3.2539 \(\int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=305 \[ \frac{3 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+3)}-\frac{(2 c d-b e) (d+e x)^{m+4} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (m+4)}+\frac{3 c (d+e x)^{m+5} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+5)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^7 (m+1)}-\frac{3 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+2)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

((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))

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Rubi [A]  time = 0.541834, antiderivative size = 305, 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 \[ \frac{3 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+3)}-\frac{(2 c d-b e) (d+e x)^{m+4} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (m+4)}+\frac{3 c (d+e x)^{m+5} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+5)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^7 (m+1)}-\frac{3 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+2)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

((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))

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Rubi in Sympy [A]  time = 105.043, size = 289, normalized size = 0.95 \[ \frac{c^{3} \left (d + e x\right )^{m + 7}}{e^{7} \left (m + 7\right )} + \frac{3 c^{2} \left (d + e x\right )^{m + 6} \left (b e - 2 c d\right )}{e^{7} \left (m + 6\right )} + \frac{3 c \left (d + e x\right )^{m + 5} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \left (m + 5\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{7} \left (m + 1\right )} + \frac{3 \left (d + e x\right )^{m + 2} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \left (m + 2\right )} + \frac{3 \left (d + e x\right )^{m + 3} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 4} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{e^{7} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.51143, size = 791, normalized size = 2.59 \[ \frac{(d+e x)^{m+1} \left (3 c e^2 \left (m^2+13 m+42\right ) \left (a^2 e^2 \left (m^2+9 m+20\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+2 a b e (m+5) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+b^2 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )+e^3 \left (m^3+18 m^2+107 m+210\right ) \left (a^3 e^3 \left (m^3+9 m^2+26 m+24\right )+3 a^2 b e^2 \left (m^2+7 m+12\right ) (e (m+1) x-d)+3 a b^2 e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+b^3 \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )+3 c^2 e (m+7) \left (a e (m+6) \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )+b \left (-120 d^5+120 d^4 e (m+1) x-60 d^3 e^2 \left (m^2+3 m+2\right ) x^2+20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3-5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )\right )+c^3 \left (720 d^6-720 d^5 e (m+1) x+360 d^4 e^2 \left (m^2+3 m+2\right ) x^2-120 d^3 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+30 d^2 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-6 d e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right )\right )}{e^7 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(c^3*(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) +
e^3*(210 + 107*m + 18*m^2 + m^3)*(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)) + 3*c*e^2*(42 + 13*m + m^2)*
(a^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) +
2*a*b*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^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)) + 3*c^2*e*(7 + m)*(a*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*(-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))))/(e^7*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*
(7 + m))

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Maple [B]  time = 0.021, size = 2927, normalized size = 9.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)*(c^3*e^6*m^6*x^6+3*b*c^2*e^6*m^6*x^5+21*c^3*e^6*m^5*x^6+3*a*c^2*e^
6*m^6*x^4+3*b^2*c*e^6*m^6*x^4+66*b*c^2*e^6*m^5*x^5-6*c^3*d*e^5*m^5*x^5+175*c^3*e
^6*m^4*x^6+6*a*b*c*e^6*m^6*x^3+69*a*c^2*e^6*m^5*x^4+b^3*e^6*m^6*x^3+69*b^2*c*e^6
*m^5*x^4-15*b*c^2*d*e^5*m^5*x^4+570*b*c^2*e^6*m^4*x^5-90*c^3*d*e^5*m^4*x^5+735*c
^3*e^6*m^3*x^6+3*a^2*c*e^6*m^6*x^2+3*a*b^2*e^6*m^6*x^2+144*a*b*c*e^6*m^5*x^3-12*
a*c^2*d*e^5*m^5*x^3+621*a*c^2*e^6*m^4*x^4+24*b^3*e^6*m^5*x^3-12*b^2*c*d*e^5*m^5*
x^3+621*b^2*c*e^6*m^4*x^4-255*b*c^2*d*e^5*m^4*x^4+2460*b*c^2*e^6*m^3*x^5+30*c^3*
d^2*e^4*m^4*x^4-510*c^3*d*e^5*m^3*x^5+1624*c^3*e^6*m^2*x^6+3*a^2*b*e^6*m^6*x+75*
a^2*c*e^6*m^5*x^2+75*a*b^2*e^6*m^5*x^2-18*a*b*c*d*e^5*m^5*x^2+1356*a*b*c*e^6*m^4
*x^3-228*a*c^2*d*e^5*m^4*x^3+2775*a*c^2*e^6*m^3*x^4-3*b^3*d*e^5*m^5*x^2+226*b^3*
e^6*m^4*x^3-228*b^2*c*d*e^5*m^4*x^3+2775*b^2*c*e^6*m^3*x^4+60*b*c^2*d^2*e^4*m^4*
x^3-1575*b*c^2*d*e^5*m^3*x^4+5547*b*c^2*e^6*m^2*x^5+300*c^3*d^2*e^4*m^3*x^4-1350
*c^3*d*e^5*m^2*x^5+1764*c^3*e^6*m*x^6+a^3*e^6*m^6+78*a^2*b*e^6*m^5*x-6*a^2*c*d*e
^5*m^5*x+741*a^2*c*e^6*m^4*x^2-6*a*b^2*d*e^5*m^5*x+741*a*b^2*e^6*m^4*x^2-378*a*b
*c*d*e^5*m^4*x^2+6336*a*b*c*e^6*m^3*x^3+36*a*c^2*d^2*e^4*m^4*x^2-1572*a*c^2*d*e^
5*m^3*x^3+6432*a*c^2*e^6*m^2*x^4-63*b^3*d*e^5*m^4*x^2+1056*b^3*e^6*m^3*x^3+36*b^
2*c*d^2*e^4*m^4*x^2-1572*b^2*c*d*e^5*m^3*x^3+6432*b^2*c*e^6*m^2*x^4+780*b*c^2*d^
2*e^4*m^3*x^3-4425*b*c^2*d*e^5*m^2*x^4+6114*b*c^2*e^6*m*x^5-120*c^3*d^3*e^3*m^3*
x^3+1050*c^3*d^2*e^4*m^2*x^4-1644*c^3*d*e^5*m*x^5+720*c^3*e^6*x^6+27*a^3*e^6*m^5
-3*a^2*b*d*e^5*m^5+810*a^2*b*e^6*m^4*x-138*a^2*c*d*e^5*m^4*x+3657*a^2*c*e^6*m^3*
x^2-138*a*b^2*d*e^5*m^4*x+3657*a*b^2*e^6*m^3*x^2+36*a*b*c*d^2*e^4*m^4*x-2934*a*b
*c*d*e^5*m^3*x^2+15270*a*b*c*e^6*m^2*x^3+576*a*c^2*d^2*e^4*m^3*x^2-4812*a*c^2*d*
e^5*m^2*x^3+7236*a*c^2*e^6*m*x^4+6*b^3*d^2*e^4*m^4*x-489*b^3*d*e^5*m^3*x^2+2545*
b^3*e^6*m^2*x^3+576*b^2*c*d^2*e^4*m^3*x^2-4812*b^2*c*d*e^5*m^2*x^3+7236*b^2*c*e^
6*m*x^4-180*b*c^2*d^3*e^3*m^3*x^2+3180*b*c^2*d^2*e^4*m^2*x^3-5610*b*c^2*d*e^5*m*
x^4+2520*b*c^2*e^6*x^5-720*c^3*d^3*e^3*m^2*x^3+1500*c^3*d^2*e^4*m*x^4-720*c^3*d*
e^5*x^5+295*a^3*e^6*m^4-75*a^2*b*d*e^5*m^4+4260*a^2*b*e^6*m^3*x+6*a^2*c*d^2*e^4*
m^4-1206*a^2*c*d*e^5*m^3*x+9336*a^2*c*e^6*m^2*x^2+6*a*b^2*d^2*e^4*m^4-1206*a*b^2
*d*e^5*m^3*x+9336*a*b^2*e^6*m^2*x^2+684*a*b*c*d^2*e^4*m^3*x-10206*a*b*c*d*e^5*m^
2*x^2+17712*a*b*c*e^6*m*x^3-72*a*c^2*d^3*e^3*m^3*x+2988*a*c^2*d^2*e^4*m^2*x^2-64
80*a*c^2*d*e^5*m*x^3+3024*a*c^2*e^6*x^4+114*b^3*d^2*e^4*m^3*x-1701*b^3*d*e^5*m^2
*x^2+2952*b^3*e^6*m*x^3-72*b^2*c*d^3*e^3*m^3*x+2988*b^2*c*d^2*e^4*m^2*x^2-6480*b
^2*c*d*e^5*m*x^3+3024*b^2*c*e^6*x^4-1800*b*c^2*d^3*e^3*m^2*x^2+4980*b*c^2*d^2*e^
4*m*x^3-2520*b*c^2*d*e^5*x^4+360*c^3*d^4*e^2*m^2*x^2-1320*c^3*d^3*e^3*m*x^3+720*
c^3*d^2*e^4*x^4+1665*a^3*e^6*m^3-735*a^2*b*d*e^5*m^3+11787*a^2*b*e^6*m^2*x+132*a
^2*c*d^2*e^4*m^3-4902*a^2*c*d*e^5*m^2*x+11388*a^2*c*e^6*m*x^2+132*a*b^2*d^2*e^4*
m^3-4902*a*b^2*d*e^5*m^2*x+11388*a*b^2*e^6*m*x^2-36*a*b*c*d^3*e^3*m^3+4500*a*b*c
*d^2*e^4*m^2*x-15192*a*b*c*d*e^5*m*x^2+7560*a*b*c*e^6*x^3-1008*a*c^2*d^3*e^3*m^2
*x+5472*a*c^2*d^2*e^4*m*x^2-3024*a*c^2*d*e^5*x^3-6*b^3*d^3*e^3*m^3+750*b^3*d^2*e
^4*m^2*x-2532*b^3*d*e^5*m*x^2+1260*b^3*e^6*x^3-1008*b^2*c*d^3*e^3*m^2*x+5472*b^2
*c*d^2*e^4*m*x^2-3024*b^2*c*d*e^5*x^3+360*b*c^2*d^4*e^2*m^2*x-4140*b*c^2*d^3*e^3
*m*x^2+2520*b*c^2*d^2*e^4*x^3+1080*c^3*d^4*e^2*m*x^2-720*c^3*d^3*e^3*x^3+5104*a^
3*e^6*m^2-3525*a^2*b*d*e^5*m^2+15822*a^2*b*e^6*m*x+1074*a^2*c*d^2*e^4*m^2-8868*a
^2*c*d*e^5*m*x+5040*a^2*c*e^6*x^2+1074*a*b^2*d^2*e^4*m^2-8868*a*b^2*d*e^5*m*x+50
40*a*b^2*e^6*x^2-648*a*b*c*d^3*e^3*m^2+11412*a*b*c*d^2*e^4*m*x-7560*a*b*c*d*e^5*
x^2+72*a*c^2*d^4*e^2*m^2-3960*a*c^2*d^3*e^3*m*x+3024*a*c^2*d^2*e^4*x^2-108*b^3*d
^3*e^3*m^2+1902*b^3*d^2*e^4*m*x-1260*b^3*d*e^5*x^2+72*b^2*c*d^4*e^2*m^2-3960*b^2
*c*d^3*e^3*m*x+3024*b^2*c*d^2*e^4*x^2+2880*b*c^2*d^4*e^2*m*x-2520*b*c^2*d^3*e^3*
x^2-720*c^3*d^5*e*m*x+720*c^3*d^4*e^2*x^2+8028*a^3*e^6*m-8262*a^2*b*d*e^5*m+7560
*a^2*b*e^6*x+3828*a^2*c*d^2*e^4*m-5040*a^2*c*d*e^5*x+3828*a*b^2*d^2*e^4*m-5040*a
*b^2*d*e^5*x-3852*a*b*c*d^3*e^3*m+7560*a*b*c*d^2*e^4*x+936*a*c^2*d^4*e^2*m-3024*
a*c^2*d^3*e^3*x-642*b^3*d^3*e^3*m+1260*b^3*d^2*e^4*x+936*b^2*c*d^4*e^2*m-3024*b^
2*c*d^3*e^3*x-360*b*c^2*d^5*e*m+2520*b*c^2*d^4*e^2*x-720*c^3*d^5*e*x+5040*a^3*e^
6-7560*a^2*b*d*e^5+5040*a^2*c*d^2*e^4+5040*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+3024*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+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)

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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*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262827, size = 3443, normalized size = 11.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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 + 5040*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 - 59*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 + a*c^2)*d*e^6 + 509*(b^3 + 6*a*b*c)*e^7)*m
^2 + 18*(10*c^3*d^3*e^4 - 35*b*c^2*d^2*e^5 + 42*(b^2*c + a*c^2)*d*e^6 + 164*(b^3
+ 6*a*b*c)*e^7)*m)*x^4 - 3*(245*a^2*b*d^2*e^5 - 555*a^3*d*e^6 + 2*(b^3 + 6*a*b*
c)*d^4*e^3 - 44*(a*b^2 + a^2*c)*d^3*e^4)*m^3 + (5040*(a*b^2 + a^2*c)*e^7 + ((b^3
+ 6*a*b*c)*d*e^6 + 3*(a*b^2 + a^2*c)*e^7)*m^6 - 3*(4*(b^2*c + a*c^2)*d^2*e^5 -
7*(b^3 + 6*a*b*c)*d*e^6 - 25*(a*b^2 + a^2*c)*e^7)*m^5 + (60*b*c^2*d^3*e^4 - 192*
(b^2*c + a*c^2)*d^2*e^5 + 163*(b^3 + 6*a*b*c)*d*e^6 + 741*(a*b^2 + a^2*c)*e^7)*m
^4 - 3*(40*c^3*d^4*e^3 - 200*b*c^2*d^3*e^4 + 332*(b^2*c + a*c^2)*d^2*e^5 - 189*(
b^3 + 6*a*b*c)*d*e^6 - 1219*(a*b^2 + a^2*c)*e^7)*m^3 - 4*(90*c^3*d^4*e^3 - 345*b
*c^2*d^3*e^4 + 456*(b^2*c + a*c^2)*d^2*e^5 - 211*(b^3 + 6*a*b*c)*d*e^6 - 2334*(a
*b^2 + a^2*c)*e^7)*m^2 - 12*(20*c^3*d^4*e^3 - 70*b*c^2*d^3*e^4 + 84*(b^2*c + a*c
^2)*d^2*e^5 - 35*(b^3 + 6*a*b*c)*d*e^6 - 949*(a*b^2 + a^2*c)*e^7)*m)*x^3 - (3525
*a^2*b*d^2*e^5 - 5104*a^3*d*e^6 - 72*(b^2*c + a*c^2)*d^5*e^2 + 108*(b^3 + 6*a*b*
c)*d^4*e^3 - 1074*(a*b^2 + a^2*c)*d^3*e^4)*m^2 + 3*(2520*a^2*b*e^7 + (a^2*b*e^7
+ (a*b^2 + a^2*c)*d*e^6)*m^6 + (26*a^2*b*e^7 - (b^3 + 6*a*b*c)*d^2*e^5 + 23*(a*b
^2 + a^2*c)*d*e^6)*m^5 + (270*a^2*b*e^7 + 12*(b^2*c + a*c^2)*d^3*e^4 - 19*(b^3 +
6*a*b*c)*d^2*e^5 + 201*(a*b^2 + a^2*c)*d*e^6)*m^4 - (60*b*c^2*d^4*e^3 - 1420*a^
2*b*e^7 - 168*(b^2*c + a*c^2)*d^3*e^4 + 125*(b^3 + 6*a*b*c)*d^2*e^5 - 817*(a*b^2
+ a^2*c)*d*e^6)*m^3 + (120*c^3*d^5*e^2 - 480*b*c^2*d^4*e^3 + 3929*a^2*b*e^7 + 6
60*(b^2*c + a*c^2)*d^3*e^4 - 317*(b^3 + 6*a*b*c)*d^2*e^5 + 1478*(a*b^2 + a^2*c)*
d*e^6)*m^2 + 6*(20*c^3*d^5*e^2 - 70*b*c^2*d^4*e^3 + 879*a^2*b*e^7 + 84*(b^2*c +
a*c^2)*d^3*e^4 - 35*(b^3 + 6*a*b*c)*d^2*e^5 + 140*(a*b^2 + a^2*c)*d*e^6)*m)*x^2
- 6*(60*b*c^2*d^6*e + 1377*a^2*b*d^2*e^5 - 1338*a^3*d*e^6 - 156*(b^2*c + a*c^2)*
d^5*e^2 + 107*(b^3 + 6*a*b*c)*d^4*e^3 - 638*(a*b^2 + a^2*c)*d^3*e^4)*m + (5040*a
^3*e^7 + (3*a^2*b*d*e^6 + a^3*e^7)*m^6 + 3*(25*a^2*b*d*e^6 + 9*a^3*e^7 - 2*(a*b^
2 + a^2*c)*d^2*e^5)*m^5 + (735*a^2*b*d*e^6 + 295*a^3*e^7 + 6*(b^3 + 6*a*b*c)*d^3
*e^4 - 132*(a*b^2 + a^2*c)*d^2*e^5)*m^4 + 3*(1175*a^2*b*d*e^6 + 555*a^3*e^7 - 24
*(b^2*c + a*c^2)*d^4*e^3 + 36*(b^3 + 6*a*b*c)*d^3*e^4 - 358*(a*b^2 + a^2*c)*d^2*
e^5)*m^3 + 2*(180*b*c^2*d^5*e^2 + 4131*a^2*b*d*e^6 + 2552*a^3*e^7 - 468*(b^2*c +
a*c^2)*d^4*e^3 + 321*(b^3 + 6*a*b*c)*d^3*e^4 - 1914*(a*b^2 + a^2*c)*d^2*e^5)*m^
2 - 36*(20*c^3*d^6*e - 70*b*c^2*d^5*e^2 - 210*a^2*b*d*e^6 - 223*a^3*e^7 + 84*(b^
2*c + a*c^2)*d^4*e^3 - 35*(b^3 + 6*a*b*c)*d^3*e^4 + 140*(a*b^2 + a^2*c)*d^2*e^5)
*m)*x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7
*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230064, 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*(e*x + d)^m,x, algorithm="giac")

[Out]

Done