\(\int (d+e x)^m (a+b x+c x^2)^3 \, dx\) [762]

Optimal result

Integrand size = 20, antiderivative size = 305 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \] Output:

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

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.56 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left ((a+x (b+c x))^3+\frac {3 (d+e x) \left (2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) (3+m) \left (60 c^2 d^2+b^2 e^2 \left (2+3 m+m^2\right )-4 c e \left (15 b d+a e \left (-13+3 m+m^2\right )\right )\right )-2 (2+m) \left (120 c^4 d^4+b^4 e^4 \left (3+4 m+m^2\right )-2 b^2 c e^3 (1+m) (b d (-3+m)+3 a e (5+m))-8 c^3 d^2 e \left (30 b d+a e \left (-33-2 m+m^2\right )\right )+2 c^2 e^2 \left (4 a b d e \left (-33-2 m+m^2\right )+b^2 d^2 \left (57-2 m+m^2\right )+4 a^2 e^2 \left (24+10 m+m^2\right )\right )\right ) (d+e x)-c e^4 (2+m) (3+m) (4+m) (5+m) (a+x (b+c x))^2 (b e (11+m)+2 c (-5 d+e (6+m) x))-2 e^2 (2+m) (3+m) (a+x (b+c x)) \left (-b^3 e^3 (1+m)+20 c^3 d^2 (-3 d+e (4+m) x)-b c e^2 \left (-2 a e (37+7 m)+b d \left (72+13 m+m^2\right )+b e \left (4+5 m+m^2\right ) x\right )+2 c^2 e \left (5 b d (d (13+m)-2 e (4+m) x)+2 a e \left (d \left (-13+3 m+m^2\right )+e \left (24+10 m+m^2\right ) x\right )\right )\right )\right )}{c e^6 (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}\right )}{e (1+m)} \] Input:

Integrate[(d + e*x)^m*(a + b*x + c*x^2)^3,x]
 

Output:

((d + e*x)^(1 + m)*((a + x*(b + c*x))^3 + (3*(d + e*x)*(2*(2*c*d - b*e)*(c 
*d^2 + e*(-(b*d) + a*e))*(3 + m)*(60*c^2*d^2 + b^2*e^2*(2 + 3*m + m^2) - 4 
*c*e*(15*b*d + a*e*(-13 + 3*m + m^2))) - 2*(2 + m)*(120*c^4*d^4 + b^4*e^4* 
(3 + 4*m + m^2) - 2*b^2*c*e^3*(1 + m)*(b*d*(-3 + m) + 3*a*e*(5 + m)) - 8*c 
^3*d^2*e*(30*b*d + a*e*(-33 - 2*m + m^2)) + 2*c^2*e^2*(4*a*b*d*e*(-33 - 2* 
m + m^2) + b^2*d^2*(57 - 2*m + m^2) + 4*a^2*e^2*(24 + 10*m + m^2)))*(d + e 
*x) - c*e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + x*(b + c*x))^2*(b*e*(11 + 
 m) + 2*c*(-5*d + e*(6 + m)*x)) - 2*e^2*(2 + m)*(3 + m)*(a + x*(b + c*x))* 
(-(b^3*e^3*(1 + m)) + 20*c^3*d^2*(-3*d + e*(4 + m)*x) - b*c*e^2*(-2*a*e*(3 
7 + 7*m) + b*d*(72 + 13*m + m^2) + b*e*(4 + 5*m + m^2)*x) + 2*c^2*e*(5*b*d 
*(d*(13 + m) - 2*e*(4 + m)*x) + 2*a*e*(d*(-13 + 3*m + m^2) + e*(24 + 10*m 
+ m^2)*x)))))/(c*e^6*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m))))/(e 
*(1 + m))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 305, 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.

\(\displaystyle \int \left (a+b x+c x^2\right )^3 (d+e x)^m \, dx\)

\(\Big \downarrow \) 1140

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d + e*x)^m*(a + b*x + c*x^2)^3,x]
 

Output:

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*(2*c*d - 
b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(2 + m))/(e^7*(2 + m)) + (3*(c*d^ 
2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(3 
+ m))/(e^7*(3 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d 
- 3*a*e))*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c 
*e*(5*b*d - a*e))*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b*e)* 
(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + 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. \(2337\) vs. \(2(305)=610\).

Time = 1.06 (sec) , antiderivative size = 2338, normalized size of antiderivative = 7.67

method	result	size
norman	\(\text {Expression too large to display}\)	\(2338\)
gosper	\(\text {Expression too large to display}\)	\(2927\)
orering	\(\text {Expression too large to display}\)	\(2930\)
risch	\(\text {Expression too large to display}\)	\(3658\)
parallelrisch	\(\text {Expression too large to display}\)	\(5366\)

Input:

int((e*x+d)^m*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

c^3/(7+m)*x^7*exp(m*ln(e*x+d))+d*(a^3*e^6*m^6+27*a^3*e^6*m^5-3*a^2*b*d*e^5 
*m^5+295*a^3*e^6*m^4-75*a^2*b*d*e^5*m^4+6*a^2*c*d^2*e^4*m^4+6*a*b^2*d^2*e^ 
4*m^4+1665*a^3*e^6*m^3-735*a^2*b*d*e^5*m^3+132*a^2*c*d^2*e^4*m^3+132*a*b^2 
*d^2*e^4*m^3-36*a*b*c*d^3*e^3*m^3-6*b^3*d^3*e^3*m^3+5104*a^3*e^6*m^2-3525* 
a^2*b*d*e^5*m^2+1074*a^2*c*d^2*e^4*m^2+1074*a*b^2*d^2*e^4*m^2-648*a*b*c*d^ 
3*e^3*m^2+72*a*c^2*d^4*e^2*m^2-108*b^3*d^3*e^3*m^2+72*b^2*c*d^4*e^2*m^2+80 
28*a^3*e^6*m-8262*a^2*b*d*e^5*m+3828*a^2*c*d^2*e^4*m+3828*a*b^2*d^2*e^4*m- 
3852*a*b*c*d^3*e^3*m+936*a*c^2*d^4*e^2*m-642*b^3*d^3*e^3*m+936*b^2*c*d^4*e 
^2*m-360*b*c^2*d^5*e*m+5040*a^3*e^6-7560*a^2*b*d*e^5+5040*a^2*c*d^2*e^4+50 
40*a*b^2*d^2*e^4-7560*a*b*c*d^3*e^3+3024*a*c^2*d^4*e^2-1260*b^3*d^3*e^3+30 
24*b^2*c*d^4*e^2-2520*b*c^2*d^5*e+720*c^3*d^6)/e^7/(m^7+28*m^6+322*m^5+196 
0*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(e*x+d))+(6*a*b*c*e^3*m^3+3 
*a*c^2*d*e^2*m^3+b^3*e^3*m^3+3*b^2*c*d*e^2*m^3+108*a*b*c*e^3*m^2+39*a*c^2* 
d*e^2*m^2+18*b^3*e^3*m^2+39*b^2*c*d*e^2*m^2-15*b*c^2*d^2*e*m^2+642*a*b*c*e 
^3*m+126*a*c^2*d*e^2*m+107*b^3*e^3*m+126*b^2*c*d*e^2*m-105*b*c^2*d^2*e*m+3 
0*c^3*d^3*m+1260*a*b*c*e^3+210*b^3*e^3)/e^3/(m^4+22*m^3+179*m^2+638*m+840) 
*x^4*exp(m*ln(e*x+d))+(3*a^2*c*e^4*m^4+3*a*b^2*e^4*m^4+6*a*b*c*d*e^3*m^4+b 
^3*d*e^3*m^4+66*a^2*c*e^4*m^3+66*a*b^2*e^4*m^3+108*a*b*c*d*e^3*m^3-12*a*c^ 
2*d^2*e^2*m^3+18*b^3*d*e^3*m^3-12*b^2*c*d^2*e^2*m^3+537*a^2*c*e^4*m^2+537* 
a*b^2*e^4*m^2+642*a*b*c*d*e^3*m^2-156*a*c^2*d^2*e^2*m^2+107*b^3*d*e^3*m...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2550 vs. \(2 (305) = 610\).

Time = 0.13 (sec) , antiderivative size = 2550, normalized size of antiderivative = 8.36 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

(a^3*d*e^6*m^6 + 720*c^3*d^7 - 2520*b*c^2*d^6*e - 7560*a^2*b*d^2*e^5 + 504 
0*a^3*d*e^6 + 3024*(b^2*c + a*c^2)*d^5*e^2 - 1260*(b^3 + 6*a*b*c)*d^4*e^3 
+ 5040*(a*b^2 + a^2*c)*d^3*e^4 + (c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e 
^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7 
)*x^7 + (2520*b*c^2*e^7 + (c^3*d*e^6 + 3*b*c^2*e^7)*m^6 + 3*(5*c^3*d*e^6 + 
 22*b*c^2*e^7)*m^5 + 5*(17*c^3*d*e^6 + 114*b*c^2*e^7)*m^4 + 15*(15*c^3*d*e 
^6 + 164*b*c^2*e^7)*m^3 + (274*c^3*d*e^6 + 5547*b*c^2*e^7)*m^2 + 6*(20*c^3 
*d*e^6 + 1019*b*c^2*e^7)*m)*x^6 - 3*(a^2*b*d^2*e^5 - 9*a^3*d*e^6)*m^5 + 3* 
(1008*(b^2*c + a*c^2)*e^7 + (b*c^2*d*e^6 + (b^2*c + a*c^2)*e^7)*m^6 - (2*c 
^3*d^2*e^5 - 17*b*c^2*d*e^6 - 23*(b^2*c + a*c^2)*e^7)*m^5 - (20*c^3*d^2*e^ 
5 - 105*b*c^2*d*e^6 - 207*(b^2*c + a*c^2)*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 5 
9*b*c^2*d*e^6 - 185*(b^2*c + a*c^2)*e^7)*m^3 - 2*(50*c^3*d^2*e^5 - 187*b*c 
^2*d*e^6 - 1072*(b^2*c + a*c^2)*e^7)*m^2 - 12*(4*c^3*d^2*e^5 - 14*b*c^2*d* 
e^6 - 201*(b^2*c + a*c^2)*e^7)*m)*x^5 - (75*a^2*b*d^2*e^5 - 295*a^3*d*e^6 
- 6*(a*b^2 + a^2*c)*d^3*e^4)*m^4 + (1260*(b^3 + 6*a*b*c)*e^7 + (3*(b^2*c + 
 a*c^2)*d*e^6 + (b^3 + 6*a*b*c)*e^7)*m^6 - 3*(5*b*c^2*d^2*e^5 - 19*(b^2*c 
+ a*c^2)*d*e^6 - 8*(b^3 + 6*a*b*c)*e^7)*m^5 + (30*c^3*d^3*e^4 - 195*b*c^2* 
d^2*e^5 + 393*(b^2*c + a*c^2)*d*e^6 + 226*(b^3 + 6*a*b*c)*e^7)*m^4 + 3*(60 
*c^3*d^3*e^4 - 265*b*c^2*d^2*e^5 + 401*(b^2*c + a*c^2)*d*e^6 + 352*(b^3 + 
6*a*b*c)*e^7)*m^3 + 5*(66*c^3*d^3*e^4 - 249*b*c^2*d^2*e^5 + 324*(b^2*c ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40049 vs. \(2 (284) = 568\).

Time = 7.74 (sec) , antiderivative size = 40049, normalized size of antiderivative = 131.31 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**m*(c*x**2+b*x+a)**3,x)
 

Output:

Piecewise((d**m*(a**3*x + 3*a**2*b*x**2/2 + a**2*c*x**3 + a*b**2*x**3 + 3* 
a*b*c*x**4/2 + 3*a*c**2*x**5/5 + b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x* 
*6/2 + c**3*x**7/7), Eq(e, 0)), (-10*a**3*e**6/(60*d**6*e**7 + 360*d**5*e* 
*8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 3 
60*d*e**12*x**5 + 60*e**13*x**6) - 6*a**2*b*d*e**5/(60*d**6*e**7 + 360*d** 
5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 
 + 360*d*e**12*x**5 + 60*e**13*x**6) - 36*a**2*b*e**6*x/(60*d**6*e**7 + 36 
0*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11 
*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 3*a**2*c*d**2*e**4/(60*d**6*e* 
*7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d** 
2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 18*a**2*c*d*e**5*x/(60* 
d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 
900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 45*a**2*c*e**6*x 
**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10 
*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 3*a*b**2 
*d**2*e**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d** 
3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 1 
8*a*b**2*d*e**5*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1 
200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x* 
*6) - 45*a*b**2*e**6*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (305) = 610\).

Time = 0.08 (sec) , antiderivative size = 1132, normalized size of antiderivative = 3.71 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^2*b/((m^2 + 3*m + 2)*e^2 
) + (e*x + d)^(m + 1)*a^3/(e*(m + 1)) + 3*((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*b^2/((m^3 + 6*m^2 + 11 
*m + 6)*e^3) + 3*((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*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + ((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*b^3/((m^4 + 10*m^3 + 35*m^2 + 5 
0*m + 24)*e^4) + 6*((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*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 3*((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^2*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) 
+ 3*((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*c^2/((m^5 + 15*m^4 + 85*m^3 + 
225*m^2 + 274*m + 120)*e^5) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m 
+ 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^ 
4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 
 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*b*c^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5388 vs. \(2 (305) = 610\).

Time = 0.63 (sec) , antiderivative size = 5388, normalized size of antiderivative = 17.67 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

((e*x + d)^m*c^3*e^7*m^6*x^7 + (e*x + d)^m*c^3*d*e^6*m^6*x^6 + 3*(e*x + d) 
^m*b*c^2*e^7*m^6*x^6 + 21*(e*x + d)^m*c^3*e^7*m^5*x^7 + 3*(e*x + d)^m*b*c^ 
2*d*e^6*m^6*x^5 + 3*(e*x + d)^m*b^2*c*e^7*m^6*x^5 + 3*(e*x + d)^m*a*c^2*e^ 
7*m^6*x^5 + 15*(e*x + d)^m*c^3*d*e^6*m^5*x^6 + 66*(e*x + d)^m*b*c^2*e^7*m^ 
5*x^6 + 175*(e*x + d)^m*c^3*e^7*m^4*x^7 + 3*(e*x + d)^m*b^2*c*d*e^6*m^6*x^ 
4 + 3*(e*x + d)^m*a*c^2*d*e^6*m^6*x^4 + (e*x + d)^m*b^3*e^7*m^6*x^4 + 6*(e 
*x + d)^m*a*b*c*e^7*m^6*x^4 - 6*(e*x + d)^m*c^3*d^2*e^5*m^5*x^5 + 51*(e*x 
+ d)^m*b*c^2*d*e^6*m^5*x^5 + 69*(e*x + d)^m*b^2*c*e^7*m^5*x^5 + 69*(e*x + 
d)^m*a*c^2*e^7*m^5*x^5 + 85*(e*x + d)^m*c^3*d*e^6*m^4*x^6 + 570*(e*x + d)^ 
m*b*c^2*e^7*m^4*x^6 + 735*(e*x + d)^m*c^3*e^7*m^3*x^7 + (e*x + d)^m*b^3*d* 
e^6*m^6*x^3 + 6*(e*x + d)^m*a*b*c*d*e^6*m^6*x^3 + 3*(e*x + d)^m*a*b^2*e^7* 
m^6*x^3 + 3*(e*x + d)^m*a^2*c*e^7*m^6*x^3 - 15*(e*x + d)^m*b*c^2*d^2*e^5*m 
^5*x^4 + 57*(e*x + d)^m*b^2*c*d*e^6*m^5*x^4 + 57*(e*x + d)^m*a*c^2*d*e^6*m 
^5*x^4 + 24*(e*x + d)^m*b^3*e^7*m^5*x^4 + 144*(e*x + d)^m*a*b*c*e^7*m^5*x^ 
4 - 60*(e*x + d)^m*c^3*d^2*e^5*m^4*x^5 + 315*(e*x + d)^m*b*c^2*d*e^6*m^4*x 
^5 + 621*(e*x + d)^m*b^2*c*e^7*m^4*x^5 + 621*(e*x + d)^m*a*c^2*e^7*m^4*x^5 
 + 225*(e*x + d)^m*c^3*d*e^6*m^3*x^6 + 2460*(e*x + d)^m*b*c^2*e^7*m^3*x^6 
+ 1624*(e*x + d)^m*c^3*e^7*m^2*x^7 + 3*(e*x + d)^m*a*b^2*d*e^6*m^6*x^2 + 3 
*(e*x + d)^m*a^2*c*d*e^6*m^6*x^2 + 3*(e*x + d)^m*a^2*b*e^7*m^6*x^2 - 12*(e 
*x + d)^m*b^2*c*d^2*e^5*m^5*x^3 - 12*(e*x + d)^m*a*c^2*d^2*e^5*m^5*x^3 ...
 

Mupad [B] (verification not implemented)

Time = 7.31 (sec) , antiderivative size = 2542, normalized size of antiderivative = 8.33 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

int((d + e*x)^m*(a + b*x + c*x^2)^3,x)
 

Output:

((d + e*x)^m*(720*c^3*d^7 + 5040*a^3*d*e^6 - 1260*b^3*d^4*e^3 + 5040*a*b^2 
*d^3*e^4 - 7560*a^2*b*d^2*e^5 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 
3024*b^2*c*d^5*e^2 + 5104*a^3*d*e^6*m^2 + 1665*a^3*d*e^6*m^3 + 295*a^3*d*e 
^6*m^4 + 27*a^3*d*e^6*m^5 + a^3*d*e^6*m^6 - 642*b^3*d^4*e^3*m - 108*b^3*d^ 
4*e^3*m^2 - 6*b^3*d^4*e^3*m^3 - 2520*b*c^2*d^6*e + 8028*a^3*d*e^6*m - 7560 
*a*b*c*d^4*e^3 - 360*b*c^2*d^6*e*m + 3828*a*b^2*d^3*e^4*m - 8262*a^2*b*d^2 
*e^5*m + 936*a*c^2*d^5*e^2*m + 3828*a^2*c*d^3*e^4*m + 936*b^2*c*d^5*e^2*m 
+ 1074*a*b^2*d^3*e^4*m^2 - 3525*a^2*b*d^2*e^5*m^2 + 132*a*b^2*d^3*e^4*m^3 
- 735*a^2*b*d^2*e^5*m^3 + 6*a*b^2*d^3*e^4*m^4 - 75*a^2*b*d^2*e^5*m^4 - 3*a 
^2*b*d^2*e^5*m^5 + 72*a*c^2*d^5*e^2*m^2 + 1074*a^2*c*d^3*e^4*m^2 + 132*a^2 
*c*d^3*e^4*m^3 + 6*a^2*c*d^3*e^4*m^4 + 72*b^2*c*d^5*e^2*m^2 - 3852*a*b*c*d 
^4*e^3*m - 648*a*b*c*d^4*e^3*m^2 - 36*a*b*c*d^4*e^3*m^3))/(e^7*(13068*m + 
13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x*(d 
+ e*x)^m*(5040*a^3*e^7 + 8028*a^3*e^7*m + 5104*a^3*e^7*m^2 + 1665*a^3*e^7* 
m^3 + 295*a^3*e^7*m^4 + 27*a^3*e^7*m^5 + a^3*e^7*m^6 + 1260*b^3*d^3*e^4*m 
+ 642*b^3*d^3*e^4*m^2 + 108*b^3*d^3*e^4*m^3 + 6*b^3*d^3*e^4*m^4 - 720*c^3* 
d^6*e*m + 7560*a^2*b*d*e^6*m - 5040*a*b^2*d^2*e^5*m + 8262*a^2*b*d*e^6*m^2 
 + 3525*a^2*b*d*e^6*m^3 + 735*a^2*b*d*e^6*m^4 + 75*a^2*b*d*e^6*m^5 + 3*a^2 
*b*d*e^6*m^6 - 3024*a*c^2*d^4*e^3*m - 5040*a^2*c*d^2*e^5*m + 2520*b*c^2*d^ 
5*e^2*m - 3024*b^2*c*d^4*e^3*m - 3828*a*b^2*d^2*e^5*m^2 - 1074*a*b^2*d^...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3657, normalized size of antiderivative = 11.99 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((e*x+d)^m*(c*x^2+b*x+a)^3,x)
 

Output:

((d + e*x)**m*(a**3*d*e**6*m**6 + 27*a**3*d*e**6*m**5 + 295*a**3*d*e**6*m* 
*4 + 1665*a**3*d*e**6*m**3 + 5104*a**3*d*e**6*m**2 + 8028*a**3*d*e**6*m + 
5040*a**3*d*e**6 + a**3*e**7*m**6*x + 27*a**3*e**7*m**5*x + 295*a**3*e**7* 
m**4*x + 1665*a**3*e**7*m**3*x + 5104*a**3*e**7*m**2*x + 8028*a**3*e**7*m* 
x + 5040*a**3*e**7*x - 3*a**2*b*d**2*e**5*m**5 - 75*a**2*b*d**2*e**5*m**4 
- 735*a**2*b*d**2*e**5*m**3 - 3525*a**2*b*d**2*e**5*m**2 - 8262*a**2*b*d** 
2*e**5*m - 7560*a**2*b*d**2*e**5 + 3*a**2*b*d*e**6*m**6*x + 75*a**2*b*d*e* 
*6*m**5*x + 735*a**2*b*d*e**6*m**4*x + 3525*a**2*b*d*e**6*m**3*x + 8262*a* 
*2*b*d*e**6*m**2*x + 7560*a**2*b*d*e**6*m*x + 3*a**2*b*e**7*m**6*x**2 + 78 
*a**2*b*e**7*m**5*x**2 + 810*a**2*b*e**7*m**4*x**2 + 4260*a**2*b*e**7*m**3 
*x**2 + 11787*a**2*b*e**7*m**2*x**2 + 15822*a**2*b*e**7*m*x**2 + 7560*a**2 
*b*e**7*x**2 + 6*a**2*c*d**3*e**4*m**4 + 132*a**2*c*d**3*e**4*m**3 + 1074* 
a**2*c*d**3*e**4*m**2 + 3828*a**2*c*d**3*e**4*m + 5040*a**2*c*d**3*e**4 - 
6*a**2*c*d**2*e**5*m**5*x - 132*a**2*c*d**2*e**5*m**4*x - 1074*a**2*c*d**2 
*e**5*m**3*x - 3828*a**2*c*d**2*e**5*m**2*x - 5040*a**2*c*d**2*e**5*m*x + 
3*a**2*c*d*e**6*m**6*x**2 + 69*a**2*c*d*e**6*m**5*x**2 + 603*a**2*c*d*e**6 
*m**4*x**2 + 2451*a**2*c*d*e**6*m**3*x**2 + 4434*a**2*c*d*e**6*m**2*x**2 + 
 2520*a**2*c*d*e**6*m*x**2 + 3*a**2*c*e**7*m**6*x**3 + 75*a**2*c*e**7*m**5 
*x**3 + 741*a**2*c*e**7*m**4*x**3 + 3657*a**2*c*e**7*m**3*x**3 + 9336*a**2 
*c*e**7*m**2*x**3 + 11388*a**2*c*e**7*m*x**3 + 5040*a**2*c*e**7*x**3 + ...