3.2146 \(\int (d+e x)^4 (a+b x+c x^2)^4 \, dx\)

Optimal. Leaf size=443 \[ \frac{(d+e x)^9 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^9}+\frac{2 c^2 (d+e x)^{11} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^9}-\frac{2 c (d+e x)^{10} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac{(d+e x)^8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}+\frac{2 (d+e x)^7 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9}-\frac{2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^4}{5 e^9}-\frac{c^3 (d+e x)^{12} (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^{13}}{13 e^9} \]

[Out]

((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^5)/(5*e^9) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^6)/(3*e
^9) + (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)^7)/(7*e^9) - ((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)^8)/(2*e^9) + ((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)^9)/(9*e^9) - (2*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)^
10)/(5*e^9) + (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^11)/(11*e^9) - (c^3*(2*c*d - b*e
)*(d + e*x)^12)/(3*e^9) + (c^4*(d + e*x)^13)/(13*e^9)

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Rubi [A]  time = 0.870615, antiderivative size = 443, 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, Rules used = {698} \[ \frac{(d+e x)^9 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^9}+\frac{2 c^2 (d+e x)^{11} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^9}-\frac{2 c (d+e x)^{10} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac{(d+e x)^8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}+\frac{2 (d+e x)^7 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9}-\frac{2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^4}{5 e^9}-\frac{c^3 (d+e x)^{12} (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^{13}}{13 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^5)/(5*e^9) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^6)/(3*e
^9) + (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)^7)/(7*e^9) - ((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)^8)/(2*e^9) + ((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)^9)/(9*e^9) - (2*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)^
10)/(5*e^9) + (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^11)/(11*e^9) - (c^3*(2*c*d - b*e
)*(d + e*x)^12)/(3*e^9) + (c^4*(d + e*x)^13)/(13*e^9)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}{e^8}+\frac{4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{e^8}+\frac{2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{e^8}+\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^7}{e^8}+\frac{\left (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 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^8}{e^8}+\frac{4 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^9}{e^8}+\frac{2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{10}}{e^8}-\frac{4 c^3 (2 c d-b e) (d+e x)^{11}}{e^8}+\frac{c^4 (d+e x)^{12}}{e^8}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^4 (d+e x)^5}{5 e^9}-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^6}{3 e^9}+\frac{2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^7}{7 e^9}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^8}{2 e^9}+\frac{\left (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 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^9}{9 e^9}-\frac{2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{10}}{5 e^9}+\frac{2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{11}}{11 e^9}-\frac{c^3 (2 c d-b e) (d+e x)^{12}}{3 e^9}+\frac{c^4 (d+e x)^{13}}{13 e^9}\\ \end{align*}

Mathematica [A]  time = 0.26029, size = 766, normalized size = 1.73 \[ \frac{1}{9} x^9 \left (6 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+4 b^2 c e^3 (3 a e+4 b d)+8 c^3 d^2 e (3 a e+2 b d)+b^4 e^4+c^4 d^4\right )+\frac{1}{2} x^8 \left (b c \left (3 a^2 e^4+18 a c d^2 e^2+c^2 d^4\right )+b^3 \left (a e^4+6 c d^2 e^2\right )+6 b^2 c d e \left (2 a e^2+c d^2\right )+2 a c^2 d e \left (3 a e^2+2 c d^2\right )+b^4 d e^3\right )+\frac{2}{7} x^7 \left (3 b^2 \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+2 a c \left (a^2 e^4+9 a c d^2 e^2+c^2 d^4\right )+8 b^3 \left (a d e^3+c d^3 e\right )+24 a b c d e \left (a e^2+c d^2\right )+3 b^4 d^2 e^2\right )+\frac{2}{3} x^6 \left (a b \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+2 a^2 c d e \left (2 a e^2+3 c d^2\right )+b^3 \left (6 a d^2 e^2+c d^4\right )+6 a b^2 d e \left (a e^2+2 c d^2\right )+b^4 d^3 e\right )+\frac{1}{5} x^5 \left (16 a^2 b d e \left (a e^2+3 c d^2\right )+a^2 \left (a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right )+12 a b^2 d^2 \left (3 a e^2+c d^2\right )+16 a b^3 d^3 e+b^4 d^4\right )+a d x^4 \left (a^2 e \left (a e^2+4 c d^2\right )+6 a b^2 d^2 e+3 a b d \left (2 a e^2+c d^2\right )+b^3 d^3\right )+\frac{2}{3} a^2 d^2 x^3 \left (8 a b d e+a \left (3 a e^2+2 c d^2\right )+3 b^2 d^2\right )+2 a^3 d^3 x^2 (a e+b d)+a^4 d^4 x+\frac{2}{11} c^2 e^2 x^{11} \left (2 c e (a e+4 b d)+3 b^2 e^2+3 c^2 d^2\right )+\frac{2}{5} c e x^{10} \left (2 c^2 d e (2 a e+3 b d)+3 b c e^2 (a e+2 b d)+b^3 e^3+c^3 d^3\right )+\frac{1}{3} c^3 e^3 x^{12} (b e+c d)+\frac{1}{13} c^4 e^4 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d^4*x + 2*a^3*d^3*(b*d + a*e)*x^2 + (2*a^2*d^2*(3*b^2*d^2 + 8*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^3)/3 + a*
d*(b^3*d^3 + 6*a*b^2*d^2*e + a^2*e*(4*c*d^2 + a*e^2) + 3*a*b*d*(c*d^2 + 2*a*e^2))*x^4 + ((b^4*d^4 + 16*a*b^3*d
^3*e + 16*a^2*b*d*e*(3*c*d^2 + a*e^2) + 12*a*b^2*d^2*(c*d^2 + 3*a*e^2) + a^2*(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2
*e^4))*x^5)/5 + (2*(b^4*d^3*e + 6*a*b^2*d*e*(2*c*d^2 + a*e^2) + 2*a^2*c*d*e*(3*c*d^2 + 2*a*e^2) + b^3*(c*d^4 +
6*a*d^2*e^2) + a*b*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^6)/3 + (2*(3*b^4*d^2*e^2 + 24*a*b*c*d*e*(c*d^2 +
a*e^2) + 8*b^3*(c*d^3*e + a*d*e^3) + 2*a*c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4) + 3*b^2*(c^2*d^4 + 12*a*c*d^2*
e^2 + a^2*e^4))*x^7)/7 + ((b^4*d*e^3 + 6*b^2*c*d*e*(c*d^2 + 2*a*e^2) + 2*a*c^2*d*e*(2*c*d^2 + 3*a*e^2) + b^3*(
6*c*d^2*e^2 + a*e^4) + b*c*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^8)/2 + ((c^4*d^4 + b^4*e^4 + 8*c^3*d^2*e*
(2*b*d + 3*a*e) + 4*b^2*c*e^3*(4*b*d + 3*a*e) + 6*c^2*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^9)/9 + (2*c*e*(
c^3*d^3 + b^3*e^3 + 3*b*c*e^2*(2*b*d + a*e) + 2*c^2*d*e*(3*b*d + 2*a*e))*x^10)/5 + (2*c^2*e^2*(3*c^2*d^2 + 3*b
^2*e^2 + 2*c*e*(4*b*d + a*e))*x^11)/11 + (c^3*e^3*(c*d + b*e)*x^12)/3 + (c^4*e^4*x^13)/13

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Maple [B]  time = 0.041, size = 949, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^4*e^4*x^13+1/12*(4*b*c^3*e^4+4*c^4*d*e^3)*x^12+1/11*(6*d^2*e^2*c^4+16*d*e^3*b*c^3+e^4*(2*(2*a*c+b^2)*c^
2+4*b^2*c^2))*x^11+1/10*(4*d^3*e*c^4+24*d^2*e^2*b*c^3+4*d*e^3*(2*(2*a*c+b^2)*c^2+4*b^2*c^2)+e^4*(4*b*a*c^2+4*(
2*a*c+b^2)*b*c))*x^10+1/9*(c^4*d^4+16*b*c^3*d^3*e+6*d^2*e^2*(2*(2*a*c+b^2)*c^2+4*b^2*c^2)+4*d*e^3*(4*b*a*c^2+4
*(2*a*c+b^2)*b*c)+e^4*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2))*x^9+1/8*(4*d^4*b*c^3+4*d^3*e*(2*(2*a*c+b^2)*c^2+4*b
^2*c^2)+6*d^2*e^2*(4*b*a*c^2+4*(2*a*c+b^2)*b*c)+4*d*e^3*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+e^4*(4*a^2*b*c+4*a
*b*(2*a*c+b^2)))*x^8+1/7*(d^4*(2*(2*a*c+b^2)*c^2+4*b^2*c^2)+4*d^3*e*(4*b*a*c^2+4*(2*a*c+b^2)*b*c)+6*d^2*e^2*(2
*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+4*d*e^3*(4*a^2*b*c+4*a*b*(2*a*c+b^2))+e^4*(2*a^2*(2*a*c+b^2)+4*b^2*a^2))*x^7
+1/6*(d^4*(4*b*a*c^2+4*(2*a*c+b^2)*b*c)+4*d^3*e*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+6*d^2*e^2*(4*a^2*b*c+4*a*b
*(2*a*c+b^2))+4*d*e^3*(2*a^2*(2*a*c+b^2)+4*b^2*a^2)+4*e^4*a^3*b)*x^6+1/5*(d^4*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)
^2)+4*d^3*e*(4*a^2*b*c+4*a*b*(2*a*c+b^2))+6*d^2*e^2*(2*a^2*(2*a*c+b^2)+4*b^2*a^2)+16*a^3*b*d*e^3+a^4*e^4)*x^5+
1/4*(d^4*(4*a^2*b*c+4*a*b*(2*a*c+b^2))+4*d^3*e*(2*a^2*(2*a*c+b^2)+4*b^2*a^2)+24*d^2*e^2*a^3*b+4*d*e^3*a^4)*x^4
+1/3*(d^4*(2*a^2*(2*a*c+b^2)+4*b^2*a^2)+16*d^3*e*a^3*b+6*d^2*e^2*a^4)*x^3+1/2*(4*a^4*d^3*e+4*a^3*b*d^4)*x^2+a^
4*d^4*x

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Maxima [A]  time = 1.11645, size = 1029, normalized size = 2.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^4*e^4*x^13 + 1/3*(c^4*d*e^3 + b*c^3*e^4)*x^12 + 2/11*(3*c^4*d^2*e^2 + 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*
c^3)*e^4)*x^11 + 2/5*(c^4*d^3*e + 6*b*c^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 + (b^3*c + 3*a*b*c^2)*e^4)*x
^10 + 1/9*(c^4*d^4 + 16*b*c^3*d^3*e + 12*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 + 16*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 +
12*a*b^2*c + 6*a^2*c^2)*e^4)*x^9 + a^4*d^4*x + 1/2*(b*c^3*d^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e + 6*(b^3*c + 3*
a*b*c^2)*d^2*e^2 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^3 + (a*b^3 + 3*a^2*b*c)*e^4)*x^8 + 2/7*((3*b^2*c^2 + 2*a
*c^3)*d^4 + 8*(b^3*c + 3*a*b*c^2)*d^3*e + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^2 + 8*(a*b^3 + 3*a^2*b*c)*d*e
^3 + (3*a^2*b^2 + 2*a^3*c)*e^4)*x^7 + 2/3*(a^3*b*e^4 + (b^3*c + 3*a*b*c^2)*d^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d^3*e + 6*(a*b^3 + 3*a^2*b*c)*d^2*e^2 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^3)*x^6 + 1/5*(16*a^3*b*d*e^3 + a^4*e^4 +
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 16*(a*b^3 + 3*a^2*b*c)*d^3*e + 12*(3*a^2*b^2 + 2*a^3*c)*d^2*e^2)*x^5 + (
6*a^3*b*d^2*e^2 + a^4*d*e^3 + (a*b^3 + 3*a^2*b*c)*d^4 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e)*x^4 + 2/3*(8*a^3*b*d^3*
e + 3*a^4*d^2*e^2 + (3*a^2*b^2 + 2*a^3*c)*d^4)*x^3 + 2*(a^3*b*d^4 + a^4*d^3*e)*x^2

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Fricas [B]  time = 1.72971, size = 2202, normalized size = 4.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*e^4*c^4 + 1/3*x^12*e^3*d*c^4 + 1/3*x^12*e^4*c^3*b + 6/11*x^11*e^2*d^2*c^4 + 16/11*x^11*e^3*d*c^3*b +
6/11*x^11*e^4*c^2*b^2 + 4/11*x^11*e^4*c^3*a + 2/5*x^10*e*d^3*c^4 + 12/5*x^10*e^2*d^2*c^3*b + 12/5*x^10*e^3*d*
c^2*b^2 + 2/5*x^10*e^4*c*b^3 + 8/5*x^10*e^3*d*c^3*a + 6/5*x^10*e^4*c^2*b*a + 1/9*x^9*d^4*c^4 + 16/9*x^9*e*d^3*
c^3*b + 4*x^9*e^2*d^2*c^2*b^2 + 16/9*x^9*e^3*d*c*b^3 + 1/9*x^9*e^4*b^4 + 8/3*x^9*e^2*d^2*c^3*a + 16/3*x^9*e^3*
d*c^2*b*a + 4/3*x^9*e^4*c*b^2*a + 2/3*x^9*e^4*c^2*a^2 + 1/2*x^8*d^4*c^3*b + 3*x^8*e*d^3*c^2*b^2 + 3*x^8*e^2*d^
2*c*b^3 + 1/2*x^8*e^3*d*b^4 + 2*x^8*e*d^3*c^3*a + 9*x^8*e^2*d^2*c^2*b*a + 6*x^8*e^3*d*c*b^2*a + 1/2*x^8*e^4*b^
3*a + 3*x^8*e^3*d*c^2*a^2 + 3/2*x^8*e^4*c*b*a^2 + 6/7*x^7*d^4*c^2*b^2 + 16/7*x^7*e*d^3*c*b^3 + 6/7*x^7*e^2*d^2
*b^4 + 4/7*x^7*d^4*c^3*a + 48/7*x^7*e*d^3*c^2*b*a + 72/7*x^7*e^2*d^2*c*b^2*a + 16/7*x^7*e^3*d*b^3*a + 36/7*x^7
*e^2*d^2*c^2*a^2 + 48/7*x^7*e^3*d*c*b*a^2 + 6/7*x^7*e^4*b^2*a^2 + 4/7*x^7*e^4*c*a^3 + 2/3*x^6*d^4*c*b^3 + 2/3*
x^6*e*d^3*b^4 + 2*x^6*d^4*c^2*b*a + 8*x^6*e*d^3*c*b^2*a + 4*x^6*e^2*d^2*b^3*a + 4*x^6*e*d^3*c^2*a^2 + 12*x^6*e
^2*d^2*c*b*a^2 + 4*x^6*e^3*d*b^2*a^2 + 8/3*x^6*e^3*d*c*a^3 + 2/3*x^6*e^4*b*a^3 + 1/5*x^5*d^4*b^4 + 12/5*x^5*d^
4*c*b^2*a + 16/5*x^5*e*d^3*b^3*a + 6/5*x^5*d^4*c^2*a^2 + 48/5*x^5*e*d^3*c*b*a^2 + 36/5*x^5*e^2*d^2*b^2*a^2 + 2
4/5*x^5*e^2*d^2*c*a^3 + 16/5*x^5*e^3*d*b*a^3 + 1/5*x^5*e^4*a^4 + x^4*d^4*b^3*a + 3*x^4*d^4*c*b*a^2 + 6*x^4*e*d
^3*b^2*a^2 + 4*x^4*e*d^3*c*a^3 + 6*x^4*e^2*d^2*b*a^3 + x^4*e^3*d*a^4 + 2*x^3*d^4*b^2*a^2 + 4/3*x^3*d^4*c*a^3 +
16/3*x^3*e*d^3*b*a^3 + 2*x^3*e^2*d^2*a^4 + 2*x^2*d^4*b*a^3 + 2*x^2*e*d^3*a^4 + x*d^4*a^4

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Sympy [B]  time = 0.18071, size = 998, normalized size = 2.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**4*x + c**4*e**4*x**13/13 + x**12*(b*c**3*e**4/3 + c**4*d*e**3/3) + x**11*(4*a*c**3*e**4/11 + 6*b**2*c*
*2*e**4/11 + 16*b*c**3*d*e**3/11 + 6*c**4*d**2*e**2/11) + x**10*(6*a*b*c**2*e**4/5 + 8*a*c**3*d*e**3/5 + 2*b**
3*c*e**4/5 + 12*b**2*c**2*d*e**3/5 + 12*b*c**3*d**2*e**2/5 + 2*c**4*d**3*e/5) + x**9*(2*a**2*c**2*e**4/3 + 4*a
*b**2*c*e**4/3 + 16*a*b*c**2*d*e**3/3 + 8*a*c**3*d**2*e**2/3 + b**4*e**4/9 + 16*b**3*c*d*e**3/9 + 4*b**2*c**2*
d**2*e**2 + 16*b*c**3*d**3*e/9 + c**4*d**4/9) + x**8*(3*a**2*b*c*e**4/2 + 3*a**2*c**2*d*e**3 + a*b**3*e**4/2 +
6*a*b**2*c*d*e**3 + 9*a*b*c**2*d**2*e**2 + 2*a*c**3*d**3*e + b**4*d*e**3/2 + 3*b**3*c*d**2*e**2 + 3*b**2*c**2
*d**3*e + b*c**3*d**4/2) + x**7*(4*a**3*c*e**4/7 + 6*a**2*b**2*e**4/7 + 48*a**2*b*c*d*e**3/7 + 36*a**2*c**2*d*
*2*e**2/7 + 16*a*b**3*d*e**3/7 + 72*a*b**2*c*d**2*e**2/7 + 48*a*b*c**2*d**3*e/7 + 4*a*c**3*d**4/7 + 6*b**4*d**
2*e**2/7 + 16*b**3*c*d**3*e/7 + 6*b**2*c**2*d**4/7) + x**6*(2*a**3*b*e**4/3 + 8*a**3*c*d*e**3/3 + 4*a**2*b**2*
d*e**3 + 12*a**2*b*c*d**2*e**2 + 4*a**2*c**2*d**3*e + 4*a*b**3*d**2*e**2 + 8*a*b**2*c*d**3*e + 2*a*b*c**2*d**4
+ 2*b**4*d**3*e/3 + 2*b**3*c*d**4/3) + x**5*(a**4*e**4/5 + 16*a**3*b*d*e**3/5 + 24*a**3*c*d**2*e**2/5 + 36*a*
*2*b**2*d**2*e**2/5 + 48*a**2*b*c*d**3*e/5 + 6*a**2*c**2*d**4/5 + 16*a*b**3*d**3*e/5 + 12*a*b**2*c*d**4/5 + b*
*4*d**4/5) + x**4*(a**4*d*e**3 + 6*a**3*b*d**2*e**2 + 4*a**3*c*d**3*e + 6*a**2*b**2*d**3*e + 3*a**2*b*c*d**4 +
a*b**3*d**4) + x**3*(2*a**4*d**2*e**2 + 16*a**3*b*d**3*e/3 + 4*a**3*c*d**4/3 + 2*a**2*b**2*d**4) + x**2*(2*a*
*4*d**3*e + 2*a**3*b*d**4)

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Giac [B]  time = 1.15786, size = 1311, normalized size = 2.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^4*x^13*e^4 + 1/3*c^4*d*x^12*e^3 + 6/11*c^4*d^2*x^11*e^2 + 2/5*c^4*d^3*x^10*e + 1/9*c^4*d^4*x^9 + 1/3*b*
c^3*x^12*e^4 + 16/11*b*c^3*d*x^11*e^3 + 12/5*b*c^3*d^2*x^10*e^2 + 16/9*b*c^3*d^3*x^9*e + 1/2*b*c^3*d^4*x^8 + 6
/11*b^2*c^2*x^11*e^4 + 4/11*a*c^3*x^11*e^4 + 12/5*b^2*c^2*d*x^10*e^3 + 8/5*a*c^3*d*x^10*e^3 + 4*b^2*c^2*d^2*x^
9*e^2 + 8/3*a*c^3*d^2*x^9*e^2 + 3*b^2*c^2*d^3*x^8*e + 2*a*c^3*d^3*x^8*e + 6/7*b^2*c^2*d^4*x^7 + 4/7*a*c^3*d^4*
x^7 + 2/5*b^3*c*x^10*e^4 + 6/5*a*b*c^2*x^10*e^4 + 16/9*b^3*c*d*x^9*e^3 + 16/3*a*b*c^2*d*x^9*e^3 + 3*b^3*c*d^2*
x^8*e^2 + 9*a*b*c^2*d^2*x^8*e^2 + 16/7*b^3*c*d^3*x^7*e + 48/7*a*b*c^2*d^3*x^7*e + 2/3*b^3*c*d^4*x^6 + 2*a*b*c^
2*d^4*x^6 + 1/9*b^4*x^9*e^4 + 4/3*a*b^2*c*x^9*e^4 + 2/3*a^2*c^2*x^9*e^4 + 1/2*b^4*d*x^8*e^3 + 6*a*b^2*c*d*x^8*
e^3 + 3*a^2*c^2*d*x^8*e^3 + 6/7*b^4*d^2*x^7*e^2 + 72/7*a*b^2*c*d^2*x^7*e^2 + 36/7*a^2*c^2*d^2*x^7*e^2 + 2/3*b^
4*d^3*x^6*e + 8*a*b^2*c*d^3*x^6*e + 4*a^2*c^2*d^3*x^6*e + 1/5*b^4*d^4*x^5 + 12/5*a*b^2*c*d^4*x^5 + 6/5*a^2*c^2
*d^4*x^5 + 1/2*a*b^3*x^8*e^4 + 3/2*a^2*b*c*x^8*e^4 + 16/7*a*b^3*d*x^7*e^3 + 48/7*a^2*b*c*d*x^7*e^3 + 4*a*b^3*d
^2*x^6*e^2 + 12*a^2*b*c*d^2*x^6*e^2 + 16/5*a*b^3*d^3*x^5*e + 48/5*a^2*b*c*d^3*x^5*e + a*b^3*d^4*x^4 + 3*a^2*b*
c*d^4*x^4 + 6/7*a^2*b^2*x^7*e^4 + 4/7*a^3*c*x^7*e^4 + 4*a^2*b^2*d*x^6*e^3 + 8/3*a^3*c*d*x^6*e^3 + 36/5*a^2*b^2
*d^2*x^5*e^2 + 24/5*a^3*c*d^2*x^5*e^2 + 6*a^2*b^2*d^3*x^4*e + 4*a^3*c*d^3*x^4*e + 2*a^2*b^2*d^4*x^3 + 4/3*a^3*
c*d^4*x^3 + 2/3*a^3*b*x^6*e^4 + 16/5*a^3*b*d*x^5*e^3 + 6*a^3*b*d^2*x^4*e^2 + 16/3*a^3*b*d^3*x^3*e + 2*a^3*b*d^
4*x^2 + 1/5*a^4*x^5*e^4 + a^4*d*x^4*e^3 + 2*a^4*d^2*x^3*e^2 + 2*a^4*d^3*x^2*e + a^4*d^4*x