*d*e**8*m*x + 
4*a**3*b*e**9*m**8*x**2 + 172*a**3*b*e**9*m**7*x**2 + 3136*a**3*b*e**9*m** 
6*x**2 + 31528*a**3*b*e**9*m**5*x**2 + 190036*a**3*b*e**9*m**4*x**2 + 6972 
28*a**3*b*e**9*m**3*x**2 + 1500264*a**3*b*e**9*m**2*x**2 + 1690272*a**3*b* 
e**9*m*x**2 + 725760*a**3*b*e**9*x**2 + 8*a**3*c*d**3*e**6*m**6 + 312*a**3 
*c*d**3*e**6*m**5 + 5000*a**3*c*d**3*e**6*m**4 + 42120*a**3*c*d**3*e**6*m* 
*3 + 196592*a**3*c*d**3*e**6*m**2 + 481728*a**3*c*d**3*e**6*m + 483840*a** 
3*c*d**3*e**6 - 8*a**3*c*d**2*e**7*m**7*x - 312*a**3*c*d**2*e**7*m**6*x - 
5000*a**3*c*d**2*e**7*m**5*x - 42120*a**3*c*d**2*e**7*m**4*x - 196592*a...