3.1720 \(\int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=206 \[ -\frac{6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}+\frac{(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{b^6 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

((b*d - a*e)^6*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*b*(b*d - a*e)^5*(d + e*x)^(
2 + m))/(e^7*(2 + m)) + (15*b^2*(b*d - a*e)^4*(d + e*x)^(3 + m))/(e^7*(3 + m)) -
(20*b^3*(b*d - a*e)^3*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (15*b^4*(b*d - a*e)^2*
(d + e*x)^(5 + m))/(e^7*(5 + m)) - (6*b^5*(b*d - a*e)*(d + e*x)^(6 + m))/(e^7*(6
+ m)) + (b^6*(d + e*x)^(7 + m))/(e^7*(7 + m))

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Rubi [A]  time = 0.303618, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}+\frac{(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{b^6 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)^6*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*b*(b*d - a*e)^5*(d + e*x)^(
2 + m))/(e^7*(2 + m)) + (15*b^2*(b*d - a*e)^4*(d + e*x)^(3 + m))/(e^7*(3 + m)) -
(20*b^3*(b*d - a*e)^3*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (15*b^4*(b*d - a*e)^2*
(d + e*x)^(5 + m))/(e^7*(5 + m)) - (6*b^5*(b*d - a*e)*(d + e*x)^(6 + m))/(e^7*(6
+ m)) + (b^6*(d + e*x)^(7 + m))/(e^7*(7 + m))

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Rubi in Sympy [A]  time = 120.628, size = 182, normalized size = 0.88 \[ \frac{b^{6} \left (d + e x\right )^{m + 7}}{e^{7} \left (m + 7\right )} + \frac{6 b^{5} \left (d + e x\right )^{m + 6} \left (a e - b d\right )}{e^{7} \left (m + 6\right )} + \frac{15 b^{4} \left (d + e x\right )^{m + 5} \left (a e - b d\right )^{2}}{e^{7} \left (m + 5\right )} + \frac{20 b^{3} \left (d + e x\right )^{m + 4} \left (a e - b d\right )^{3}}{e^{7} \left (m + 4\right )} + \frac{15 b^{2} \left (d + e x\right )^{m + 3} \left (a e - b d\right )^{4}}{e^{7} \left (m + 3\right )} + \frac{6 b \left (d + e x\right )^{m + 2} \left (a e - b d\right )^{5}}{e^{7} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right )^{6}}{e^{7} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**6*(d + e*x)**(m + 7)/(e**7*(m + 7)) + 6*b**5*(d + e*x)**(m + 6)*(a*e - b*d)/(
e**7*(m + 6)) + 15*b**4*(d + e*x)**(m + 5)*(a*e - b*d)**2/(e**7*(m + 5)) + 20*b*
*3*(d + e*x)**(m + 4)*(a*e - b*d)**3/(e**7*(m + 4)) + 15*b**2*(d + e*x)**(m + 3)
*(a*e - b*d)**4/(e**7*(m + 3)) + 6*b*(d + e*x)**(m + 2)*(a*e - b*d)**5/(e**7*(m
+ 2)) + (d + e*x)**(m + 1)*(a*e - b*d)**6/(e**7*(m + 1))

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Mathematica [B]  time = 0.891793, size = 646, normalized size = 3.14 \[ \frac{(d+e x)^{m+1} \left (a^6 e^6 \left (m^6+27 m^5+295 m^4+1665 m^3+5104 m^2+8028 m+5040\right )-6 a^5 b e^5 \left (m^5+25 m^4+245 m^3+1175 m^2+2754 m+2520\right ) (d-e (m+1) x)+15 a^4 b^2 e^4 \left (m^4+22 m^3+179 m^2+638 m+840\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+20 a^3 b^3 e^3 \left (m^3+18 m^2+107 m+210\right ) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+15 a^2 b^4 e^2 \left (m^2+13 m+42\right ) \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )+6 a b^5 e (m+7) \left (-120 d^5+120 d^4 e (m+1) x-60 d^3 e^2 \left (m^2+3 m+2\right ) x^2+20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3-5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )+b^6 \left (720 d^6-720 d^5 e (m+1) x+360 d^4 e^2 \left (m^2+3 m+2\right ) x^2-120 d^3 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+30 d^2 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-6 d e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right )\right )}{e^7 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a^6*e^6*(5040 + 8028*m + 5104*m^2 + 1665*m^3 + 295*m^4 + 27*
m^5 + m^6) - 6*a^5*b*e^5*(2520 + 2754*m + 1175*m^2 + 245*m^3 + 25*m^4 + m^5)*(d
- e*(1 + m)*x) + 15*a^4*b^2*e^4*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)*(2*d^2 -
2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 20*a^3*b^3*e^3*(210 + 107*m + 18*m^
2 + m^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) + 15*a^2*b^4*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) + 6*a*b^5*e*(7 + m)*(-120*d^5 + 12
0*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^6*(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^7*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7
+ m))

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Maple [B]  time = 0.023, size = 2157, normalized size = 10.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)*(b^6*e^6*m^6*x^6+6*a*b^5*e^6*m^6*x^5+21*b^6*e^6*m^5*x^6+15*a^2*b^4
*e^6*m^6*x^4+132*a*b^5*e^6*m^5*x^5-6*b^6*d*e^5*m^5*x^5+175*b^6*e^6*m^4*x^6+20*a^
3*b^3*e^6*m^6*x^3+345*a^2*b^4*e^6*m^5*x^4-30*a*b^5*d*e^5*m^5*x^4+1140*a*b^5*e^6*
m^4*x^5-90*b^6*d*e^5*m^4*x^5+735*b^6*e^6*m^3*x^6+15*a^4*b^2*e^6*m^6*x^2+480*a^3*
b^3*e^6*m^5*x^3-60*a^2*b^4*d*e^5*m^5*x^3+3105*a^2*b^4*e^6*m^4*x^4-510*a*b^5*d*e^
5*m^4*x^4+4920*a*b^5*e^6*m^3*x^5+30*b^6*d^2*e^4*m^4*x^4-510*b^6*d*e^5*m^3*x^5+16
24*b^6*e^6*m^2*x^6+6*a^5*b*e^6*m^6*x+375*a^4*b^2*e^6*m^5*x^2-60*a^3*b^3*d*e^5*m^
5*x^2+4520*a^3*b^3*e^6*m^4*x^3-1140*a^2*b^4*d*e^5*m^4*x^3+13875*a^2*b^4*e^6*m^3*
x^4+120*a*b^5*d^2*e^4*m^4*x^3-3150*a*b^5*d*e^5*m^3*x^4+11094*a*b^5*e^6*m^2*x^5+3
00*b^6*d^2*e^4*m^3*x^4-1350*b^6*d*e^5*m^2*x^5+1764*b^6*e^6*m*x^6+a^6*e^6*m^6+156
*a^5*b*e^6*m^5*x-30*a^4*b^2*d*e^5*m^5*x+3705*a^4*b^2*e^6*m^4*x^2-1260*a^3*b^3*d*
e^5*m^4*x^2+21120*a^3*b^3*e^6*m^3*x^3+180*a^2*b^4*d^2*e^4*m^4*x^2-7860*a^2*b^4*d
*e^5*m^3*x^3+32160*a^2*b^4*e^6*m^2*x^4+1560*a*b^5*d^2*e^4*m^3*x^3-8850*a*b^5*d*e
^5*m^2*x^4+12228*a*b^5*e^6*m*x^5-120*b^6*d^3*e^3*m^3*x^3+1050*b^6*d^2*e^4*m^2*x^
4-1644*b^6*d*e^5*m*x^5+720*b^6*e^6*x^6+27*a^6*e^6*m^5-6*a^5*b*d*e^5*m^5+1620*a^5
*b*e^6*m^4*x-690*a^4*b^2*d*e^5*m^4*x+18285*a^4*b^2*e^6*m^3*x^2+120*a^3*b^3*d^2*e
^4*m^4*x-9780*a^3*b^3*d*e^5*m^3*x^2+50900*a^3*b^3*e^6*m^2*x^3+2880*a^2*b^4*d^2*e
^4*m^3*x^2-24060*a^2*b^4*d*e^5*m^2*x^3+36180*a^2*b^4*e^6*m*x^4-360*a*b^5*d^3*e^3
*m^3*x^2+6360*a*b^5*d^2*e^4*m^2*x^3-11220*a*b^5*d*e^5*m*x^4+5040*a*b^5*e^6*x^5-7
20*b^6*d^3*e^3*m^2*x^3+1500*b^6*d^2*e^4*m*x^4-720*b^6*d*e^5*x^5+295*a^6*e^6*m^4-
150*a^5*b*d*e^5*m^4+8520*a^5*b*e^6*m^3*x+30*a^4*b^2*d^2*e^4*m^4-6030*a^4*b^2*d*e
^5*m^3*x+46680*a^4*b^2*e^6*m^2*x^2+2280*a^3*b^3*d^2*e^4*m^3*x-34020*a^3*b^3*d*e^
5*m^2*x^2+59040*a^3*b^3*e^6*m*x^3-360*a^2*b^4*d^3*e^3*m^3*x+14940*a^2*b^4*d^2*e^
4*m^2*x^2-32400*a^2*b^4*d*e^5*m*x^3+15120*a^2*b^4*e^6*x^4-3600*a*b^5*d^3*e^3*m^2
*x^2+9960*a*b^5*d^2*e^4*m*x^3-5040*a*b^5*d*e^5*x^4+360*b^6*d^4*e^2*m^2*x^2-1320*
b^6*d^3*e^3*m*x^3+720*b^6*d^2*e^4*x^4+1665*a^6*e^6*m^3-1470*a^5*b*d*e^5*m^3+2357
4*a^5*b*e^6*m^2*x+660*a^4*b^2*d^2*e^4*m^3-24510*a^4*b^2*d*e^5*m^2*x+56940*a^4*b^
2*e^6*m*x^2-120*a^3*b^3*d^3*e^3*m^3+15000*a^3*b^3*d^2*e^4*m^2*x-50640*a^3*b^3*d*
e^5*m*x^2+25200*a^3*b^3*e^6*x^3-5040*a^2*b^4*d^3*e^3*m^2*x+27360*a^2*b^4*d^2*e^4
*m*x^2-15120*a^2*b^4*d*e^5*x^3+720*a*b^5*d^4*e^2*m^2*x-8280*a*b^5*d^3*e^3*m*x^2+
5040*a*b^5*d^2*e^4*x^3+1080*b^6*d^4*e^2*m*x^2-720*b^6*d^3*e^3*x^3+5104*a^6*e^6*m
^2-7050*a^5*b*d*e^5*m^2+31644*a^5*b*e^6*m*x+5370*a^4*b^2*d^2*e^4*m^2-44340*a^4*b
^2*d*e^5*m*x+25200*a^4*b^2*e^6*x^2-2160*a^3*b^3*d^3*e^3*m^2+38040*a^3*b^3*d^2*e^
4*m*x-25200*a^3*b^3*d*e^5*x^2+360*a^2*b^4*d^4*e^2*m^2-19800*a^2*b^4*d^3*e^3*m*x+
15120*a^2*b^4*d^2*e^4*x^2+5760*a*b^5*d^4*e^2*m*x-5040*a*b^5*d^3*e^3*x^2-720*b^6*
d^5*e*m*x+720*b^6*d^4*e^2*x^2+8028*a^6*e^6*m-16524*a^5*b*d*e^5*m+15120*a^5*b*e^6
*x+19140*a^4*b^2*d^2*e^4*m-25200*a^4*b^2*d*e^5*x-12840*a^3*b^3*d^3*e^3*m+25200*a
^3*b^3*d^2*e^4*x+4680*a^2*b^4*d^4*e^2*m-15120*a^2*b^4*d^3*e^3*x-720*a*b^5*d^5*e*
m+5040*a*b^5*d^4*e^2*x-720*b^6*d^5*e*x+5040*a^6*e^6-15120*a^5*b*d*e^5+25200*a^4*
b^2*d^2*e^4-25200*a^3*b^3*d^3*e^3+15120*a^2*b^4*d^4*e^2-5040*a*b^5*d^5*e+720*b^6
*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((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234653, size = 3011, normalized size = 14.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^6*d*e^6*m^6 + 720*b^6*d^7 - 5040*a*b^5*d^6*e + 15120*a^2*b^4*d^5*e^2 - 25200*
a^3*b^3*d^4*e^3 + 25200*a^4*b^2*d^3*e^4 - 15120*a^5*b*d^2*e^5 + 5040*a^6*d*e^6 +
(b^6*e^7*m^6 + 21*b^6*e^7*m^5 + 175*b^6*e^7*m^4 + 735*b^6*e^7*m^3 + 1624*b^6*e^
7*m^2 + 1764*b^6*e^7*m + 720*b^6*e^7)*x^7 + (5040*a*b^5*e^7 + (b^6*d*e^6 + 6*a*b
^5*e^7)*m^6 + 3*(5*b^6*d*e^6 + 44*a*b^5*e^7)*m^5 + 5*(17*b^6*d*e^6 + 228*a*b^5*e
^7)*m^4 + 15*(15*b^6*d*e^6 + 328*a*b^5*e^7)*m^3 + 2*(137*b^6*d*e^6 + 5547*a*b^5*
e^7)*m^2 + 12*(10*b^6*d*e^6 + 1019*a*b^5*e^7)*m)*x^6 - 3*(2*a^5*b*d^2*e^5 - 9*a^
6*d*e^6)*m^5 + 3*(5040*a^2*b^4*e^7 + (2*a*b^5*d*e^6 + 5*a^2*b^4*e^7)*m^6 - (2*b^
6*d^2*e^5 - 34*a*b^5*d*e^6 - 115*a^2*b^4*e^7)*m^5 - 5*(4*b^6*d^2*e^5 - 42*a*b^5*
d*e^6 - 207*a^2*b^4*e^7)*m^4 - 5*(14*b^6*d^2*e^5 - 118*a*b^5*d*e^6 - 925*a^2*b^4
*e^7)*m^3 - 4*(25*b^6*d^2*e^5 - 187*a*b^5*d*e^6 - 2680*a^2*b^4*e^7)*m^2 - 12*(4*
b^6*d^2*e^5 - 28*a*b^5*d*e^6 - 1005*a^2*b^4*e^7)*m)*x^5 + 5*(6*a^4*b^2*d^3*e^4 -
30*a^5*b*d^2*e^5 + 59*a^6*d*e^6)*m^4 + 5*(5040*a^3*b^3*e^7 + (3*a^2*b^4*d*e^6 +
4*a^3*b^3*e^7)*m^6 - 3*(2*a*b^5*d^2*e^5 - 19*a^2*b^4*d*e^6 - 32*a^3*b^3*e^7)*m^
5 + (6*b^6*d^3*e^4 - 78*a*b^5*d^2*e^5 + 393*a^2*b^4*d*e^6 + 904*a^3*b^3*e^7)*m^4
+ 3*(12*b^6*d^3*e^4 - 106*a*b^5*d^2*e^5 + 401*a^2*b^4*d*e^6 + 1408*a^3*b^3*e^7)
*m^3 + 2*(33*b^6*d^3*e^4 - 249*a*b^5*d^2*e^5 + 810*a^2*b^4*d*e^6 + 5090*a^3*b^3*
e^7)*m^2 + 36*(b^6*d^3*e^4 - 7*a*b^5*d^2*e^5 + 21*a^2*b^4*d*e^6 + 328*a^3*b^3*e^
7)*m)*x^4 - 15*(8*a^3*b^3*d^4*e^3 - 44*a^4*b^2*d^3*e^4 + 98*a^5*b*d^2*e^5 - 111*
a^6*d*e^6)*m^3 + 5*(5040*a^4*b^2*e^7 + (4*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*m^6 - 3
*(4*a^2*b^4*d^2*e^5 - 28*a^3*b^3*d*e^6 - 25*a^4*b^2*e^7)*m^5 + (24*a*b^5*d^3*e^4
- 192*a^2*b^4*d^2*e^5 + 652*a^3*b^3*d*e^6 + 741*a^4*b^2*e^7)*m^4 - 3*(8*b^6*d^4
*e^3 - 80*a*b^5*d^3*e^4 + 332*a^2*b^4*d^2*e^5 - 756*a^3*b^3*d*e^6 - 1219*a^4*b^2
*e^7)*m^3 - 8*(9*b^6*d^4*e^3 - 69*a*b^5*d^3*e^4 + 228*a^2*b^4*d^2*e^5 - 422*a^3*
b^3*d*e^6 - 1167*a^4*b^2*e^7)*m^2 - 12*(4*b^6*d^4*e^3 - 28*a*b^5*d^3*e^4 + 84*a^
2*b^4*d^2*e^5 - 140*a^3*b^3*d*e^6 - 949*a^4*b^2*e^7)*m)*x^3 + 2*(180*a^2*b^4*d^5
*e^2 - 1080*a^3*b^3*d^4*e^3 + 2685*a^4*b^2*d^3*e^4 - 3525*a^5*b*d^2*e^5 + 2552*a
^6*d*e^6)*m^2 + 3*(5040*a^5*b*e^7 + (5*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*m^6 - (20*a^
3*b^3*d^2*e^5 - 115*a^4*b^2*d*e^6 - 52*a^5*b*e^7)*m^5 + 5*(12*a^2*b^4*d^3*e^4 -
76*a^3*b^3*d^2*e^5 + 201*a^4*b^2*d*e^6 + 108*a^5*b*e^7)*m^4 - 5*(24*a*b^5*d^4*e^
3 - 168*a^2*b^4*d^3*e^4 + 500*a^3*b^3*d^2*e^5 - 817*a^4*b^2*d*e^6 - 568*a^5*b*e^
7)*m^3 + 2*(60*b^6*d^5*e^2 - 480*a*b^5*d^4*e^3 + 1650*a^2*b^4*d^3*e^4 - 3170*a^3
*b^3*d^2*e^5 + 3695*a^4*b^2*d*e^6 + 3929*a^5*b*e^7)*m^2 + 12*(10*b^6*d^5*e^2 - 7
0*a*b^5*d^4*e^3 + 210*a^2*b^4*d^3*e^4 - 350*a^3*b^3*d^2*e^5 + 350*a^4*b^2*d*e^6
+ 879*a^5*b*e^7)*m)*x^2 - 12*(60*a*b^5*d^6*e - 390*a^2*b^4*d^5*e^2 + 1070*a^3*b^
3*d^4*e^3 - 1595*a^4*b^2*d^3*e^4 + 1377*a^5*b*d^2*e^5 - 669*a^6*d*e^6)*m + (5040
*a^6*e^7 + (6*a^5*b*d*e^6 + a^6*e^7)*m^6 - 3*(10*a^4*b^2*d^2*e^5 - 50*a^5*b*d*e^
6 - 9*a^6*e^7)*m^5 + 5*(24*a^3*b^3*d^3*e^4 - 132*a^4*b^2*d^2*e^5 + 294*a^5*b*d*e
^6 + 59*a^6*e^7)*m^4 - 15*(24*a^2*b^4*d^4*e^3 - 144*a^3*b^3*d^3*e^4 + 358*a^4*b^
2*d^2*e^5 - 470*a^5*b*d*e^6 - 111*a^6*e^7)*m^3 + 4*(180*a*b^5*d^5*e^2 - 1170*a^2
*b^4*d^4*e^3 + 3210*a^3*b^3*d^3*e^4 - 4785*a^4*b^2*d^2*e^5 + 4131*a^5*b*d*e^6 +
1276*a^6*e^7)*m^2 - 36*(20*b^6*d^6*e - 140*a*b^5*d^5*e^2 + 420*a^2*b^4*d^4*e^3 -
700*a^3*b^3*d^3*e^4 + 700*a^4*b^2*d^2*e^5 - 420*a^5*b*d*e^6 - 223*a^6*e^7)*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)

_______________________________________________________________________________________

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*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222842, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done