\(\int (b+2 c x) (d+e x)^m (a+b x+c x^2)^3 \, dx\) [676]

Optimal result

Integrand size = 26, antiderivative size = 449 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{2+m}}{e^8 (2+m)}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}-\frac {5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5+m}}{e^8 (5+m)}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac {7 c^3 (2 c d-b e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {2 c^4 (d+e x)^{8+m}}{e^8 (8+m)} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1112\) vs. \(2(449)=898\).

Time = 3.47 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.48 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)^(1 + m)*(-(c^2*(14*c*d - b*e*(14 + m) - 2*c*e*(7 + m)*x)*(a + x 
*(b + c*x))^3) + (3*(-2*((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(2 + m)* 
(840*c^4*d^4 - b^4*e^4*m*(3 + 4*m + m^2) + 4*b^2*c*e^3*m*(1 + m)*(-5*b*d + 
 a*e*(11 + 2*m)) - 80*c^3*d^2*e*(21*b*d + a*e*(-21 + m + m^2)) + 4*c^2*e^2 
*(20*a*b*d*e*(-21 + m + m^2) + 5*b^2*d^2*(42 + m + m^2) - 2*a^2*e^2*(-105 
+ 16*m + 18*m^2 + 2*m^3))) + (1 + m)*(-1680*c^6*d^6 + b^6*e^6*m*(6 + 5*m + 
 m^2) - b^4*c*e^5*m*(2 + m)*(b*d*(-11 + 3*m) + a*e*(47 + 9*m)) + 80*c^5*d^ 
4*e*(63*b*d + a*e*(-63 - 5*m + 2*m^2)) + b^2*c^2*e^4*m*(3*b^2*d^2*(26 - 15 
*m + m^2) + 12*a^2*e^2*(47 + 20*m + 2*m^2) + 8*a*b*d*e*(-47 + 5*m + 3*m^2) 
) - 4*c^4*d^2*e^2*(40*a*b*d*e*(-63 - 5*m + 2*m^2) + 5*b^2*d^2*(252 - 5*m + 
 2*m^2) - 12*a^2*e^2*(-105 - 24*m + 5*m^2 + m^3)) - 8*c^3*e^3*(-5*b^3*d^3* 
(42 - 5*m + 2*m^2) + 3*a*b^2*d^2*e*(210 + m - 5*m^2 + m^3) + 6*a^2*b*d*e^2 
*(-105 - 24*m + 5*m^2 + m^3) + 2*a^3*e^3*(105 + 71*m + 15*m^2 + m^3)))*(d 
+ e*x) + e^2*(1 + m)*(2 + m)*(c*e*(4 + m)*(c*e*(b*d - 2*a*e)*(6 + m)*(14*b 
*(c*d^2 + a*e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (b*d*(5*c*d - 2*b*e) 
+ a*e*(2*c*d*m - b*e*(1 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d + a 
*e*(7 + m)))) - (3*c*d - b*e)*(c*e*(2*c*d - b*e)*(6 + m)*(14*b*(c*d^2 + a* 
e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (10*c^2*d^2 - b^2*e^2*(3 + m) + c 
*e*(b*d*(-4 + m) + 2*a*e*(5 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d 
 + a*e*(7 + m)))) + c*e*(3 + m)*(c*e*(2*c*d - b*e)*(6 + m)*(14*b*(c*d^2...
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(4283\) vs. \(2(449)=898\).

Time = 1.59 (sec) , antiderivative size = 4284, normalized size of antiderivative = 9.54

method	result	size
norman	\(\text {Expression too large to display}\)	\(4284\)
gosper	\(\text {Expression too large to display}\)	\(5439\)
orering	\(\text {Expression too large to display}\)	\(5442\)
risch	\(\text {Expression too large to display}\)	\(6726\)
parallelrisch	\(\text {Expression too large to display}\)	\(9666\)

Input:

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

Output:

d*(a^3*b*e^7*m^7+35*a^3*b*e^7*m^6-2*a^3*c*d*e^6*m^6-3*a^2*b^2*d*e^6*m^6+51 
1*a^3*b*e^7*m^5-66*a^3*c*d*e^6*m^5-99*a^2*b^2*d*e^6*m^5+18*a^2*b*c*d^2*e^5 
*m^5+6*a*b^3*d^2*e^5*m^5+4025*a^3*b*e^7*m^4-890*a^3*c*d*e^6*m^4-1335*a^2*b 
^2*d*e^6*m^4+540*a^2*b*c*d^2*e^5*m^4-36*a^2*c^2*d^3*e^4*m^4+180*a*b^3*d^2* 
e^5*m^4-72*a*b^2*c*d^3*e^4*m^4-6*b^4*d^3*e^4*m^4+18424*a^3*b*e^7*m^3-6270* 
a^3*c*d*e^6*m^3-9405*a^2*b^2*d*e^6*m^3+6390*a^2*b*c*d^2*e^5*m^3-936*a^2*c^ 
2*d^3*e^4*m^3+2130*a*b^3*d^2*e^5*m^3-1872*a*b^2*c*d^3*e^4*m^3+360*a*b*c^2* 
d^4*e^3*m^3-156*b^4*d^3*e^4*m^3+120*b^3*c*d^4*e^3*m^3+48860*a^3*b*e^7*m^2- 
24308*a^3*c*d*e^6*m^2-36462*a^2*b^2*d*e^6*m^2+37260*a^2*b*c*d^2*e^5*m^2-90 
36*a^2*c^2*d^3*e^4*m^2+12420*a*b^3*d^2*e^5*m^2-18072*a*b^2*c*d^3*e^4*m^2+7 
560*a*b*c^2*d^4*e^3*m^2-720*a*c^3*d^5*e^2*m^2-1506*b^4*d^3*e^4*m^2+2520*b^ 
3*c*d^4*e^3*m^2-1080*b^2*c^2*d^5*e^2*m^2+69264*a^3*b*e^7*m-49104*a^3*c*d*e 
^6*m-73656*a^2*b^2*d*e^6*m+106992*a^2*b*c*d^2*e^5*m-38376*a^2*c^2*d^3*e^4* 
m+35664*a*b^3*d^2*e^5*m-76752*a*b^2*c*d^3*e^4*m+52560*a*b*c^2*d^4*e^3*m-10 
800*a*c^3*d^5*e^2*m-6396*b^4*d^3*e^4*m+17520*b^3*c*d^4*e^3*m-16200*b^2*c^2 
*d^5*e^2*m+5040*b*c^3*d^6*e*m+40320*a^3*b*e^7-40320*a^3*c*d*e^6-60480*a^2* 
b^2*d*e^6+120960*a^2*b*c*d^2*e^5-60480*a^2*c^2*d^3*e^4+40320*a*b^3*d^2*e^5 
-120960*a*b^2*c*d^3*e^4+120960*a*b*c^2*d^4*e^3-40320*a*c^3*d^5*e^2-10080*b 
^4*d^3*e^4+40320*b^3*c*d^4*e^3-60480*b^2*c^2*d^5*e^2+40320*b*c^3*d^6*e-100 
80*c^4*d^7)/e^8/(m^8+36*m^7+546*m^6+4536*m^5+22449*m^4+67284*m^3+118124...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4607 vs. \(2 (449) = 898\).

Time = 0.15 (sec) , antiderivative size = 4607, normalized size of antiderivative = 10.26 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a)^3,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. 76621 vs. \(2 (432) = 864\).

Time = 16.06 (sec) , antiderivative size = 76621, normalized size of antiderivative = 170.65 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((d**m*(a**3*b*x + a**3*c*x**2 + 3*a**2*b**2*x**2/2 + 3*a**2*b*c* 
x**3 + 3*a**2*c**2*x**4/2 + a*b**3*x**3 + 3*a*b**2*c*x**4 + 3*a*b*c**2*x** 
5 + a*c**3*x**6 + b**4*x**4/4 + b**3*c*x**5 + 3*b**2*c**2*x**6/2 + b*c**3* 
x**7 + c**4*x**8/4), Eq(e, 0)), (-60*a**3*b*e**7/(420*d**7*e**8 + 2940*d** 
6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12 
*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 20*a* 
*3*c*d*e**6/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 147 
00*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d 
*e**14*x**6 + 420*e**15*x**7) - 140*a**3*c*e**7*x/(420*d**7*e**8 + 2940*d* 
*6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**1 
2*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 30*a 
**2*b**2*d*e**6/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 
 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 29 
40*d*e**14*x**6 + 420*e**15*x**7) - 210*a**2*b**2*e**7*x/(420*d**7*e**8 + 
2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d* 
*3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) 
 - 36*a**2*b*c*d**2*e**5/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e** 
10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13* 
x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 252*a**2*b*c*d*e**6*x/(420*d* 
*7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1772 vs. \(2 (449) = 898\).

Time = 0.10 (sec) , antiderivative size = 1772, normalized size of antiderivative = 3.95 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(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^2/((m^2 + 3*m + 2)*e 
^2) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^3*c/((m^2 + 3*m + 
2)*e^2) + (e*x + d)^(m + 1)*a^3*b/(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^3/((m^3 + 6* 
m^2 + 11*m + 6)*e^3) + 9*((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*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^4/((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*b^2*c/((m^4 + 10*m^3 + 35*m^2 + 50*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^2*c^2/((m^4 + 10*m^3 + 35*m^2 
+ 50*m + 24)*e^4) + 5*((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^3*c/((m^5 + 
15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 15*((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*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9691 vs. \(2 (449) = 898\).

Time = 0.21 (sec) , antiderivative size = 9691, normalized size of antiderivative = 21.58 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(2*(e*x + d)^m*c^4*e^8*m^7*x^8 + 2*(e*x + d)^m*c^4*d*e^7*m^7*x^7 + 7*(e*x 
+ d)^m*b*c^3*e^8*m^7*x^7 + 56*(e*x + d)^m*c^4*e^8*m^6*x^8 + 7*(e*x + d)^m* 
b*c^3*d*e^7*m^7*x^6 + 9*(e*x + d)^m*b^2*c^2*e^8*m^7*x^6 + 6*(e*x + d)^m*a* 
c^3*e^8*m^7*x^6 + 42*(e*x + d)^m*c^4*d*e^7*m^6*x^7 + 203*(e*x + d)^m*b*c^3 
*e^8*m^6*x^7 + 644*(e*x + d)^m*c^4*e^8*m^5*x^8 + 9*(e*x + d)^m*b^2*c^2*d*e 
^7*m^7*x^5 + 6*(e*x + d)^m*a*c^3*d*e^7*m^7*x^5 + 5*(e*x + d)^m*b^3*c*e^8*m 
^7*x^5 + 15*(e*x + d)^m*a*b*c^2*e^8*m^7*x^5 - 14*(e*x + d)^m*c^4*d^2*e^6*m 
^6*x^6 + 161*(e*x + d)^m*b*c^3*d*e^7*m^6*x^6 + 270*(e*x + d)^m*b^2*c^2*e^8 
*m^6*x^6 + 180*(e*x + d)^m*a*c^3*e^8*m^6*x^6 + 350*(e*x + d)^m*c^4*d*e^7*m 
^5*x^7 + 2401*(e*x + d)^m*b*c^3*e^8*m^5*x^7 + 3920*(e*x + d)^m*c^4*e^8*m^4 
*x^8 + 5*(e*x + d)^m*b^3*c*d*e^7*m^7*x^4 + 15*(e*x + d)^m*a*b*c^2*d*e^7*m^ 
7*x^4 + (e*x + d)^m*b^4*e^8*m^7*x^4 + 12*(e*x + d)^m*a*b^2*c*e^8*m^7*x^4 + 
 6*(e*x + d)^m*a^2*c^2*e^8*m^7*x^4 - 42*(e*x + d)^m*b*c^3*d^2*e^6*m^6*x^5 
+ 225*(e*x + d)^m*b^2*c^2*d*e^7*m^6*x^5 + 150*(e*x + d)^m*a*c^3*d*e^7*m^6* 
x^5 + 155*(e*x + d)^m*b^3*c*e^8*m^6*x^5 + 465*(e*x + d)^m*a*b*c^2*e^8*m^6* 
x^5 - 210*(e*x + d)^m*c^4*d^2*e^6*m^5*x^6 + 1435*(e*x + d)^m*b*c^3*d*e^7*m 
^5*x^6 + 3294*(e*x + d)^m*b^2*c^2*e^8*m^5*x^6 + 2196*(e*x + d)^m*a*c^3*e^8 
*m^5*x^6 + 1470*(e*x + d)^m*c^4*d*e^7*m^4*x^7 + 14945*(e*x + d)^m*b*c^3*e^ 
8*m^4*x^7 + 13538*(e*x + d)^m*c^4*e^8*m^3*x^8 + (e*x + d)^m*b^4*d*e^7*m^7* 
x^3 + 12*(e*x + d)^m*a*b^2*c*d*e^7*m^7*x^3 + 6*(e*x + d)^m*a^2*c^2*d*e^...
 

Mupad [B] (verification not implemented)

Time = 14.53 (sec) , antiderivative size = 4573, normalized size of antiderivative = 10.18 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(x*(d + e*x)^m*(40320*a^3*b*e^8 + 48860*a^3*b*e^8*m^2 + 18424*a^3*b*e^8*m^ 
3 + 4025*a^3*b*e^8*m^4 + 511*a^3*b*e^8*m^5 + 35*a^3*b*e^8*m^6 + a^3*b*e^8* 
m^7 + 10080*b^4*d^3*e^5*m + 6396*b^4*d^3*e^5*m^2 + 1506*b^4*d^3*e^5*m^3 + 
156*b^4*d^3*e^5*m^4 + 6*b^4*d^3*e^5*m^5 + 69264*a^3*b*e^8*m + 10080*c^4*d^ 
7*e*m + 40320*a^3*c*d*e^7*m + 38376*a^2*c^2*d^3*e^5*m^2 + 9036*a^2*c^2*d^3 
*e^5*m^3 + 936*a^2*c^2*d^3*e^5*m^4 + 36*a^2*c^2*d^3*e^5*m^5 + 16200*b^2*c^ 
2*d^5*e^3*m^2 + 1080*b^2*c^2*d^5*e^3*m^3 - 40320*a*b^3*d^2*e^6*m + 60480*a 
^2*b^2*d*e^7*m + 40320*a*c^3*d^5*e^3*m + 49104*a^3*c*d*e^7*m^2 + 24308*a^3 
*c*d*e^7*m^3 + 6270*a^3*c*d*e^7*m^4 + 890*a^3*c*d*e^7*m^5 + 66*a^3*c*d*e^7 
*m^6 + 2*a^3*c*d*e^7*m^7 - 40320*b*c^3*d^6*e^2*m - 40320*b^3*c*d^4*e^4*m - 
 35664*a*b^3*d^2*e^6*m^2 + 73656*a^2*b^2*d*e^7*m^2 - 12420*a*b^3*d^2*e^6*m 
^3 + 36462*a^2*b^2*d*e^7*m^3 - 2130*a*b^3*d^2*e^6*m^4 + 9405*a^2*b^2*d*e^7 
*m^4 - 180*a*b^3*d^2*e^6*m^5 + 1335*a^2*b^2*d*e^7*m^5 - 6*a*b^3*d^2*e^6*m^ 
6 + 99*a^2*b^2*d*e^7*m^6 + 3*a^2*b^2*d*e^7*m^7 + 60480*a^2*c^2*d^3*e^5*m + 
 10800*a*c^3*d^5*e^3*m^2 + 720*a*c^3*d^5*e^3*m^3 + 60480*b^2*c^2*d^5*e^3*m 
 - 5040*b*c^3*d^6*e^2*m^2 - 17520*b^3*c*d^4*e^4*m^2 - 2520*b^3*c*d^4*e^4*m 
^3 - 120*b^3*c*d^4*e^4*m^4 - 52560*a*b*c^2*d^4*e^4*m^2 + 76752*a*b^2*c*d^3 
*e^5*m^2 - 106992*a^2*b*c*d^2*e^6*m^2 - 7560*a*b*c^2*d^4*e^4*m^3 + 18072*a 
*b^2*c*d^3*e^5*m^3 - 37260*a^2*b*c*d^2*e^6*m^3 - 360*a*b*c^2*d^4*e^4*m^4 + 
 1872*a*b^2*c*d^3*e^5*m^4 - 6390*a^2*b*c*d^2*e^6*m^4 + 72*a*b^2*c*d^3*e...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 6725, normalized size of antiderivative = 14.98 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)**m*(a**3*b*d*e**7*m**7 + 35*a**3*b*d*e**7*m**6 + 511*a**3*b*d*e 
**7*m**5 + 4025*a**3*b*d*e**7*m**4 + 18424*a**3*b*d*e**7*m**3 + 48860*a**3 
*b*d*e**7*m**2 + 69264*a**3*b*d*e**7*m + 40320*a**3*b*d*e**7 + a**3*b*e**8 
*m**7*x + 35*a**3*b*e**8*m**6*x + 511*a**3*b*e**8*m**5*x + 4025*a**3*b*e** 
8*m**4*x + 18424*a**3*b*e**8*m**3*x + 48860*a**3*b*e**8*m**2*x + 69264*a** 
3*b*e**8*m*x + 40320*a**3*b*e**8*x - 2*a**3*c*d**2*e**6*m**6 - 66*a**3*c*d 
**2*e**6*m**5 - 890*a**3*c*d**2*e**6*m**4 - 6270*a**3*c*d**2*e**6*m**3 - 2 
4308*a**3*c*d**2*e**6*m**2 - 49104*a**3*c*d**2*e**6*m - 40320*a**3*c*d**2* 
e**6 + 2*a**3*c*d*e**7*m**7*x + 66*a**3*c*d*e**7*m**6*x + 890*a**3*c*d*e** 
7*m**5*x + 6270*a**3*c*d*e**7*m**4*x + 24308*a**3*c*d*e**7*m**3*x + 49104* 
a**3*c*d*e**7*m**2*x + 40320*a**3*c*d*e**7*m*x + 2*a**3*c*e**8*m**7*x**2 + 
 68*a**3*c*e**8*m**6*x**2 + 956*a**3*c*e**8*m**5*x**2 + 7160*a**3*c*e**8*m 
**4*x**2 + 30578*a**3*c*e**8*m**3*x**2 + 73412*a**3*c*e**8*m**2*x**2 + 894 
24*a**3*c*e**8*m*x**2 + 40320*a**3*c*e**8*x**2 - 3*a**2*b**2*d**2*e**6*m** 
6 - 99*a**2*b**2*d**2*e**6*m**5 - 1335*a**2*b**2*d**2*e**6*m**4 - 9405*a** 
2*b**2*d**2*e**6*m**3 - 36462*a**2*b**2*d**2*e**6*m**2 - 73656*a**2*b**2*d 
**2*e**6*m - 60480*a**2*b**2*d**2*e**6 + 3*a**2*b**2*d*e**7*m**7*x + 99*a* 
*2*b**2*d*e**7*m**6*x + 1335*a**2*b**2*d*e**7*m**5*x + 9405*a**2*b**2*d*e* 
*7*m**4*x + 36462*a**2*b**2*d*e**7*m**3*x + 73656*a**2*b**2*d*e**7*m**2*x 
+ 60480*a**2*b**2*d*e**7*m*x + 3*a**2*b**2*e**8*m**7*x**2 + 102*a**2*b*...