3.2538 \(\int (d+e x)^m \left (a+b x+c x^2\right )^4 \, dx\)

Optimal. Leaf size=485 \[ \frac{(d+e x)^{m+5} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9 (m+5)}+\frac{2 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (m+3)}-\frac{4 (2 c d-b e) (d+e x)^{m+4} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (m+4)}-\frac{4 c (2 c d-b e) (d+e x)^{m+6} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (m+6)}+\frac{2 c^2 (d+e x)^{m+7} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (m+7)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^4}{e^9 (m+1)}-\frac{4 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^3}{e^9 (m+2)}-\frac{4 c^3 (2 c d-b e) (d+e x)^{m+8}}{e^9 (m+8)}+\frac{c^4 (d+e x)^{m+9}}{e^9 (m+9)} \]

[Out]

((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))

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Rubi [A]  time = 1.08914, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(d+e x)^{m+5} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9 (m+5)}+\frac{2 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (m+3)}-\frac{4 (2 c d-b e) (d+e x)^{m+4} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (m+4)}-\frac{4 c (2 c d-b e) (d+e x)^{m+6} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (m+6)}+\frac{2 c^2 (d+e x)^{m+7} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (m+7)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^4}{e^9 (m+1)}-\frac{4 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^3}{e^9 (m+2)}-\frac{4 c^3 (2 c d-b e) (d+e x)^{m+8}}{e^9 (m+8)}+\frac{c^4 (d+e x)^{m+9}}{e^9 (m+9)} \]

Antiderivative was successfully verified.

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

[Out]

((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))

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

[Out]

Timed out

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Mathematica [B]  time = 4.88777, size = 1708, normalized size = 3.52 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(c^4*(40320*d^8 - 40320*d^7*e*(1 + m)*x + 20160*d^6*e^2*(2 +
3*m + m^2)*x^2 - 6720*d^5*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + 1680*d^4*e^4*(24 +
50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 336*d^3*e^5*(120 + 274*m + 225*m^2 + 85*m^3
+ 15*m^4 + m^5)*x^5 + 56*d^2*e^6*(720 + 1764*m + 1624*m^2 + 735*m^3 + 175*m^4 +
21*m^5 + m^6)*x^6 - 8*d*e^7*(5040 + 13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 +
322*m^5 + 28*m^6 + m^7)*x^7 + e^8*(40320 + 109584*m + 118124*m^2 + 67284*m^3 + 2
2449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8)*x^8) + e^4*(3024 + 1650*m + 335*m^
2 + 30*m^3 + m^4)*(a^4*e^4*(120 + 154*m + 71*m^2 + 14*m^3 + m^4) + 4*a^3*b*e^3*(
60 + 47*m + 12*m^2 + m^3)*(-d + e*(1 + m)*x) + 6*a^2*b^2*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) + 4*a*b^3*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) + b^4*(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)) +
4*c*e^3*(504 + 191*m + 24*m^2 + m^3)*(a^3*e^3*(120 + 74*m + 15*m^2 + m^3)*(2*d^
2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 3*a^2*b*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) + 3*a*b^2*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) + b^3*(-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)) + 6*c^2*e^2*(72 + 17*m + m^2)*(a^2*e^2*(42 + 13*m + m^2)*(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) + 2*a*b*e*(7 + m)*(-120*d^5 + 1
20*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) + b^2*(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 + 3
0*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*d*e^5*(120 + 274*m + 225*m
^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2 + 735*m^3 + 175*m
^4 + 21*m^5 + m^6)*x^6)) + 4*c^3*e*(9 + m)*(a*e*(8 + m)*(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
+ 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2 + 735*m^3
+ 175*m^4 + 21*m^5 + m^6)*x^6) + b*(-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 + 8
5*m^3 + 15*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 + 3
22*m^5 + 28*m^6 + m^7)*x^7))))/(e^9*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 +
m)*(7 + m)*(8 + m)*(9 + m))

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Maple [B]  time = 0.04, size = 7696, normalized size = 15.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x+a)^4,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)^4*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283687, size = 8351, normalized size = 17.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^4*d*e^8*m^8 + 40320*c^4*d^9 - 181440*b*c^3*d^8*e - 725760*a^3*b*d^2*e^7 + 362
880*a^4*d*e^8 + 103680*(3*b^2*c^2 + 2*a*c^3)*d^7*e^2 - 241920*(b^3*c + 3*a*b*c^2
)*d^6*e^3 + 72576*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^4 - 362880*(a*b^3 + 3*a^2
*b*c)*d^4*e^5 + 241920*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6 + (c^4*e^9*m^8 + 36*c^4*e^9
*m^7 + 546*c^4*e^9*m^6 + 4536*c^4*e^9*m^5 + 22449*c^4*e^9*m^4 + 67284*c^4*e^9*m^
3 + 118124*c^4*e^9*m^2 + 109584*c^4*e^9*m + 40320*c^4*e^9)*x^9 + (181440*b*c^3*e
^9 + (c^4*d*e^8 + 4*b*c^3*e^9)*m^8 + 4*(7*c^4*d*e^8 + 37*b*c^3*e^9)*m^7 + 14*(23
*c^4*d*e^8 + 164*b*c^3*e^9)*m^6 + 56*(35*c^4*d*e^8 + 347*b*c^3*e^9)*m^5 + 7*(967
*c^4*d*e^8 + 13948*b*c^3*e^9)*m^4 + 28*(469*c^4*d*e^8 + 10579*b*c^3*e^9)*m^3 + 3
6*(363*c^4*d*e^8 + 14584*b*c^3*e^9)*m^2 + 144*(35*c^4*d*e^8 + 3407*b*c^3*e^9)*m)
*x^8 - 4*(a^3*b*d^2*e^7 - 11*a^4*d*e^8)*m^7 + 2*(51840*(3*b^2*c^2 + 2*a*c^3)*e^9
+ (2*b*c^3*d*e^8 + (3*b^2*c^2 + 2*a*c^3)*e^9)*m^8 - 2*(2*c^4*d^2*e^7 - 30*b*c^3
*d*e^8 - 19*(3*b^2*c^2 + 2*a*c^3)*e^9)*m^7 - 4*(21*c^4*d^2*e^7 - 182*b*c^3*d*e^8
- 151*(3*b^2*c^2 + 2*a*c^3)*e^9)*m^6 - 14*(50*c^4*d^2*e^7 - 330*b*c^3*d*e^8 - 3
73*(3*b^2*c^2 + 2*a*c^3)*e^9)*m^5 - 7*(420*c^4*d^2*e^7 - 2354*b*c^3*d*e^8 - 3817
*(3*b^2*c^2 + 2*a*c^3)*e^9)*m^4 - 28*(232*c^4*d^2*e^7 - 1170*b*c^3*d*e^8 - 2939*
(3*b^2*c^2 + 2*a*c^3)*e^9)*m^3 - 36*(196*c^4*d^2*e^7 - 922*b*c^3*d*e^8 - 4101*(3
*b^2*c^2 + 2*a*c^3)*e^9)*m^2 - 144*(20*c^4*d^2*e^7 - 90*b*c^3*d*e^8 - 967*(3*b^2
*c^2 + 2*a*c^3)*e^9)*m)*x^7 - 2*(84*a^3*b*d^2*e^7 - 413*a^4*d*e^8 - 2*(3*a^2*b^2
+ 2*a^3*c)*d^3*e^6)*m^6 + 2*(120960*(b^3*c + 3*a*b*c^2)*e^9 + ((3*b^2*c^2 + 2*a
*c^3)*d*e^8 + 2*(b^3*c + 3*a*b*c^2)*e^9)*m^8 - 2*(7*b*c^3*d^2*e^7 - 16*(3*b^2*c^
2 + 2*a*c^3)*d*e^8 - 39*(b^3*c + 3*a*b*c^2)*e^9)*m^7 + 4*(7*c^4*d^3*e^6 - 84*b*c
^3*d^2*e^7 + 103*(3*b^2*c^2 + 2*a*c^3)*d*e^8 + 318*(b^3*c + 3*a*b*c^2)*e^9)*m^6
+ 2*(210*c^4*d^3*e^6 - 1540*b*c^3*d^2*e^7 + 1375*(3*b^2*c^2 + 2*a*c^3)*d*e^8 + 5
634*(b^3*c + 3*a*b*c^2)*e^9)*m^5 + (2380*c^4*d^3*e^6 - 13860*b*c^3*d^2*e^7 + 102
19*(3*b^2*c^2 + 2*a*c^3)*d*e^8 + 58938*(b^3*c + 3*a*b*c^2)*e^9)*m^4 + 2*(3150*c^
4*d^3*e^6 - 16093*b*c^3*d^2*e^7 + 10489*(3*b^2*c^2 + 2*a*c^3)*d*e^8 + 92511*(b^3
*c + 3*a*b*c^2)*e^9)*m^3 + 4*(1918*c^4*d^3*e^6 - 9051*b*c^3*d^2*e^7 + 5442*(3*b^
2*c^2 + 2*a*c^3)*d*e^8 + 84307*(b^3*c + 3*a*b*c^2)*e^9)*m^2 + 48*(70*c^4*d^3*e^6
- 315*b*c^3*d^2*e^7 + 180*(3*b^2*c^2 + 2*a*c^3)*d*e^8 + 6709*(b^3*c + 3*a*b*c^2
)*e^9)*m)*x^6 - 4*(742*a^3*b*d^2*e^7 - 2156*a^4*d*e^8 + 6*(a*b^3 + 3*a^2*b*c)*d^
4*e^5 - 39*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6)*m^5 + (72576*(b^4 + 12*a*b^2*c + 6*a^2
*c^2)*e^9 + (4*(b^3*c + 3*a*b*c^2)*d*e^8 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^9)*m
^8 - 4*(3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^7 - 34*(b^3*c + 3*a*b*c^2)*d*e^8 - 10*(b^4
+ 12*a*b^2*c + 6*a^2*c^2)*e^9)*m^7 + 2*(84*b*c^3*d^3*e^6 - 162*(3*b^2*c^2 + 2*a
*c^3)*d^2*e^7 + 932*(b^3*c + 3*a*b*c^2)*d*e^8 + 335*(b^4 + 12*a*b^2*c + 6*a^2*c^
2)*e^9)*m^6 - 4*(84*c^4*d^4*e^5 - 798*b*c^3*d^3*e^6 + 831*(3*b^2*c^2 + 2*a*c^3)*
d^2*e^7 - 3304*(b^3*c + 3*a*b*c^2)*d*e^8 - 1525*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e
^9)*m^5 - (3360*c^4*d^4*e^5 - 21000*b*c^3*d^3*e^6 + 16380*(3*b^2*c^2 + 2*a*c^3)*
d^2*e^7 - 51796*(b^3*c + 3*a*b*c^2)*d*e^8 - 32773*(b^4 + 12*a*b^2*c + 6*a^2*c^2)
*e^9)*m^4 - 4*(2940*c^4*d^4*e^5 - 15330*b*c^3*d^3*e^6 + 10182*(3*b^2*c^2 + 2*a*c
^3)*d^2*e^7 - 27766*(b^3*c + 3*a*b*c^2)*d*e^8 - 26365*(b^4 + 12*a*b^2*c + 6*a^2*
c^2)*e^9)*m^3 - 12*(1400*c^4*d^4*e^5 - 6636*b*c^3*d^3*e^6 + 4008*(3*b^2*c^2 + 2*
a*c^3)*d^2*e^7 - 9928*(b^3*c + 3*a*b*c^2)*d*e^8 - 16365*(b^4 + 12*a*b^2*c + 6*a^
2*c^2)*e^9)*m^2 - 144*(56*c^4*d^4*e^5 - 252*b*c^3*d^3*e^6 + 144*(3*b^2*c^2 + 2*a
*c^3)*d^2*e^7 - 336*(b^3*c + 3*a*b*c^2)*d*e^8 - 1325*(b^4 + 12*a*b^2*c + 6*a^2*c
^2)*e^9)*m)*x^5 - (28560*a^3*b*d^2*e^7 - 54649*a^4*d*e^8 - 24*(b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^5*e^4 + 840*(a*b^3 + 3*a^2*b*c)*d^4*e^5 - 2500*(3*a^2*b^2 + 2*a^3
*c)*d^3*e^6)*m^4 + (362880*(a*b^3 + 3*a^2*b*c)*e^9 + ((b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d*e^8 + 4*(a*b^3 + 3*a^2*b*c)*e^9)*m^8 - 4*(5*(b^3*c + 3*a*b*c^2)*d^2*e^7 -
9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 - 41*(a*b^3 + 3*a^2*b*c)*e^9)*m^7 + 2*(3
0*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 - 300*(b^3*c + 3*a*b*c^2)*d^2*e^7 + 263*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d*e^8 + 1412*(a*b^3 + 3*a^2*b*c)*e^9)*m^6 - 4*(210*b*c^3*
d^4*e^5 - 345*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 + 1730*(b^3*c + 3*a*b*c^2)*d^2*e^7 -
999*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 - 6626*(a*b^3 + 3*a^2*b*c)*e^9)*m^5 +
(1680*c^4*d^5*e^4 - 12600*b*c^3*d^4*e^5 + 11100*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 -
38400*(b^3*c + 3*a*b*c^2)*d^2*e^7 + 16789*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 +
147076*(a*b^3 + 3*a^2*b*c)*e^9)*m^4 + 4*(2520*c^4*d^5*e^4 - 13650*b*c^3*d^4*e^5
+ 9375*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 - 26345*(b^3*c + 3*a*b*c^2)*d^2*e^7 + 9576
*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 + 122249*(a*b^3 + 3*a^2*b*c)*e^9)*m^3 + 12
*(1540*c^4*d^5*e^4 - 7350*b*c^3*d^4*e^5 + 4470*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 - 1
1150*(b^3*c + 3*a*b*c^2)*d^2*e^7 + 3597*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 + 7
8228*(a*b^3 + 3*a^2*b*c)*e^9)*m^2 + 144*(70*c^4*d^5*e^4 - 315*b*c^3*d^4*e^5 + 18
0*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 - 420*(b^3*c + 3*a*b*c^2)*d^2*e^7 + 126*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d*e^8 + 6499*(a*b^3 + 3*a^2*b*c)*e^9)*m)*x^4 - 4*(40369*a
^3*b*d^2*e^7 - 53669*a^4*d*e^8 + 120*(b^3*c + 3*a*b*c^2)*d^6*e^3 - 180*(b^4 + 12
*a*b^2*c + 6*a^2*c^2)*d^5*e^4 + 2910*(a*b^3 + 3*a^2*b*c)*d^4*e^5 - 5265*(3*a^2*b
^2 + 2*a^3*c)*d^3*e^6)*m^3 + 2*(120960*(3*a^2*b^2 + 2*a^3*c)*e^9 + (2*(a*b^3 + 3
*a^2*b*c)*d*e^8 + (3*a^2*b^2 + 2*a^3*c)*e^9)*m^8 - 2*((b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d^2*e^7 - 38*(a*b^3 + 3*a^2*b*c)*d*e^8 - 21*(3*a^2*b^2 + 2*a^3*c)*e^9)*m^7
+ 2*(20*(b^3*c + 3*a*b*c^2)*d^3*e^6 - 33*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7
+ 592*(a*b^3 + 3*a^2*b*c)*d*e^8 + 372*(3*a^2*b^2 + 2*a^3*c)*e^9)*m^6 - 2*(60*(3*
b^2*c^2 + 2*a*c^3)*d^4*e^5 - 540*(b^3*c + 3*a*b*c^2)*d^3*e^6 + 427*(b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^2*e^7 - 4850*(a*b^3 + 3*a^2*b*c)*d*e^8 - 3609*(3*a^2*b^2 + 2
*a^3*c)*e^9)*m^5 + (1680*b*c^3*d^5*e^4 - 2400*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 + 10
600*(b^3*c + 3*a*b*c^2)*d^3*e^6 - 5430*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7 +
44438*(a*b^3 + 3*a^2*b*c)*d*e^8 + 41619*(3*a^2*b^2 + 2*a^3*c)*e^9)*m^4 - 4*(840*
c^4*d^6*e^3 - 5040*b*c^3*d^5*e^4 + 3750*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 - 11250*(b
^3*c + 3*a*b*c^2)*d^3*e^6 + 4322*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7 - 27796*
(a*b^3 + 3*a^2*b*c)*d*e^8 - 36117*(3*a^2*b^2 + 2*a^3*c)*e^9)*m^3 - 4*(2520*c^4*d
^6*e^3 - 12180*b*c^3*d^5*e^4 + 7500*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 - 18940*(b^3*c
+ 3*a*b*c^2)*d^3*e^6 + 6186*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7 - 33954*(a*b
^3 + 3*a^2*b*c)*d*e^8 - 72569*(3*a^2*b^2 + 2*a^3*c)*e^9)*m^2 - 48*(140*c^4*d^6*e
^3 - 630*b*c^3*d^5*e^4 + 360*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 - 840*(b^3*c + 3*a*b*
c^2)*d^3*e^6 + 252*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7 - 1260*(a*b^3 + 3*a^2*
b*c)*d*e^8 - 6289*(3*a^2*b^2 + 2*a^3*c)*e^9)*m)*x^3 - 4*(133938*a^3*b*d^2*e^7 -
127251*a^4*d*e^8 - 360*(3*b^2*c^2 + 2*a*c^3)*d^7*e^2 + 2880*(b^3*c + 3*a*b*c^2)*
d^6*e^3 - 2010*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^4 + 19950*(a*b^3 + 3*a^2*b*c
)*d^4*e^5 - 24574*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6)*m^2 + 2*(362880*a^3*b*e^9 + (2*
a^3*b*e^9 + (3*a^2*b^2 + 2*a^3*c)*d*e^8)*m^8 + 2*(43*a^3*b*e^9 - 3*(a*b^3 + 3*a^
2*b*c)*d^2*e^7 + 20*(3*a^2*b^2 + 2*a^3*c)*d*e^8)*m^7 + 2*(784*a^3*b*e^9 + 3*(b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 108*(a*b^3 + 3*a^2*b*c)*d^2*e^7 + 332*(3*a^
2*b^2 + 2*a^3*c)*d*e^8)*m^6 + 2*(7882*a^3*b*e^9 - 60*(b^3*c + 3*a*b*c^2)*d^4*e^5
+ 93*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 1560*(a*b^3 + 3*a^2*b*c)*d^2*e^7
+ 2945*(3*a^2*b^2 + 2*a^3*c)*d*e^8)*m^5 + (95018*a^3*b*e^9 + 360*(3*b^2*c^2 + 2*
a*c^3)*d^5*e^4 - 3000*(b^3*c + 3*a*b*c^2)*d^4*e^5 + 2190*(b^4 + 12*a*b^2*c + 6*a
^2*c^2)*d^3*e^6 - 22860*(a*b^3 + 3*a^2*b*c)*d^2*e^7 + 29839*(3*a^2*b^2 + 2*a^3*c
)*d*e^8)*m^4 - 2*(2520*b*c^3*d^6*e^3 - 174307*a^3*b*e^9 - 3240*(3*b^2*c^2 + 2*a*
c^3)*d^5*e^4 + 12900*(b^3*c + 3*a*b*c^2)*d^4*e^5 - 5955*(b^4 + 12*a*b^2*c + 6*a^
2*c^2)*d^3*e^6 + 43797*(a*b^3 + 3*a^2*b*c)*d^2*e^7 - 42395*(3*a^2*b^2 + 2*a^3*c)
*d*e^8)*m^3 + 12*(840*c^4*d^7*e^2 - 4200*b*c^3*d^6*e^3 + 62511*a^3*b*e^9 + 2670*
(3*b^2*c^2 + 2*a*c^3)*d^5*e^4 - 6950*(b^3*c + 3*a*b*c^2)*d^4*e^5 + 2337*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 13197*(a*b^3 + 3*a^2*b*c)*d^2*e^7 + 10058*(3*a^
2*b^2 + 2*a^3*c)*d*e^8)*m^2 + 144*(70*c^4*d^7*e^2 - 315*b*c^3*d^6*e^3 + 5869*a^3
*b*e^9 + 180*(3*b^2*c^2 + 2*a*c^3)*d^5*e^4 - 420*(b^3*c + 3*a*b*c^2)*d^4*e^5 + 1
26*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 630*(a*b^3 + 3*a^2*b*c)*d^2*e^7 + 42
0*(3*a^2*b^2 + 2*a^3*c)*d*e^8)*m)*x^2 - 48*(420*b*c^3*d^8*e + 20094*a^3*b*d^2*e^
7 - 13827*a^4*d*e^8 - 510*(3*b^2*c^2 + 2*a*c^3)*d^7*e^2 + 1910*(b^3*c + 3*a*b*c^
2)*d^6*e^3 - 825*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^4 + 5637*(a*b^3 + 3*a^2*b*
c)*d^4*e^5 - 5018*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6)*m + (362880*a^4*e^9 + (4*a^3*b*
d*e^8 + a^4*e^9)*m^8 + 4*(42*a^3*b*d*e^8 + 11*a^4*e^9 - (3*a^2*b^2 + 2*a^3*c)*d^
2*e^7)*m^7 + 2*(1484*a^3*b*d*e^8 + 413*a^4*e^9 + 12*(a*b^3 + 3*a^2*b*c)*d^3*e^6
- 78*(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*m^6 + 4*(7140*a^3*b*d*e^8 + 2156*a^4*e^9 - 6
*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^5 + 210*(a*b^3 + 3*a^2*b*c)*d^3*e^6 - 625*
(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*m^5 + (161476*a^3*b*d*e^8 + 54649*a^4*e^9 + 480*(
b^3*c + 3*a*b*c^2)*d^5*e^4 - 720*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^5 + 11640*
(a*b^3 + 3*a^2*b*c)*d^3*e^6 - 21060*(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*m^4 + 4*(1339
38*a^3*b*d*e^8 + 53669*a^4*e^9 - 360*(3*b^2*c^2 + 2*a*c^3)*d^6*e^3 + 2880*(b^3*c
+ 3*a*b*c^2)*d^5*e^4 - 2010*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^5 + 19950*(a*b
^3 + 3*a^2*b*c)*d^3*e^6 - 24574*(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*m^3 + 12*(1680*b*
c^3*d^7*e^2 + 80376*a^3*b*d*e^8 + 42417*a^4*e^9 - 2040*(3*b^2*c^2 + 2*a*c^3)*d^6
*e^3 + 7640*(b^3*c + 3*a*b*c^2)*d^5*e^4 - 3300*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
4*e^5 + 22548*(a*b^3 + 3*a^2*b*c)*d^3*e^6 - 20072*(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)
*m^2 - 144*(280*c^4*d^8*e - 1260*b*c^3*d^7*e^2 - 5040*a^3*b*d*e^8 - 4609*a^4*e^9
+ 720*(3*b^2*c^2 + 2*a*c^3)*d^6*e^3 - 1680*(b^3*c + 3*a*b*c^2)*d^5*e^4 + 504*(b
^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^5 - 2520*(a*b^3 + 3*a^2*b*c)*d^3*e^6 + 1680*(
3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*m)*x)*(e*x + d)^m/(e^9*m^9 + 45*e^9*m^8 + 870*e^9*
m^7 + 9450*e^9*m^6 + 63273*e^9*m^5 + 269325*e^9*m^4 + 723680*e^9*m^3 + 1172700*e
^9*m^2 + 1026576*e^9*m + 362880*e^9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277179, 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)^4*(e*x + d)^m,x, algorithm="giac")

[Out]

Done