3.608 \(\int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx\)

Optimal. Leaf size=89 \[ \frac{a^2 (d+e x)^4}{4 e}+\frac{\left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac{a b (d+e x)^6}{3 e}+\frac{b c (d+e x)^{10}}{5 e}+\frac{c^2 (d+e x)^{12}}{12 e} \]

[Out]

(a^2*(d + e*x)^4)/(4*e) + (a*b*(d + e*x)^6)/(3*e) + ((b^2 + 2*a*c)*(d + e*x)^8)/
(8*e) + (b*c*(d + e*x)^10)/(5*e) + (c^2*(d + e*x)^12)/(12*e)

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Rubi [A]  time = 0.396266, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^2 (d+e x)^4}{4 e}+\frac{\left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac{a b (d+e x)^6}{3 e}+\frac{b c (d+e x)^{10}}{5 e}+\frac{c^2 (d+e x)^{12}}{12 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]

[Out]

(a^2*(d + e*x)^4)/(4*e) + (a*b*(d + e*x)^6)/(3*e) + ((b^2 + 2*a*c)*(d + e*x)^8)/
(8*e) + (b*c*(d + e*x)^10)/(5*e) + (c^2*(d + e*x)^12)/(12*e)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \int ^{\left (d + e x\right )^{2}} x\, dx}{2 e} + \frac{a b \left (d + e x\right )^{6}}{3 e} + \frac{b c \left (d + e x\right )^{10}}{5 e} + \frac{c^{2} \left (d + e x\right )^{12}}{12 e} + \frac{\left (d + e x\right )^{8} \left (2 a c + b^{2}\right )}{8 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

a**2*Integral(x, (x, (d + e*x)**2))/(2*e) + a*b*(d + e*x)**6/(3*e) + b*c*(d + e*
x)**10/(5*e) + c**2*(d + e*x)**12/(12*e) + (d + e*x)**8*(2*a*c + b**2)/(8*e)

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Mathematica [B]  time = 0.21506, size = 401, normalized size = 4.51 \[ \frac{1}{4} e^3 x^4 \left (a^2+20 a b d^2+70 a c d^4+35 b^2 d^4+168 b c d^6+165 c^2 d^8\right )+\frac{1}{3} d e^2 x^3 \left (3 a^2+20 a b d^2+42 a c d^4+21 b^2 d^4+72 b c d^6+55 c^2 d^8\right )+\frac{1}{2} d^2 e x^2 \left (3 a^2+10 a b d^2+14 a c d^4+7 b^2 d^4+18 b c d^6+11 c^2 d^8\right )+\frac{1}{8} e^7 x^8 \left (2 a c+b^2+72 b c d^2+330 c^2 d^4\right )+d e^6 x^7 \left (2 a c+b^2+24 b c d^2+66 c^2 d^4\right )+\frac{1}{6} e^5 x^6 \left (2 a b+42 a c d^2+21 b^2 d^2+252 b c d^4+462 c^2 d^6\right )+\frac{1}{5} d e^4 x^5 \left (10 a b+70 a c d^2+35 b^2 d^2+252 b c d^4+330 c^2 d^6\right )+d^3 x \left (a+b d^2+c d^4\right )^2+\frac{1}{10} c e^9 x^{10} \left (2 b+55 c d^2\right )+\frac{1}{3} c d e^8 x^9 \left (6 b+55 c d^2\right )+c^2 d e^{10} x^{11}+\frac{1}{12} c^2 e^{11} x^{12} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]

[Out]

d^3*(a + b*d^2 + c*d^4)^2*x + (d^2*(3*a^2 + 10*a*b*d^2 + 7*b^2*d^4 + 14*a*c*d^4
+ 18*b*c*d^6 + 11*c^2*d^8)*e*x^2)/2 + (d*(3*a^2 + 20*a*b*d^2 + 21*b^2*d^4 + 42*a
*c*d^4 + 72*b*c*d^6 + 55*c^2*d^8)*e^2*x^3)/3 + ((a^2 + 20*a*b*d^2 + 35*b^2*d^4 +
70*a*c*d^4 + 168*b*c*d^6 + 165*c^2*d^8)*e^3*x^4)/4 + (d*(10*a*b + 35*b^2*d^2 +
70*a*c*d^2 + 252*b*c*d^4 + 330*c^2*d^6)*e^4*x^5)/5 + ((2*a*b + 21*b^2*d^2 + 42*a
*c*d^2 + 252*b*c*d^4 + 462*c^2*d^6)*e^5*x^6)/6 + d*(b^2 + 2*a*c + 24*b*c*d^2 + 6
6*c^2*d^4)*e^6*x^7 + ((b^2 + 2*a*c + 72*b*c*d^2 + 330*c^2*d^4)*e^7*x^8)/8 + (c*d
*(6*b + 55*c*d^2)*e^8*x^9)/3 + (c*(2*b + 55*c*d^2)*e^9*x^10)/10 + c^2*d*e^10*x^1
1 + (c^2*e^11*x^12)/12

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Maple [B]  time = 0.002, size = 1314, normalized size = 14.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x)

[Out]

1/12*e^11*c^2*x^12+e^10*d*c^2*x^11+1/10*(27*d^2*e^9*c^2+e^3*(2*(6*c*d^2*e^2+b*e^
2)*c*e^4+16*c^2*d^2*e^6))*x^10+1/9*(25*d^3*c^2*e^8+3*e^2*d*(2*(6*c*d^2*e^2+b*e^2
)*c*e^4+16*c^2*d^2*e^6)+e^3*(2*(4*c*d^3*e+2*b*d*e)*c*e^4+8*(6*c*d^2*e^2+b*e^2)*c
*d*e^3))*x^9+1/8*(8*d^4*c^2*e^7+3*d^2*e*(2*(6*c*d^2*e^2+b*e^2)*c*e^4+16*c^2*d^2*
e^6)+3*e^2*d*(2*(4*c*d^3*e+2*b*d*e)*c*e^4+8*(6*c*d^2*e^2+b*e^2)*c*d*e^3)+e^3*(2*
(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*e+2*b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2))*x^8+
1/7*(d^3*(2*(6*c*d^2*e^2+b*e^2)*c*e^4+16*c^2*d^2*e^6)+3*d^2*e*(2*(4*c*d^3*e+2*b*
d*e)*c*e^4+8*(6*c*d^2*e^2+b*e^2)*c*d*e^3)+3*e^2*d*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*
c*d^3*e+2*b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2)+e^3*(8*(c*d^4+b*d^2+a)*c*d*e^3+2
*(4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2)))*x^7+1/6*(d^3*(2*(4*c*d^3*e+2*b*d*e)*c
*e^4+8*(6*c*d^2*e^2+b*e^2)*c*d*e^3)+3*d^2*e*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*
e+2*b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2)+3*e^2*d*(8*(c*d^4+b*d^2+a)*c*d*e^3+2*(
4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2))+e^3*(2*(c*d^4+b*d^2+a)*(6*c*d^2*e^2+b*e^
2)+(4*c*d^3*e+2*b*d*e)^2))*x^6+1/5*(d^3*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*e+2*
b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2)+3*d^2*e*(8*(c*d^4+b*d^2+a)*c*d*e^3+2*(4*c*
d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2))+3*e^2*d*(2*(c*d^4+b*d^2+a)*(6*c*d^2*e^2+b*e^
2)+(4*c*d^3*e+2*b*d*e)^2)+2*e^3*(c*d^4+b*d^2+a)*(4*c*d^3*e+2*b*d*e))*x^5+1/4*(d^
3*(8*(c*d^4+b*d^2+a)*c*d*e^3+2*(4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2))+3*d^2*e*
(2*(c*d^4+b*d^2+a)*(6*c*d^2*e^2+b*e^2)+(4*c*d^3*e+2*b*d*e)^2)+6*e^2*d*(c*d^4+b*d
^2+a)*(4*c*d^3*e+2*b*d*e)+e^3*(c*d^4+b*d^2+a)^2)*x^4+1/3*(d^3*(2*(c*d^4+b*d^2+a)
*(6*c*d^2*e^2+b*e^2)+(4*c*d^3*e+2*b*d*e)^2)+6*d^2*e*(c*d^4+b*d^2+a)*(4*c*d^3*e+2
*b*d*e)+3*e^2*d*(c*d^4+b*d^2+a)^2)*x^3+1/2*(2*d^3*(c*d^4+b*d^2+a)*(4*c*d^3*e+2*b
*d*e)+3*d^2*e*(c*d^4+b*d^2+a)^2)*x^2+d^3*(c*d^4+b*d^2+a)^2*x

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Maxima [A]  time = 0.752595, size = 544, normalized size = 6.11 \[ \frac{1}{12} \, c^{2} e^{11} x^{12} + c^{2} d e^{10} x^{11} + \frac{1}{10} \,{\left (55 \, c^{2} d^{2} + 2 \, b c\right )} e^{9} x^{10} + \frac{1}{3} \,{\left (55 \, c^{2} d^{3} + 6 \, b c d\right )} e^{8} x^{9} + \frac{1}{8} \,{\left (330 \, c^{2} d^{4} + 72 \, b c d^{2} + b^{2} + 2 \, a c\right )} e^{7} x^{8} +{\left (66 \, c^{2} d^{5} + 24 \, b c d^{3} +{\left (b^{2} + 2 \, a c\right )} d\right )} e^{6} x^{7} + \frac{1}{6} \,{\left (462 \, c^{2} d^{6} + 252 \, b c d^{4} + 21 \,{\left (b^{2} + 2 \, a c\right )} d^{2} + 2 \, a b\right )} e^{5} x^{6} + \frac{1}{5} \,{\left (330 \, c^{2} d^{7} + 252 \, b c d^{5} + 35 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 10 \, a b d\right )} e^{4} x^{5} + \frac{1}{4} \,{\left (165 \, c^{2} d^{8} + 168 \, b c d^{6} + 35 \,{\left (b^{2} + 2 \, a c\right )} d^{4} + 20 \, a b d^{2} + a^{2}\right )} e^{3} x^{4} + \frac{1}{3} \,{\left (55 \, c^{2} d^{9} + 72 \, b c d^{7} + 21 \,{\left (b^{2} + 2 \, a c\right )} d^{5} + 20 \, a b d^{3} + 3 \, a^{2} d\right )} e^{2} x^{3} + \frac{1}{2} \,{\left (11 \, c^{2} d^{10} + 18 \, b c d^{8} + 7 \,{\left (b^{2} + 2 \, a c\right )} d^{6} + 10 \, a b d^{4} + 3 \, a^{2} d^{2}\right )} e x^{2} +{\left (c^{2} d^{11} + 2 \, b c d^{9} +{\left (b^{2} + 2 \, a c\right )} d^{7} + 2 \, a b d^{5} + a^{2} d^{3}\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)^3,x, algorithm="maxima")

[Out]

1/12*c^2*e^11*x^12 + c^2*d*e^10*x^11 + 1/10*(55*c^2*d^2 + 2*b*c)*e^9*x^10 + 1/3*
(55*c^2*d^3 + 6*b*c*d)*e^8*x^9 + 1/8*(330*c^2*d^4 + 72*b*c*d^2 + b^2 + 2*a*c)*e^
7*x^8 + (66*c^2*d^5 + 24*b*c*d^3 + (b^2 + 2*a*c)*d)*e^6*x^7 + 1/6*(462*c^2*d^6 +
252*b*c*d^4 + 21*(b^2 + 2*a*c)*d^2 + 2*a*b)*e^5*x^6 + 1/5*(330*c^2*d^7 + 252*b*
c*d^5 + 35*(b^2 + 2*a*c)*d^3 + 10*a*b*d)*e^4*x^5 + 1/4*(165*c^2*d^8 + 168*b*c*d^
6 + 35*(b^2 + 2*a*c)*d^4 + 20*a*b*d^2 + a^2)*e^3*x^4 + 1/3*(55*c^2*d^9 + 72*b*c*
d^7 + 21*(b^2 + 2*a*c)*d^5 + 20*a*b*d^3 + 3*a^2*d)*e^2*x^3 + 1/2*(11*c^2*d^10 +
18*b*c*d^8 + 7*(b^2 + 2*a*c)*d^6 + 10*a*b*d^4 + 3*a^2*d^2)*e*x^2 + (c^2*d^11 + 2
*b*c*d^9 + (b^2 + 2*a*c)*d^7 + 2*a*b*d^5 + a^2*d^3)*x

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Fricas [A]  time = 0.266645, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} e^{11} c^{2} + x^{11} e^{10} d c^{2} + \frac{11}{2} x^{10} e^{9} d^{2} c^{2} + \frac{55}{3} x^{9} e^{8} d^{3} c^{2} + \frac{165}{4} x^{8} e^{7} d^{4} c^{2} + \frac{1}{5} x^{10} e^{9} c b + 66 x^{7} e^{6} d^{5} c^{2} + 2 x^{9} e^{8} d c b + 77 x^{6} e^{5} d^{6} c^{2} + 9 x^{8} e^{7} d^{2} c b + 66 x^{5} e^{4} d^{7} c^{2} + 24 x^{7} e^{6} d^{3} c b + \frac{165}{4} x^{4} e^{3} d^{8} c^{2} + 42 x^{6} e^{5} d^{4} c b + \frac{1}{8} x^{8} e^{7} b^{2} + \frac{1}{4} x^{8} e^{7} c a + \frac{55}{3} x^{3} e^{2} d^{9} c^{2} + \frac{252}{5} x^{5} e^{4} d^{5} c b + x^{7} e^{6} d b^{2} + 2 x^{7} e^{6} d c a + \frac{11}{2} x^{2} e d^{10} c^{2} + 42 x^{4} e^{3} d^{6} c b + \frac{7}{2} x^{6} e^{5} d^{2} b^{2} + 7 x^{6} e^{5} d^{2} c a + x d^{11} c^{2} + 24 x^{3} e^{2} d^{7} c b + 7 x^{5} e^{4} d^{3} b^{2} + 14 x^{5} e^{4} d^{3} c a + 9 x^{2} e d^{8} c b + \frac{35}{4} x^{4} e^{3} d^{4} b^{2} + \frac{35}{2} x^{4} e^{3} d^{4} c a + \frac{1}{3} x^{6} e^{5} b a + 2 x d^{9} c b + 7 x^{3} e^{2} d^{5} b^{2} + 14 x^{3} e^{2} d^{5} c a + 2 x^{5} e^{4} d b a + \frac{7}{2} x^{2} e d^{6} b^{2} + 7 x^{2} e d^{6} c a + 5 x^{4} e^{3} d^{2} b a + x d^{7} b^{2} + 2 x d^{7} c a + \frac{20}{3} x^{3} e^{2} d^{3} b a + 5 x^{2} e d^{4} b a + \frac{1}{4} x^{4} e^{3} a^{2} + 2 x d^{5} b a + x^{3} e^{2} d a^{2} + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^11*c^2 + x^11*e^10*d*c^2 + 11/2*x^10*e^9*d^2*c^2 + 55/3*x^9*e^8*d^3*
c^2 + 165/4*x^8*e^7*d^4*c^2 + 1/5*x^10*e^9*c*b + 66*x^7*e^6*d^5*c^2 + 2*x^9*e^8*
d*c*b + 77*x^6*e^5*d^6*c^2 + 9*x^8*e^7*d^2*c*b + 66*x^5*e^4*d^7*c^2 + 24*x^7*e^6
*d^3*c*b + 165/4*x^4*e^3*d^8*c^2 + 42*x^6*e^5*d^4*c*b + 1/8*x^8*e^7*b^2 + 1/4*x^
8*e^7*c*a + 55/3*x^3*e^2*d^9*c^2 + 252/5*x^5*e^4*d^5*c*b + x^7*e^6*d*b^2 + 2*x^7
*e^6*d*c*a + 11/2*x^2*e*d^10*c^2 + 42*x^4*e^3*d^6*c*b + 7/2*x^6*e^5*d^2*b^2 + 7*
x^6*e^5*d^2*c*a + x*d^11*c^2 + 24*x^3*e^2*d^7*c*b + 7*x^5*e^4*d^3*b^2 + 14*x^5*e
^4*d^3*c*a + 9*x^2*e*d^8*c*b + 35/4*x^4*e^3*d^4*b^2 + 35/2*x^4*e^3*d^4*c*a + 1/3
*x^6*e^5*b*a + 2*x*d^9*c*b + 7*x^3*e^2*d^5*b^2 + 14*x^3*e^2*d^5*c*a + 2*x^5*e^4*
d*b*a + 7/2*x^2*e*d^6*b^2 + 7*x^2*e*d^6*c*a + 5*x^4*e^3*d^2*b*a + x*d^7*b^2 + 2*
x*d^7*c*a + 20/3*x^3*e^2*d^3*b*a + 5*x^2*e*d^4*b*a + 1/4*x^4*e^3*a^2 + 2*x*d^5*b
*a + x^3*e^2*d*a^2 + 3/2*x^2*e*d^2*a^2 + x*d^3*a^2

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Sympy [A]  time = 0.469583, size = 559, normalized size = 6.28 \[ c^{2} d e^{10} x^{11} + \frac{c^{2} e^{11} x^{12}}{12} + x^{10} \left (\frac{b c e^{9}}{5} + \frac{11 c^{2} d^{2} e^{9}}{2}\right ) + x^{9} \left (2 b c d e^{8} + \frac{55 c^{2} d^{3} e^{8}}{3}\right ) + x^{8} \left (\frac{a c e^{7}}{4} + \frac{b^{2} e^{7}}{8} + 9 b c d^{2} e^{7} + \frac{165 c^{2} d^{4} e^{7}}{4}\right ) + x^{7} \left (2 a c d e^{6} + b^{2} d e^{6} + 24 b c d^{3} e^{6} + 66 c^{2} d^{5} e^{6}\right ) + x^{6} \left (\frac{a b e^{5}}{3} + 7 a c d^{2} e^{5} + \frac{7 b^{2} d^{2} e^{5}}{2} + 42 b c d^{4} e^{5} + 77 c^{2} d^{6} e^{5}\right ) + x^{5} \left (2 a b d e^{4} + 14 a c d^{3} e^{4} + 7 b^{2} d^{3} e^{4} + \frac{252 b c d^{5} e^{4}}{5} + 66 c^{2} d^{7} e^{4}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + 5 a b d^{2} e^{3} + \frac{35 a c d^{4} e^{3}}{2} + \frac{35 b^{2} d^{4} e^{3}}{4} + 42 b c d^{6} e^{3} + \frac{165 c^{2} d^{8} e^{3}}{4}\right ) + x^{3} \left (a^{2} d e^{2} + \frac{20 a b d^{3} e^{2}}{3} + 14 a c d^{5} e^{2} + 7 b^{2} d^{5} e^{2} + 24 b c d^{7} e^{2} + \frac{55 c^{2} d^{9} e^{2}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e}{2} + 5 a b d^{4} e + 7 a c d^{6} e + \frac{7 b^{2} d^{6} e}{2} + 9 b c d^{8} e + \frac{11 c^{2} d^{10} e}{2}\right ) + x \left (a^{2} d^{3} + 2 a b d^{5} + 2 a c d^{7} + b^{2} d^{7} + 2 b c d^{9} + c^{2} d^{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

c**2*d*e**10*x**11 + c**2*e**11*x**12/12 + x**10*(b*c*e**9/5 + 11*c**2*d**2*e**9
/2) + x**9*(2*b*c*d*e**8 + 55*c**2*d**3*e**8/3) + x**8*(a*c*e**7/4 + b**2*e**7/8
+ 9*b*c*d**2*e**7 + 165*c**2*d**4*e**7/4) + x**7*(2*a*c*d*e**6 + b**2*d*e**6 +
24*b*c*d**3*e**6 + 66*c**2*d**5*e**6) + x**6*(a*b*e**5/3 + 7*a*c*d**2*e**5 + 7*b
**2*d**2*e**5/2 + 42*b*c*d**4*e**5 + 77*c**2*d**6*e**5) + x**5*(2*a*b*d*e**4 + 1
4*a*c*d**3*e**4 + 7*b**2*d**3*e**4 + 252*b*c*d**5*e**4/5 + 66*c**2*d**7*e**4) +
x**4*(a**2*e**3/4 + 5*a*b*d**2*e**3 + 35*a*c*d**4*e**3/2 + 35*b**2*d**4*e**3/4 +
42*b*c*d**6*e**3 + 165*c**2*d**8*e**3/4) + x**3*(a**2*d*e**2 + 20*a*b*d**3*e**2
/3 + 14*a*c*d**5*e**2 + 7*b**2*d**5*e**2 + 24*b*c*d**7*e**2 + 55*c**2*d**9*e**2/
3) + x**2*(3*a**2*d**2*e/2 + 5*a*b*d**4*e + 7*a*c*d**6*e + 7*b**2*d**6*e/2 + 9*b
*c*d**8*e + 11*c**2*d**10*e/2) + x*(a**2*d**3 + 2*a*b*d**5 + 2*a*c*d**7 + b**2*d
**7 + 2*b*c*d**9 + c**2*d**11)

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GIAC/XCAS [A]  time = 0.268788, size = 730, normalized size = 8.2 \[ \frac{1}{12} \, c^{2} x^{12} e^{11} + c^{2} d x^{11} e^{10} + \frac{11}{2} \, c^{2} d^{2} x^{10} e^{9} + \frac{55}{3} \, c^{2} d^{3} x^{9} e^{8} + \frac{165}{4} \, c^{2} d^{4} x^{8} e^{7} + 66 \, c^{2} d^{5} x^{7} e^{6} + 77 \, c^{2} d^{6} x^{6} e^{5} + 66 \, c^{2} d^{7} x^{5} e^{4} + \frac{165}{4} \, c^{2} d^{8} x^{4} e^{3} + \frac{55}{3} \, c^{2} d^{9} x^{3} e^{2} + \frac{11}{2} \, c^{2} d^{10} x^{2} e + c^{2} d^{11} x + \frac{1}{5} \, b c x^{10} e^{9} + 2 \, b c d x^{9} e^{8} + 9 \, b c d^{2} x^{8} e^{7} + 24 \, b c d^{3} x^{7} e^{6} + 42 \, b c d^{4} x^{6} e^{5} + \frac{252}{5} \, b c d^{5} x^{5} e^{4} + 42 \, b c d^{6} x^{4} e^{3} + 24 \, b c d^{7} x^{3} e^{2} + 9 \, b c d^{8} x^{2} e + 2 \, b c d^{9} x + \frac{1}{8} \, b^{2} x^{8} e^{7} + \frac{1}{4} \, a c x^{8} e^{7} + b^{2} d x^{7} e^{6} + 2 \, a c d x^{7} e^{6} + \frac{7}{2} \, b^{2} d^{2} x^{6} e^{5} + 7 \, a c d^{2} x^{6} e^{5} + 7 \, b^{2} d^{3} x^{5} e^{4} + 14 \, a c d^{3} x^{5} e^{4} + \frac{35}{4} \, b^{2} d^{4} x^{4} e^{3} + \frac{35}{2} \, a c d^{4} x^{4} e^{3} + 7 \, b^{2} d^{5} x^{3} e^{2} + 14 \, a c d^{5} x^{3} e^{2} + \frac{7}{2} \, b^{2} d^{6} x^{2} e + 7 \, a c d^{6} x^{2} e + b^{2} d^{7} x + 2 \, a c d^{7} x + \frac{1}{3} \, a b x^{6} e^{5} + 2 \, a b d x^{5} e^{4} + 5 \, a b d^{2} x^{4} e^{3} + \frac{20}{3} \, a b d^{3} x^{3} e^{2} + 5 \, a b d^{4} x^{2} e + 2 \, a b d^{5} x + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*x + d)^3,x, algorithm="giac")

[Out]

1/12*c^2*x^12*e^11 + c^2*d*x^11*e^10 + 11/2*c^2*d^2*x^10*e^9 + 55/3*c^2*d^3*x^9*
e^8 + 165/4*c^2*d^4*x^8*e^7 + 66*c^2*d^5*x^7*e^6 + 77*c^2*d^6*x^6*e^5 + 66*c^2*d
^7*x^5*e^4 + 165/4*c^2*d^8*x^4*e^3 + 55/3*c^2*d^9*x^3*e^2 + 11/2*c^2*d^10*x^2*e
+ c^2*d^11*x + 1/5*b*c*x^10*e^9 + 2*b*c*d*x^9*e^8 + 9*b*c*d^2*x^8*e^7 + 24*b*c*d
^3*x^7*e^6 + 42*b*c*d^4*x^6*e^5 + 252/5*b*c*d^5*x^5*e^4 + 42*b*c*d^6*x^4*e^3 + 2
4*b*c*d^7*x^3*e^2 + 9*b*c*d^8*x^2*e + 2*b*c*d^9*x + 1/8*b^2*x^8*e^7 + 1/4*a*c*x^
8*e^7 + b^2*d*x^7*e^6 + 2*a*c*d*x^7*e^6 + 7/2*b^2*d^2*x^6*e^5 + 7*a*c*d^2*x^6*e^
5 + 7*b^2*d^3*x^5*e^4 + 14*a*c*d^3*x^5*e^4 + 35/4*b^2*d^4*x^4*e^3 + 35/2*a*c*d^4
*x^4*e^3 + 7*b^2*d^5*x^3*e^2 + 14*a*c*d^5*x^3*e^2 + 7/2*b^2*d^6*x^2*e + 7*a*c*d^
6*x^2*e + b^2*d^7*x + 2*a*c*d^7*x + 1/3*a*b*x^6*e^5 + 2*a*b*d*x^5*e^4 + 5*a*b*d^
2*x^4*e^3 + 20/3*a*b*d^3*x^3*e^2 + 5*a*b*d^4*x^2*e + 2*a*b*d^5*x + 1/4*a^2*x^4*e
^3 + a^2*d*x^3*e^2 + 3/2*a^2*d^2*x^2*e + a^2*d^3*x