3.2073 \(\int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx\)

Optimal. Leaf size=130 \[ -\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+6}}{e^4 (m+6)}-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac{c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)} \]

[Out]

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

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Rubi [A]  time = 0.273802, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+6}}{e^4 (m+6)}-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac{c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 61.152, size = 114, normalized size = 0.88 \[ \frac{c^{3} d^{3} \left (d + e x\right )^{m + 7}}{e^{4} \left (m + 7\right )} + \frac{3 c^{2} d^{2} \left (d + e x\right )^{m + 6} \left (a e^{2} - c d^{2}\right )}{e^{4} \left (m + 6\right )} + \frac{3 c d \left (d + e x\right )^{m + 5} \left (a e^{2} - c d^{2}\right )^{2}}{e^{4} \left (m + 5\right )} + \frac{\left (d + e x\right )^{m + 4} \left (a e^{2} - c d^{2}\right )^{3}}{e^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.328631, size = 187, normalized size = 1.44 \[ \frac{(d+e x)^{m+4} \left (a^3 e^6 \left (m^3+18 m^2+107 m+210\right )-3 a^2 c d e^4 \left (m^2+13 m+42\right ) (d-e (m+4) x)+3 a c^2 d^2 e^2 (m+7) \left (2 d^2-2 d e (m+4) x+e^2 \left (m^2+9 m+20\right ) x^2\right )-c^3 d^3 \left (6 d^3-6 d^2 e (m+4) x+3 d e^2 \left (m^2+9 m+20\right ) x^2-e^3 \left (m^3+15 m^2+74 m+120\right ) x^3\right )\right )}{e^4 (m+4) (m+5) (m+6) (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(4 + m)*(a^3*e^6*(210 + 107*m + 18*m^2 + m^3) - 3*a^2*c*d*e^4*(42 + 1
3*m + m^2)*(d - e*(4 + m)*x) + 3*a*c^2*d^2*e^2*(7 + m)*(2*d^2 - 2*d*e*(4 + m)*x
+ e^2*(20 + 9*m + m^2)*x^2) - c^3*d^3*(6*d^3 - 6*d^2*e*(4 + m)*x + 3*d*e^2*(20 +
9*m + m^2)*x^2 - e^3*(120 + 74*m + 15*m^2 + m^3)*x^3)))/(e^4*(4 + m)*(5 + m)*(6
+ m)*(7 + m))

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Maple [B]  time = 0.016, size = 436, normalized size = 3.4 \[{\frac{ \left ( ex+d \right ) ^{4+m} \left ({c}^{3}{d}^{3}{e}^{3}{m}^{3}{x}^{3}+3\,a{c}^{2}{d}^{2}{e}^{4}{m}^{3}{x}^{2}+15\,{c}^{3}{d}^{3}{e}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}cd{e}^{5}{m}^{3}x+48\,a{c}^{2}{d}^{2}{e}^{4}{m}^{2}{x}^{2}-3\,{c}^{3}{d}^{4}{e}^{2}{m}^{2}{x}^{2}+74\,{c}^{3}{d}^{3}{e}^{3}m{x}^{3}+{a}^{3}{e}^{6}{m}^{3}+51\,{a}^{2}cd{e}^{5}{m}^{2}x-6\,a{c}^{2}{d}^{3}{e}^{3}{m}^{2}x+249\,a{c}^{2}{d}^{2}{e}^{4}m{x}^{2}-27\,{c}^{3}{d}^{4}{e}^{2}m{x}^{2}+120\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+18\,{a}^{3}{e}^{6}{m}^{2}-3\,{a}^{2}c{d}^{2}{e}^{4}{m}^{2}+282\,{a}^{2}cd{e}^{5}mx-66\,a{c}^{2}{d}^{3}{e}^{3}mx+420\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}+6\,{c}^{3}{d}^{5}emx-60\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+107\,{a}^{3}{e}^{6}m-39\,{a}^{2}c{d}^{2}{e}^{4}m+504\,x{a}^{2}cd{e}^{5}+6\,a{c}^{2}{d}^{4}{e}^{2}m-168\,xa{c}^{2}{d}^{3}{e}^{3}+24\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-126\,{a}^{2}c{d}^{2}{e}^{4}+42\,{c}^{2}{d}^{4}a{e}^{2}-6\,{c}^{3}{d}^{6} \right ) }{{e}^{4} \left ({m}^{4}+22\,{m}^{3}+179\,{m}^{2}+638\,m+840 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(4+m)*(c^3*d^3*e^3*m^3*x^3+3*a*c^2*d^2*e^4*m^3*x^2+15*c^3*d^3*e^3*m^2*x^
3+3*a^2*c*d*e^5*m^3*x+48*a*c^2*d^2*e^4*m^2*x^2-3*c^3*d^4*e^2*m^2*x^2+74*c^3*d^3*
e^3*m*x^3+a^3*e^6*m^3+51*a^2*c*d*e^5*m^2*x-6*a*c^2*d^3*e^3*m^2*x+249*a*c^2*d^2*e
^4*m*x^2-27*c^3*d^4*e^2*m*x^2+120*c^3*d^3*e^3*x^3+18*a^3*e^6*m^2-3*a^2*c*d^2*e^4
*m^2+282*a^2*c*d*e^5*m*x-66*a*c^2*d^3*e^3*m*x+420*a*c^2*d^2*e^4*x^2+6*c^3*d^5*e*
m*x-60*c^3*d^4*e^2*x^2+107*a^3*e^6*m-39*a^2*c*d^2*e^4*m+504*a^2*c*d*e^5*x+6*a*c^
2*d^4*e^2*m-168*a*c^2*d^3*e^3*x+24*c^3*d^5*e*x+210*a^3*e^6-126*a^2*c*d^2*e^4+42*
a*c^2*d^4*e^2-6*c^3*d^6)/e^4/(m^4+22*m^3+179*m^2+638*m+840)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236388, size = 1561, normalized size = 12.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^3*d^4*e^6*m^3 - 6*c^3*d^10 + 42*a*c^2*d^8*e^2 - 126*a^2*c*d^6*e^4 + 210*a^3*d
^4*e^6 + (c^3*d^3*e^7*m^3 + 15*c^3*d^3*e^7*m^2 + 74*c^3*d^3*e^7*m + 120*c^3*d^3*
e^7)*x^7 + (420*c^3*d^4*e^6 + 420*a*c^2*d^2*e^8 + (4*c^3*d^4*e^6 + 3*a*c^2*d^2*e
^8)*m^3 + 3*(19*c^3*d^4*e^6 + 16*a*c^2*d^2*e^8)*m^2 + (269*c^3*d^4*e^6 + 249*a*c
^2*d^2*e^8)*m)*x^6 + 3*(168*c^3*d^5*e^5 + 504*a*c^2*d^3*e^7 + 168*a^2*c*d*e^9 +
(2*c^3*d^5*e^5 + 4*a*c^2*d^3*e^7 + a^2*c*d*e^9)*m^3 + (26*c^3*d^5*e^5 + 62*a*c^2
*d^3*e^7 + 17*a^2*c*d*e^9)*m^2 + 2*(57*c^3*d^5*e^5 + 155*a*c^2*d^3*e^7 + 47*a^2*
c*d*e^9)*m)*x^5 + (210*c^3*d^6*e^4 + 1890*a*c^2*d^4*e^6 + 1890*a^2*c*d^2*e^8 + 2
10*a^3*e^10 + (4*c^3*d^6*e^4 + 18*a*c^2*d^4*e^6 + 12*a^2*c*d^2*e^8 + a^3*e^10)*m
^3 + 3*(14*c^3*d^6*e^4 + 88*a*c^2*d^4*e^6 + 67*a^2*c*d^2*e^8 + 6*a^3*e^10)*m^2 +
(158*c^3*d^6*e^4 + 1236*a*c^2*d^4*e^6 + 1089*a^2*c*d^2*e^8 + 107*a^3*e^10)*m)*x
^4 + (840*a*c^2*d^5*e^5 + 2520*a^2*c*d^3*e^7 + 840*a^3*d*e^9 + (c^3*d^7*e^3 + 12
*a*c^2*d^5*e^5 + 18*a^2*c*d^3*e^7 + 4*a^3*d*e^9)*m^3 + 3*(c^3*d^7*e^3 + 52*a*c^2
*d^5*e^5 + 98*a^2*c*d^3*e^7 + 24*a^3*d*e^9)*m^2 + 2*(c^3*d^7*e^3 + 312*a*c^2*d^5
*e^5 + 768*a^2*c*d^3*e^7 + 214*a^3*d*e^9)*m)*x^3 - 3*(a^2*c*d^6*e^4 - 6*a^3*d^4*
e^6)*m^2 + 3*(420*a^2*c*d^4*e^6 + 420*a^3*d^2*e^8 + (a*c^2*d^6*e^4 + 4*a^2*c*d^4
*e^6 + 2*a^3*d^2*e^8)*m^3 - (c^3*d^8*e^2 - 8*a*c^2*d^6*e^4 - 62*a^2*c*d^4*e^6 -
36*a^3*d^2*e^8)*m^2 - (c^3*d^8*e^2 - 7*a*c^2*d^6*e^4 - 298*a^2*c*d^4*e^6 - 214*a
^3*d^2*e^8)*m)*x^2 + (6*a*c^2*d^8*e^2 - 39*a^2*c*d^6*e^4 + 107*a^3*d^4*e^6)*m +
(840*a^3*d^3*e^7 + (3*a^2*c*d^5*e^5 + 4*a^3*d^3*e^7)*m^3 - 3*(2*a*c^2*d^7*e^3 -
13*a^2*c*d^5*e^5 - 24*a^3*d^3*e^7)*m^2 + 2*(3*c^3*d^9*e - 21*a*c^2*d^7*e^3 + 63*
a^2*c*d^5*e^5 + 214*a^3*d^3*e^7)*m)*x)*(e*x + d)^m/(e^4*m^4 + 22*e^4*m^3 + 179*e
^4*m^2 + 638*e^4*m + 840*e^4)

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Sympy [A]  time = 58.1979, size = 7164, normalized size = 55.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((c**3*d**6*d**m*x**4/4, Eq(e, 0)), (-2*a**3*e**6/(6*d**3*e**4 + 18*d**
2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 3*a**2*c*d**2*e**4/(6*d**3*e**4 + 18*
d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 9*a**2*c*d*e**5*x/(6*d**3*e**4 + 1
8*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*a*c**2*d**4*e**2/(6*d**3*e**4
+ 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18*a*c**2*d**3*e**3*x/(6*d**3
*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18*a*c**2*d**2*e**4*x**
2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*c**3*d**6*lo
g(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*c*
*3*d**6/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*
d**5*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x*
*3) + 27*c**3*d**5*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x
**3) + 18*c**3*d**4*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*
e**6*x**2 + 6*e**7*x**3) + 18*c**3*d**4*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x
+ 18*d*e**6*x**2 + 6*e**7*x**3) + 6*c**3*d**3*e**3*x**3*log(d/e + x)/(6*d**3*e**
4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -7)), (-a**3*e**6/(2*d
**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 3*a**2*c*d**2*e**4/(2*d**2*e**4 + 4*d*e**
5*x + 2*e**6*x**2) - 6*a**2*c*d*e**5*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)
+ 6*a*c**2*d**4*e**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 9*a
*c**2*d**4*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*a*c**2*d**3*e**3*x
*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*a*c**2*d**3*e**3*x/(
2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*a*c**2*d**2*e**4*x**2*log(d/e + x)/(
2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d**6*log(d/e + x)/(2*d**2*e**4
+ 4*d*e**5*x + 2*e**6*x**2) - 15*c**3*d**6/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x*
*2) - 12*c**3*d**5*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2
4*c**3*d**5*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d**4*e**2*x**2
*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d**4*e**2*x**2/(
2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c**3*d**3*e**3*x**3/(2*d**2*e**4 + 4
*d*e**5*x + 2*e**6*x**2), Eq(m, -6)), (-2*a**3*e**6/(2*d*e**4 + 2*e**5*x) + 6*a*
*2*c*d**2*e**4*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*a**2*c*d**2*e**4/(2*d*e**4
+ 2*e**5*x) + 6*a**2*c*d*e**5*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 12*a*c**2*
d**4*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 30*a*c**2*d**4*e**2/(2*d*e**4 + 2
*e**5*x) - 12*a*c**2*d**3*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 18*a*c**2*
d**3*e**3*x/(2*d*e**4 + 2*e**5*x) + 6*a*c**2*d**2*e**4*x**2/(2*d*e**4 + 2*e**5*x
) + 6*c**3*d**6*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 12*c**3*d**6/(2*d*e**4 + 2*
e**5*x) + 6*c**3*d**5*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*c**3*d**5*e*x/(
2*d*e**4 + 2*e**5*x) - 3*c**3*d**4*e**2*x**2/(2*d*e**4 + 2*e**5*x) + c**3*d**3*e
**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -5)), (a**3*e**2*log(d/e + x) - 3*a**2*c*d
**2*log(d/e + x) + 3*a**2*c*d*e*x + 3*a*c**2*d**4*log(d/e + x)/e**2 - 3*a*c**2*d
**3*x/e + 3*a*c**2*d**2*x**2/2 - c**3*d**6*log(d/e + x)/e**4 + c**3*d**5*x/e**3
- c**3*d**4*x**2/(2*e**2) + c**3*d**3*x**3/(3*e), Eq(m, -4)), (a**3*d**4*e**6*m*
*3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**
4) + 18*a**3*d**4*e**6*m**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m*
*2 + 638*e**4*m + 840*e**4) + 107*a**3*d**4*e**6*m*(d + e*x)**m/(e**4*m**4 + 22*
e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 210*a**3*d**4*e**6*(d + e*x
)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*a**3
*d**3*e**7*m**3*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e
**4*m + 840*e**4) + 72*a**3*d**3*e**7*m**2*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m
**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 428*a**3*d**3*e**7*m*x*(d + e*x)*
*m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 840*a**3
*d**3*e**7*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m
+ 840*e**4) + 6*a**3*d**2*e**8*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3
+ 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 108*a**3*d**2*e**8*m**2*x**2*(d + e*
x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 642*a
**3*d**2*e**8*m*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 63
8*e**4*m + 840*e**4) + 1260*a**3*d**2*e**8*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**
4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*a**3*d*e**9*m**3*x**3*(d + e
*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 72*a
**3*d*e**9*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 63
8*e**4*m + 840*e**4) + 428*a**3*d*e**9*m*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*
m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 840*a**3*d*e**9*x**3*(d + e*x)**
m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + a**3*e**1
0*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m
+ 840*e**4) + 18*a**3*e**10*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 1
79*e**4*m**2 + 638*e**4*m + 840*e**4) + 107*a**3*e**10*m*x**4*(d + e*x)**m/(e**4
*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 210*a**3*e**10*x
**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e*
*4) - 3*a**2*c*d**6*e**4*m**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*
m**2 + 638*e**4*m + 840*e**4) - 39*a**2*c*d**6*e**4*m*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 126*a**2*c*d**6*e**4*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3
*a**2*c*d**5*e**5*m**3*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 39*a**2*c*d**5*e**5*m**2*x*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 126*a**2*c*d**5*e**5*m*x
*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4)
+ 12*a**2*c*d**4*e**6*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e*
*4*m**2 + 638*e**4*m + 840*e**4) + 186*a**2*c*d**4*e**6*m**2*x**2*(d + e*x)**m/(
e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 894*a**2*c*d
**4*e**6*m*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**
4*m + 840*e**4) + 1260*a**2*c*d**4*e**6*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m
**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 18*a**2*c*d**3*e**7*m**3*x**3*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 2
94*a**2*c*d**3*e**7*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*
m**2 + 638*e**4*m + 840*e**4) + 1536*a**2*c*d**3*e**7*m*x**3*(d + e*x)**m/(e**4*
m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 2520*a**2*c*d**3*
e**7*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m +
840*e**4) + 12*a**2*c*d**2*e**8*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3
+ 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 201*a**2*c*d**2*e**8*m**2*x**4*(d +
e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 108
9*a**2*c*d**2*e**8*m*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 1890*a**2*c*d**2*e**8*x**4*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3*a**2*c*d*e**9*m**3*x*
*5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**
4) + 51*a**2*c*d*e**9*m**2*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**
4*m**2 + 638*e**4*m + 840*e**4) + 282*a**2*c*d*e**9*m*x**5*(d + e*x)**m/(e**4*m*
*4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 504*a**2*c*d*e**9*x
**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e*
*4) + 6*a*c**2*d**8*e**2*m*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**
2 + 638*e**4*m + 840*e**4) + 42*a*c**2*d**8*e**2*(d + e*x)**m/(e**4*m**4 + 22*e*
*4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 6*a*c**2*d**7*e**3*m**2*x*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 4
2*a*c**2*d**7*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 +
638*e**4*m + 840*e**4) + 3*a*c**2*d**6*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 24*a*c**2*d**6*e**4*m**2
*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*
e**4) + 21*a*c**2*d**6*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*
e**4*m**2 + 638*e**4*m + 840*e**4) + 12*a*c**2*d**5*e**5*m**3*x**3*(d + e*x)**m/
(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 156*a*c**2*
d**5*e**5*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638
*e**4*m + 840*e**4) + 624*a*c**2*d**5*e**5*m*x**3*(d + e*x)**m/(e**4*m**4 + 22*e
**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 840*a*c**2*d**5*e**5*x**3*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) +
18*a*c**2*d**4*e**6*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*
m**2 + 638*e**4*m + 840*e**4) + 264*a*c**2*d**4*e**6*m**2*x**4*(d + e*x)**m/(e**
4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1236*a*c**2*d**
4*e**6*m*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*
m + 840*e**4) + 1890*a*c**2*d**4*e**6*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**
3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 12*a*c**2*d**3*e**7*m**3*x**5*(d +
e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 186
*a*c**2*d**3*e**7*m**2*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m*
*2 + 638*e**4*m + 840*e**4) + 930*a*c**2*d**3*e**7*m*x**5*(d + e*x)**m/(e**4*m**
4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 1512*a*c**2*d**3*e**
7*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840
*e**4) + 3*a*c**2*d**2*e**8*m**3*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 1
79*e**4*m**2 + 638*e**4*m + 840*e**4) + 48*a*c**2*d**2*e**8*m**2*x**6*(d + e*x)*
*m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 249*a*c*
*2*d**2*e**8*m*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638
*e**4*m + 840*e**4) + 420*a*c**2*d**2*e**8*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**
4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 6*c**3*d**10*(d + e*x)**m/(e**
4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*c**3*d**9*e*m
*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**
4) - 3*c**3*d**8*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**
4*m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**8*e**2*m*x**2*(d + e*x)**m/(e**4*m**
4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + c**3*d**7*e**3*m**3*
x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e
**4) + 3*c**3*d**7*e**3*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e
**4*m**2 + 638*e**4*m + 840*e**4) + 2*c**3*d**7*e**3*m*x**3*(d + e*x)**m/(e**4*m
**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*c**3*d**6*e**4*m
**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 8
40*e**4) + 42*c**3*d**6*e**4*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 +
179*e**4*m**2 + 638*e**4*m + 840*e**4) + 158*c**3*d**6*e**4*m*x**4*(d + e*x)**m/
(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 210*c**3*d*
*6*e**4*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m
+ 840*e**4) + 6*c**3*d**5*e**5*m**3*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3
+ 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 78*c**3*d**5*e**5*m**2*x**5*(d + e*x
)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 342*c*
*3*d**5*e**5*m*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638
*e**4*m + 840*e**4) + 504*c**3*d**5*e**5*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*
m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*c**3*d**4*e**6*m**3*x**6*(d +
e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 57*
c**3*d**4*e**6*m**2*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 269*c**3*d**4*e**6*m*x**6*(d + e*x)**m/(e**4*m**4 + 2
2*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 420*c**3*d**4*e**6*x**6*(
d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) +
c**3*d**3*e**7*m**3*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2
+ 638*e**4*m + 840*e**4) + 15*c**3*d**3*e**7*m**2*x**7*(d + e*x)**m/(e**4*m**4
+ 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 74*c**3*d**3*e**7*m*x*
*7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**
4) + 120*c**3*d**3*e**7*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m
**2 + 638*e**4*m + 840*e**4), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227489, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done