3.1654 \(\int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=449 \[ \frac{(d+e x)^{m+4} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8 (m+4)}+\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+2)}-\frac{3 (2 c d-b e) (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+3)}-\frac{5 c (2 c d-b e) (d+e x)^{m+5} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+5)}+\frac{3 c^2 (d+e x)^{m+6} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+6)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^8 (m+1)}-\frac{7 c^3 (2 c d-b e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{2 c^4 (d+e x)^{m+8}}{e^8 (m+8)} \]

[Out]

-(((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))

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Rubi [A]  time = 0.835911, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{(d+e x)^{m+4} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8 (m+4)}+\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+2)}-\frac{3 (2 c d-b e) (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+3)}-\frac{5 c (2 c d-b e) (d+e x)^{m+5} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+5)}+\frac{3 c^2 (d+e x)^{m+6} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+6)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^8 (m+1)}-\frac{7 c^3 (2 c d-b e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{2 c^4 (d+e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

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

[Out]

-(((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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 3.23948, size = 1259, normalized size = 2.8 \[ \frac{(d+e x)^{m+1} \left (-2 \left (5040 d^7-5040 e (m+1) x d^6+2520 e^2 \left (m^2+3 m+2\right ) x^2 d^5-840 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^4+210 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^3-42 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d^2+7 e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6 d-e^7 \left (m^7+28 m^6+322 m^5+1960 m^4+6769 m^3+13132 m^2+13068 m+5040\right ) x^7\right ) c^4+e (m+8) \left (6 a e (m+7) \left (-120 d^5+120 e (m+1) x d^4-60 e^2 \left (m^2+3 m+2\right ) x^2 d^3+20 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^2-5 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )+7 b \left (720 d^6-720 e (m+1) x d^5+360 e^2 \left (m^2+3 m+2\right ) x^2 d^4-120 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^3+30 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^2-6 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right )\right ) c^3+3 e^2 \left (m^2+15 m+56\right ) \left (3 \left (-120 d^5+120 e (m+1) x d^4-60 e^2 \left (m^2+3 m+2\right ) x^2 d^3+20 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^2-5 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right ) b^2+5 a e (m+6) \left (24 d^4-24 e (m+1) x d^3+12 e^2 \left (m^2+3 m+2\right ) x^2 d^2-4 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right ) b+2 a^2 e^2 \left (m^2+11 m+30\right ) \left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right ) c^2+e^3 \left (m^3+21 m^2+146 m+336\right ) \left (5 \left (24 d^4-24 e (m+1) x d^3+12 e^2 \left (m^2+3 m+2\right ) x^2 d^2-4 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right ) b^3+12 a e (m+5) \left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right ) b^2+9 a^2 e^2 \left (m^2+9 m+20\right ) \left (2 d^2-2 e (m+1) x d+e^2 \left (m^2+3 m+2\right ) x^2\right ) b+2 a^3 e^3 \left (m^3+12 m^2+47 m+60\right ) (e (m+1) x-d)\right ) c+b e^4 \left (m^4+26 m^3+251 m^2+1066 m+1680\right ) \left (\left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right ) b^3+3 a e (m+4) \left (2 d^2-2 e (m+1) x d+e^2 \left (m^2+3 m+2\right ) x^2\right ) b^2+3 a^2 e^2 \left (m^2+7 m+12\right ) (e (m+1) x-d) b+a^3 e^3 \left (m^3+9 m^2+26 m+24\right )\right )\right )}{e^8 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(-2*c^4*(5040*d^7 - 5040*d^6*e*(1 + m)*x + 2520*d^5*e^2*(2 +
3*m + m^2)*x^2 - 840*d^4*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + 210*d^3*e^4*(24 + 50
*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 42*d^2*e^5*(120 + 274*m + 225*m^2 + 85*m^3 + 1
5*m^4 + m^5)*x^5 + 7*d*e^6*(720 + 1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5
+ m^6)*x^6 - e^7*(5040 + 13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 +
28*m^6 + m^7)*x^7) + b*e^4*(1680 + 1066*m + 251*m^2 + 26*m^3 + m^4)*(a^3*e^3*(24
+ 26*m + 9*m^2 + m^3) + 3*a^2*b*e^2*(12 + 7*m + m^2)*(-d + e*(1 + m)*x) + 3*a*b
^2*e*(4 + m)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + b^3*(-6*d^3 +
6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)*
x^3)) + c*e^3*(336 + 146*m + 21*m^2 + m^3)*(2*a^3*e^3*(60 + 47*m + 12*m^2 + m^3)
*(-d + e*(1 + m)*x) + 9*a^2*b*e^2*(20 + 9*m + m^2)*(2*d^2 - 2*d*e*(1 + m)*x + e^
2*(2 + 3*m + m^2)*x^2) + 12*a*b^2*e*(5 + m)*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^
2*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)*x^3) + 5*b^3*(24*d^4 - 24*d
^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6 + 11*m + 6*m^2 + m^
3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4)) + 3*c^2*e^2*(56 + 15*m +
m^2)*(2*a^2*e^2*(30 + 11*m + m^2)*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m
+ m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)*x^3) + 5*a*b*e*(6 + m)*(24*d^4 - 24*d
^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6 + 11*m + 6*m^2 + m^
3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4) + 3*b^2*(-120*d^5 + 120*d^
4*e*(1 + m)*x - 60*d^3*e^2*(2 + 3*m + m^2)*x^2 + 20*d^2*e^3*(6 + 11*m + 6*m^2 +
m^3)*x^3 - 5*d*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 + e^5*(120 + 274*m +
225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5)) + c^3*e*(8 + m)*(6*a*e*(7 + m)*(-120*d^5
+ 120*d^4*e*(1 + m)*x - 60*d^3*e^2*(2 + 3*m + m^2)*x^2 + 20*d^2*e^3*(6 + 11*m +
6*m^2 + m^3)*x^3 - 5*d*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 + e^5*(120 +
274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5) + 7*b*(720*d^6 - 720*d^5*e*(1 + m)
*x + 360*d^4*e^2*(2 + 3*m + m^2)*x^2 - 120*d^3*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3
+ 30*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*d*e^5*(120 + 274*m + 22
5*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2 + 735*m^3 + 17
5*m^4 + 21*m^5 + m^6)*x^6))))/(e^8*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 +
m)*(7 + m)*(8 + m))

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Maple [B]  time = 0.033, size = 5439, normalized size = 12.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.325479, size = 6219, normalized size = 13.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^3*b*d*e^7*m^7 - 10080*c^4*d^8 + 40320*b*c^3*d^7*e + 40320*a^3*b*d*e^7 - 20160
*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 40320*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 10080*(b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 40320*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 20160*(3
*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2*(c^4*e^8*m^7 + 28*c^4*e^8*m^6 + 322*c^4*e^8*m^5
+ 1960*c^4*e^8*m^4 + 6769*c^4*e^8*m^3 + 13132*c^4*e^8*m^2 + 13068*c^4*e^8*m + 50
40*c^4*e^8)*x^8 + (40320*b*c^3*e^8 + (2*c^4*d*e^7 + 7*b*c^3*e^8)*m^7 + 7*(6*c^4*
d*e^7 + 29*b*c^3*e^8)*m^6 + 7*(50*c^4*d*e^7 + 343*b*c^3*e^8)*m^5 + 245*(6*c^4*d*
e^7 + 61*b*c^3*e^8)*m^4 + 112*(29*c^4*d*e^7 + 469*b*c^3*e^8)*m^3 + 196*(18*c^4*d
*e^7 + 527*b*c^3*e^8)*m^2 + 144*(10*c^4*d*e^7 + 721*b*c^3*e^8)*m)*x^7 + (35*a^3*
b*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^6 + (20160*(3*b^2*c^2 + 2*a*c^3)*e^8
+ (7*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^7 - (14*c^4*d^2*e^6 - 161*b*c^
3*d*e^7 - 90*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^6 - (210*c^4*d^2*e^6 - 1435*b*c^3*d*e^
7 - 1098*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^5 - 5*(238*c^4*d^2*e^6 - 1267*b*c^3*d*e^7
- 1404*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^4 - (3150*c^4*d^2*e^6 - 14518*b*c^3*d*e^7 -
25227*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^3 - 2*(1918*c^4*d^2*e^6 - 8092*b*c^3*d*e^7 -
25245*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^2 - 24*(70*c^4*d^2*e^6 - 280*b*c^3*d*e^7 - 21
43*(3*b^2*c^2 + 2*a*c^3)*e^8)*m)*x^6 + (511*a^3*b*d*e^7 + 6*(a*b^3 + 3*a^2*b*c)*
d^3*e^5 - 33*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^5 + (40320*(b^3*c + 3*a*b*c^2)*e^8
+ (3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 5*(b^3*c + 3*a*b*c^2)*e^8)*m^7 - (42*b*c^3*d
^2*e^6 - 75*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 155*(b^3*c + 3*a*b*c^2)*e^8)*m^6 + (84
*c^4*d^3*e^5 - 756*b*c^3*d^2*e^6 + 723*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 1955*(b^3*c
+ 3*a*b*c^2)*e^8)*m^5 + 5*(168*c^4*d^3*e^5 - 966*b*c^3*d^2*e^6 + 681*(3*b^2*c^2
+ 2*a*c^3)*d*e^7 + 2581*(b^3*c + 3*a*b*c^2)*e^8)*m^4 + 2*(1470*c^4*d^3*e^5 - 69
30*b*c^3*d^2*e^6 + 4101*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 23860*(b^3*c + 3*a*b*c^2)*
e^8)*m^3 + 4*(1050*c^4*d^3*e^5 - 4452*b*c^3*d^2*e^6 + 2370*(3*b^2*c^2 + 2*a*c^3)
*d*e^7 + 24455*(b^3*c + 3*a*b*c^2)*e^8)*m^2 + 144*(14*c^4*d^3*e^5 - 56*b*c^3*d^2
*e^6 + 28*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 705*(b^3*c + 3*a*b*c^2)*e^8)*m)*x^5 + (4
025*a^3*b*d*e^7 - 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 180*(a*b^3 + 3*a^2*
b*c)*d^3*e^5 - 445*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^4 + (10080*(b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*e^8 + (5*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2
)*e^8)*m^7 - (15*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 135*(b^3*c + 3*a*b*c^2)*d*e^7 -
32*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*m^6 + (210*b*c^3*d^3*e^5 - 315*(3*b^2*c^
2 + 2*a*c^3)*d^2*e^6 + 1415*(b^3*c + 3*a*b*c^2)*d*e^7 + 418*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*e^8)*m^5 - (420*c^4*d^4*e^4 - 2940*b*c^3*d^3*e^5 + 2355*(3*b^2*c^2 +
2*a*c^3)*d^2*e^6 - 7245*(b^3*c + 3*a*b*c^2)*d*e^7 - 2864*(b^4 + 12*a*b^2*c + 6*a
^2*c^2)*e^8)*m^4 - (2520*c^4*d^4*e^4 - 12390*b*c^3*d^3*e^5 + 7605*(3*b^2*c^2 + 2
*a*c^3)*d^2*e^6 - 18740*(b^3*c + 3*a*b*c^2)*d*e^7 - 10993*(b^4 + 12*a*b^2*c + 6*
a^2*c^2)*e^8)*m^3 - 2*(2310*c^4*d^4*e^4 - 9870*b*c^3*d^3*e^5 + 5295*(3*b^2*c^2 +
2*a*c^3)*d^2*e^6 - 11430*(b^3*c + 3*a*b*c^2)*d*e^7 - 11656*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*e^8)*m^2 - 36*(70*c^4*d^4*e^4 - 280*b*c^3*d^3*e^5 + 140*(3*b^2*c^2 +
2*a*c^3)*d^2*e^6 - 280*(b^3*c + 3*a*b*c^2)*d*e^7 - 691*(b^4 + 12*a*b^2*c + 6*a^2
*c^2)*e^8)*m)*x^4 + (18424*a^3*b*d*e^7 + 120*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 156*(
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 2130*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 3135*
(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^3 + (40320*(a*b^3 + 3*a^2*b*c)*e^8 + ((b^4 + 12
*a*b^2*c + 6*a^2*c^2)*d*e^7 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*m^7 - (20*(b^3*c + 3*a*
b*c^2)*d^2*e^6 - 29*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 99*(a*b^3 + 3*a^2*b*c
)*e^8)*m^6 + (60*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 480*(b^3*c + 3*a*b*c^2)*d^2*e^6
+ 331*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 1341*(a*b^3 + 3*a^2*b*c)*e^8)*m^5
- (840*b*c^3*d^4*e^4 - 1080*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4220*(b^3*c + 3*a*b*
c^2)*d^2*e^6 - 1871*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 9585*(a*b^3 + 3*a^2*b
*c)*e^8)*m^4 + 4*(420*c^4*d^5*e^3 - 2310*b*c^3*d^4*e^4 + 1545*(3*b^2*c^2 + 2*a*c
^3)*d^3*e^5 - 4080*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 1345*(b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d*e^7 + 9648*(a*b^3 + 3*a^2*b*c)*e^8)*m^3 + 4*(1260*c^4*d^5*e^3 - 5460*b*c^
3*d^4*e^4 + 2970*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 6500*(b^3*c + 3*a*b*c^2)*d^2*e^
6 + 1793*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 21519*(a*b^3 + 3*a^2*b*c)*e^8)*m
^2 + 48*(70*c^4*d^5*e^3 - 280*b*c^3*d^4*e^4 + 140*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5
- 280*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 20
03*(a*b^3 + 3*a^2*b*c)*e^8)*m)*x^3 + 2*(24430*a^3*b*d*e^7 - 180*(3*b^2*c^2 + 2*a
*c^3)*d^6*e^2 + 1260*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 753*(b^4 + 12*a*b^2*c + 6*a^2
*c^2)*d^4*e^4 + 6210*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 6077*(3*a^2*b^2 + 2*a^3*c)*d^
2*e^6)*m^2 + (20160*(3*a^2*b^2 + 2*a^3*c)*e^8 + (3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (
3*a^2*b^2 + 2*a^3*c)*e^8)*m^7 - (3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 93*(
a*b^3 + 3*a^2*b*c)*d*e^7 - 34*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^6 + (60*(b^3*c + 3*a*
b*c^2)*d^3*e^5 - 81*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 1155*(a*b^3 + 3*a^2
*b*c)*d*e^7 + 478*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^5 - (180*(3*b^2*c^2 + 2*a*c^3)*d^
4*e^4 - 1320*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 831*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
2*e^6 - 7275*(a*b^3 + 3*a^2*b*c)*d*e^7 - 3580*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^4 + (
2520*b*c^3*d^5*e^3 - 2880*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 10020*(b^3*c + 3*a*b*c
^2)*d^3*e^5 - 3951*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 24042*(a*b^3 + 3*a^2
*b*c)*d*e^7 + 15289*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^3 - 2*(2520*c^4*d^6*e^2 - 11340
*b*c^3*d^5*e^3 + 6390*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 14460*(b^3*c + 3*a*b*c^2)*
d^3*e^5 + 4119*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 18996*(a*b^3 + 3*a^2*b*c
)*d*e^7 - 18353*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^2 - 72*(70*c^4*d^6*e^2 - 280*b*c^3*
d^5*e^3 + 140*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 280*(b^3*c + 3*a*b*c^2)*d^3*e^5 +
70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 280*(a*b^3 + 3*a^2*b*c)*d*e^7 - 621*
(3*a^2*b^2 + 2*a^3*c)*e^8)*m)*x^2 + 12*(420*b*c^3*d^7*e + 5772*a^3*b*d*e^7 - 450
*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 1460*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 533*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 2972*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 2046*(3*a^2*
b^2 + 2*a^3*c)*d^2*e^6)*m + (40320*a^3*b*e^8 + (a^3*b*e^8 + (3*a^2*b^2 + 2*a^3*c
)*d*e^7)*m^7 + (35*a^3*b*e^8 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 33*(3*a^2*b^2 + 2
*a^3*c)*d*e^7)*m^6 + (511*a^3*b*e^8 + 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 -
180*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 445*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m^5 + (4025*
a^3*b*e^8 - 120*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 156*(b^4 + 12*a*b^2*c + 6*a^2*c^2)
*d^3*e^5 - 2130*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 3135*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*
m^4 + 2*(9212*a^3*b*e^8 + 180*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 1260*(b^3*c + 3*a*
b*c^2)*d^4*e^4 + 753*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 6210*(a*b^3 + 3*a^
2*b*c)*d^2*e^6 + 6077*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m^3 - 4*(1260*b*c^3*d^6*e^2 -
12215*a^3*b*e^8 - 1350*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 4380*(b^3*c + 3*a*b*c^2)
*d^4*e^4 - 1599*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 8916*(a*b^3 + 3*a^2*b*c
)*d^2*e^6 - 6138*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m^2 + 144*(70*c^4*d^7*e - 280*b*c^
3*d^6*e^2 + 481*a^3*b*e^8 + 140*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 280*(b^3*c + 3*a
*b*c^2)*d^4*e^4 + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 280*(a*b^3 + 3*a^2
*b*c)*d^2*e^6 + 140*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m)*x)*(e*x + d)^m/(e^8*m^8 + 36
*e^8*m^7 + 546*e^8*m^6 + 4536*e^8*m^5 + 22449*e^8*m^4 + 67284*e^8*m^3 + 118124*e
^8*m^2 + 109584*e^8*m + 40320*e^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.324281, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done