3.2147 \(\int (d+e x)^3 (a+b x+c x^2)^4 \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.673569, 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} \[ \frac{(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{8 e^9}+\frac{c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac{4 c (d+e x)^9 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^9}-\frac{4 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac{4 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}{4 e^9}-\frac{4 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^9}+\frac{c^4 (d+e x)^{12}}{12 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.203673, size = 611, normalized size = 1.38 \[ \frac{1}{6} x^6 \left (2 a^2 c e \left (2 a e^2+9 c d^2\right )+4 b^3 \left (3 a d e^2+c d^3\right )+6 a b^2 e \left (a e^2+6 c d^2\right )+12 a b c d \left (3 a e^2+c d^2\right )+3 b^4 d^2 e\right )+\frac{1}{5} x^5 \left (4 a^2 b e \left (a e^2+9 c d^2\right )+6 a^2 c d \left (2 a e^2+c d^2\right )+6 a b^2 d \left (3 a e^2+2 c d^2\right )+12 a b^3 d^2 e+b^4 d^3\right )+\frac{1}{4} a x^4 \left (a^2 e \left (a e^2+12 c d^2\right )+18 a b^2 d^2 e+12 a b d \left (a e^2+c d^2\right )+4 b^3 d^3\right )+\frac{1}{3} a^2 d x^3 \left (12 a b d e+a \left (3 a e^2+4 c d^2\right )+6 b^2 d^2\right )+\frac{1}{2} a^3 d^2 x^2 (3 a e+4 b d)+a^4 d^3 x+\frac{1}{10} c^2 e x^{10} \left (4 c e (a e+3 b d)+6 b^2 e^2+3 c^2 d^2\right )+\frac{1}{9} c x^9 \left (12 c^2 d e (a e+b d)+6 b c e^2 (2 a e+3 b d)+4 b^3 e^3+c^3 d^3\right )+\frac{1}{8} x^8 \left (6 b^2 c e \left (2 a e^2+3 c d^2\right )+4 b c^2 d \left (9 a e^2+c d^2\right )+6 a c^2 e \left (a e^2+2 c d^2\right )+12 b^3 c d e^2+b^4 e^3\right )+\frac{1}{7} x^7 \left (4 b^3 \left (a e^3+3 c d^2 e\right )+6 b^2 c d \left (6 a e^2+c d^2\right )+12 a b c e \left (a e^2+3 c d^2\right )+2 a c^2 d \left (9 a e^2+2 c d^2\right )+3 b^4 d e^2\right )+\frac{1}{11} c^3 e^2 x^{11} (4 b e+3 c d)+\frac{1}{12} c^4 e^3 x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 747, normalized size = 1.7 \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)^3*(c*x^2+b*x+a)^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.999234, size = 829, normalized size = 1.87 \begin{align*} \frac{1}{12} \, c^{4} e^{3} x^{12} + \frac{1}{11} \,{\left (3 \, c^{4} d e^{2} + 4 \, b c^{3} e^{3}\right )} x^{11} + \frac{1}{10} \,{\left (3 \, c^{4} d^{2} e + 12 \, b c^{3} d e^{2} + 2 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{10} + \frac{1}{9} \,{\left (c^{4} d^{3} + 12 \, b c^{3} d^{2} e + 6 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} + 4 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (4 \, b c^{3} d^{3} + 6 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e + 12 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{2} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac{1}{7} \,{\left (2 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 12 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e + 3 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{2} + 4 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (4 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} + 3 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e + 12 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{2} + 2 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, a^{3} b e^{3} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 12 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e + 6 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (12 \, a^{3} b d e^{2} + a^{4} e^{3} + 4 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} + 6 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e\right )} x^{4} + \frac{1}{3} \,{\left (12 \, a^{3} b d^{2} e + 3 \, a^{4} d e^{2} + 2 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.73839, size = 1704, normalized size = 3.85 \begin{align*} \frac{1}{12} x^{12} e^{3} c^{4} + \frac{3}{11} x^{11} e^{2} d c^{4} + \frac{4}{11} x^{11} e^{3} c^{3} b + \frac{3}{10} x^{10} e d^{2} c^{4} + \frac{6}{5} x^{10} e^{2} d c^{3} b + \frac{3}{5} x^{10} e^{3} c^{2} b^{2} + \frac{2}{5} x^{10} e^{3} c^{3} a + \frac{1}{9} x^{9} d^{3} c^{4} + \frac{4}{3} x^{9} e d^{2} c^{3} b + 2 x^{9} e^{2} d c^{2} b^{2} + \frac{4}{9} x^{9} e^{3} c b^{3} + \frac{4}{3} x^{9} e^{2} d c^{3} a + \frac{4}{3} x^{9} e^{3} c^{2} b a + \frac{1}{2} x^{8} d^{3} c^{3} b + \frac{9}{4} x^{8} e d^{2} c^{2} b^{2} + \frac{3}{2} x^{8} e^{2} d c b^{3} + \frac{1}{8} x^{8} e^{3} b^{4} + \frac{3}{2} x^{8} e d^{2} c^{3} a + \frac{9}{2} x^{8} e^{2} d c^{2} b a + \frac{3}{2} x^{8} e^{3} c b^{2} a + \frac{3}{4} x^{8} e^{3} c^{2} a^{2} + \frac{6}{7} x^{7} d^{3} c^{2} b^{2} + \frac{12}{7} x^{7} e d^{2} c b^{3} + \frac{3}{7} x^{7} e^{2} d b^{4} + \frac{4}{7} x^{7} d^{3} c^{3} a + \frac{36}{7} x^{7} e d^{2} c^{2} b a + \frac{36}{7} x^{7} e^{2} d c b^{2} a + \frac{4}{7} x^{7} e^{3} b^{3} a + \frac{18}{7} x^{7} e^{2} d c^{2} a^{2} + \frac{12}{7} x^{7} e^{3} c b a^{2} + \frac{2}{3} x^{6} d^{3} c b^{3} + \frac{1}{2} x^{6} e d^{2} b^{4} + 2 x^{6} d^{3} c^{2} b a + 6 x^{6} e d^{2} c b^{2} a + 2 x^{6} e^{2} d b^{3} a + 3 x^{6} e d^{2} c^{2} a^{2} + 6 x^{6} e^{2} d c b a^{2} + x^{6} e^{3} b^{2} a^{2} + \frac{2}{3} x^{6} e^{3} c a^{3} + \frac{1}{5} x^{5} d^{3} b^{4} + \frac{12}{5} x^{5} d^{3} c b^{2} a + \frac{12}{5} x^{5} e d^{2} b^{3} a + \frac{6}{5} x^{5} d^{3} c^{2} a^{2} + \frac{36}{5} x^{5} e d^{2} c b a^{2} + \frac{18}{5} x^{5} e^{2} d b^{2} a^{2} + \frac{12}{5} x^{5} e^{2} d c a^{3} + \frac{4}{5} x^{5} e^{3} b a^{3} + x^{4} d^{3} b^{3} a + 3 x^{4} d^{3} c b a^{2} + \frac{9}{2} x^{4} e d^{2} b^{2} a^{2} + 3 x^{4} e d^{2} c a^{3} + 3 x^{4} e^{2} d b a^{3} + \frac{1}{4} x^{4} e^{3} a^{4} + 2 x^{3} d^{3} b^{2} a^{2} + \frac{4}{3} x^{3} d^{3} c a^{3} + 4 x^{3} e d^{2} b a^{3} + x^{3} e^{2} d a^{4} + 2 x^{2} d^{3} b a^{3} + \frac{3}{2} x^{2} e d^{2} a^{4} + x d^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*e^3*c^4 + 3/11*x^11*e^2*d*c^4 + 4/11*x^11*e^3*c^3*b + 3/10*x^10*e*d^2*c^4 + 6/5*x^10*e^2*d*c^3*b + 3
/5*x^10*e^3*c^2*b^2 + 2/5*x^10*e^3*c^3*a + 1/9*x^9*d^3*c^4 + 4/3*x^9*e*d^2*c^3*b + 2*x^9*e^2*d*c^2*b^2 + 4/9*x
^9*e^3*c*b^3 + 4/3*x^9*e^2*d*c^3*a + 4/3*x^9*e^3*c^2*b*a + 1/2*x^8*d^3*c^3*b + 9/4*x^8*e*d^2*c^2*b^2 + 3/2*x^8
*e^2*d*c*b^3 + 1/8*x^8*e^3*b^4 + 3/2*x^8*e*d^2*c^3*a + 9/2*x^8*e^2*d*c^2*b*a + 3/2*x^8*e^3*c*b^2*a + 3/4*x^8*e
^3*c^2*a^2 + 6/7*x^7*d^3*c^2*b^2 + 12/7*x^7*e*d^2*c*b^3 + 3/7*x^7*e^2*d*b^4 + 4/7*x^7*d^3*c^3*a + 36/7*x^7*e*d
^2*c^2*b*a + 36/7*x^7*e^2*d*c*b^2*a + 4/7*x^7*e^3*b^3*a + 18/7*x^7*e^2*d*c^2*a^2 + 12/7*x^7*e^3*c*b*a^2 + 2/3*
x^6*d^3*c*b^3 + 1/2*x^6*e*d^2*b^4 + 2*x^6*d^3*c^2*b*a + 6*x^6*e*d^2*c*b^2*a + 2*x^6*e^2*d*b^3*a + 3*x^6*e*d^2*
c^2*a^2 + 6*x^6*e^2*d*c*b*a^2 + x^6*e^3*b^2*a^2 + 2/3*x^6*e^3*c*a^3 + 1/5*x^5*d^3*b^4 + 12/5*x^5*d^3*c*b^2*a +
12/5*x^5*e*d^2*b^3*a + 6/5*x^5*d^3*c^2*a^2 + 36/5*x^5*e*d^2*c*b*a^2 + 18/5*x^5*e^2*d*b^2*a^2 + 12/5*x^5*e^2*d
*c*a^3 + 4/5*x^5*e^3*b*a^3 + x^4*d^3*b^3*a + 3*x^4*d^3*c*b*a^2 + 9/2*x^4*e*d^2*b^2*a^2 + 3*x^4*e*d^2*c*a^3 + 3
*x^4*e^2*d*b*a^3 + 1/4*x^4*e^3*a^4 + 2*x^3*d^3*b^2*a^2 + 4/3*x^3*d^3*c*a^3 + 4*x^3*e*d^2*b*a^3 + x^3*e^2*d*a^4
+ 2*x^2*d^3*b*a^3 + 3/2*x^2*e*d^2*a^4 + x*d^3*a^4

________________________________________________________________________________________

Sympy [A]  time = 0.151718, size = 777, normalized size = 1.75 \begin{align*} a^{4} d^{3} x + \frac{c^{4} e^{3} x^{12}}{12} + x^{11} \left (\frac{4 b c^{3} e^{3}}{11} + \frac{3 c^{4} d e^{2}}{11}\right ) + x^{10} \left (\frac{2 a c^{3} e^{3}}{5} + \frac{3 b^{2} c^{2} e^{3}}{5} + \frac{6 b c^{3} d e^{2}}{5} + \frac{3 c^{4} d^{2} e}{10}\right ) + x^{9} \left (\frac{4 a b c^{2} e^{3}}{3} + \frac{4 a c^{3} d e^{2}}{3} + \frac{4 b^{3} c e^{3}}{9} + 2 b^{2} c^{2} d e^{2} + \frac{4 b c^{3} d^{2} e}{3} + \frac{c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac{3 a^{2} c^{2} e^{3}}{4} + \frac{3 a b^{2} c e^{3}}{2} + \frac{9 a b c^{2} d e^{2}}{2} + \frac{3 a c^{3} d^{2} e}{2} + \frac{b^{4} e^{3}}{8} + \frac{3 b^{3} c d e^{2}}{2} + \frac{9 b^{2} c^{2} d^{2} e}{4} + \frac{b c^{3} d^{3}}{2}\right ) + x^{7} \left (\frac{12 a^{2} b c e^{3}}{7} + \frac{18 a^{2} c^{2} d e^{2}}{7} + \frac{4 a b^{3} e^{3}}{7} + \frac{36 a b^{2} c d e^{2}}{7} + \frac{36 a b c^{2} d^{2} e}{7} + \frac{4 a c^{3} d^{3}}{7} + \frac{3 b^{4} d e^{2}}{7} + \frac{12 b^{3} c d^{2} e}{7} + \frac{6 b^{2} c^{2} d^{3}}{7}\right ) + x^{6} \left (\frac{2 a^{3} c e^{3}}{3} + a^{2} b^{2} e^{3} + 6 a^{2} b c d e^{2} + 3 a^{2} c^{2} d^{2} e + 2 a b^{3} d e^{2} + 6 a b^{2} c d^{2} e + 2 a b c^{2} d^{3} + \frac{b^{4} d^{2} e}{2} + \frac{2 b^{3} c d^{3}}{3}\right ) + x^{5} \left (\frac{4 a^{3} b e^{3}}{5} + \frac{12 a^{3} c d e^{2}}{5} + \frac{18 a^{2} b^{2} d e^{2}}{5} + \frac{36 a^{2} b c d^{2} e}{5} + \frac{6 a^{2} c^{2} d^{3}}{5} + \frac{12 a b^{3} d^{2} e}{5} + \frac{12 a b^{2} c d^{3}}{5} + \frac{b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + 3 a^{3} c d^{2} e + \frac{9 a^{2} b^{2} d^{2} e}{2} + 3 a^{2} b c d^{3} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + \frac{4 a^{3} c d^{3}}{3} + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**3*x + c**4*e**3*x**12/12 + x**11*(4*b*c**3*e**3/11 + 3*c**4*d*e**2/11) + x**10*(2*a*c**3*e**3/5 + 3*b*
*2*c**2*e**3/5 + 6*b*c**3*d*e**2/5 + 3*c**4*d**2*e/10) + x**9*(4*a*b*c**2*e**3/3 + 4*a*c**3*d*e**2/3 + 4*b**3*
c*e**3/9 + 2*b**2*c**2*d*e**2 + 4*b*c**3*d**2*e/3 + c**4*d**3/9) + x**8*(3*a**2*c**2*e**3/4 + 3*a*b**2*c*e**3/
2 + 9*a*b*c**2*d*e**2/2 + 3*a*c**3*d**2*e/2 + b**4*e**3/8 + 3*b**3*c*d*e**2/2 + 9*b**2*c**2*d**2*e/4 + b*c**3*
d**3/2) + x**7*(12*a**2*b*c*e**3/7 + 18*a**2*c**2*d*e**2/7 + 4*a*b**3*e**3/7 + 36*a*b**2*c*d*e**2/7 + 36*a*b*c
**2*d**2*e/7 + 4*a*c**3*d**3/7 + 3*b**4*d*e**2/7 + 12*b**3*c*d**2*e/7 + 6*b**2*c**2*d**3/7) + x**6*(2*a**3*c*e
**3/3 + a**2*b**2*e**3 + 6*a**2*b*c*d*e**2 + 3*a**2*c**2*d**2*e + 2*a*b**3*d*e**2 + 6*a*b**2*c*d**2*e + 2*a*b*
c**2*d**3 + b**4*d**2*e/2 + 2*b**3*c*d**3/3) + x**5*(4*a**3*b*e**3/5 + 12*a**3*c*d*e**2/5 + 18*a**2*b**2*d*e**
2/5 + 36*a**2*b*c*d**2*e/5 + 6*a**2*c**2*d**3/5 + 12*a*b**3*d**2*e/5 + 12*a*b**2*c*d**3/5 + b**4*d**3/5) + x**
4*(a**4*e**3/4 + 3*a**3*b*d*e**2 + 3*a**3*c*d**2*e + 9*a**2*b**2*d**2*e/2 + 3*a**2*b*c*d**3 + a*b**3*d**3) + x
**3*(a**4*d*e**2 + 4*a**3*b*d**2*e + 4*a**3*c*d**3/3 + 2*a**2*b**2*d**3) + x**2*(3*a**4*d**2*e/2 + 2*a**3*b*d*
*3)

________________________________________________________________________________________

Giac [A]  time = 1.103, size = 1018, normalized size = 2.3 \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)^3*(c*x^2+b*x+a)^4,x, algorithm="giac")

[Out]

1/12*c^4*x^12*e^3 + 3/11*c^4*d*x^11*e^2 + 3/10*c^4*d^2*x^10*e + 1/9*c^4*d^3*x^9 + 4/11*b*c^3*x^11*e^3 + 6/5*b*
c^3*d*x^10*e^2 + 4/3*b*c^3*d^2*x^9*e + 1/2*b*c^3*d^3*x^8 + 3/5*b^2*c^2*x^10*e^3 + 2/5*a*c^3*x^10*e^3 + 2*b^2*c
^2*d*x^9*e^2 + 4/3*a*c^3*d*x^9*e^2 + 9/4*b^2*c^2*d^2*x^8*e + 3/2*a*c^3*d^2*x^8*e + 6/7*b^2*c^2*d^3*x^7 + 4/7*a
*c^3*d^3*x^7 + 4/9*b^3*c*x^9*e^3 + 4/3*a*b*c^2*x^9*e^3 + 3/2*b^3*c*d*x^8*e^2 + 9/2*a*b*c^2*d*x^8*e^2 + 12/7*b^
3*c*d^2*x^7*e + 36/7*a*b*c^2*d^2*x^7*e + 2/3*b^3*c*d^3*x^6 + 2*a*b*c^2*d^3*x^6 + 1/8*b^4*x^8*e^3 + 3/2*a*b^2*c
*x^8*e^3 + 3/4*a^2*c^2*x^8*e^3 + 3/7*b^4*d*x^7*e^2 + 36/7*a*b^2*c*d*x^7*e^2 + 18/7*a^2*c^2*d*x^7*e^2 + 1/2*b^4
*d^2*x^6*e + 6*a*b^2*c*d^2*x^6*e + 3*a^2*c^2*d^2*x^6*e + 1/5*b^4*d^3*x^5 + 12/5*a*b^2*c*d^3*x^5 + 6/5*a^2*c^2*
d^3*x^5 + 4/7*a*b^3*x^7*e^3 + 12/7*a^2*b*c*x^7*e^3 + 2*a*b^3*d*x^6*e^2 + 6*a^2*b*c*d*x^6*e^2 + 12/5*a*b^3*d^2*
x^5*e + 36/5*a^2*b*c*d^2*x^5*e + a*b^3*d^3*x^4 + 3*a^2*b*c*d^3*x^4 + a^2*b^2*x^6*e^3 + 2/3*a^3*c*x^6*e^3 + 18/
5*a^2*b^2*d*x^5*e^2 + 12/5*a^3*c*d*x^5*e^2 + 9/2*a^2*b^2*d^2*x^4*e + 3*a^3*c*d^2*x^4*e + 2*a^2*b^2*d^3*x^3 + 4
/3*a^3*c*d^3*x^3 + 4/5*a^3*b*x^5*e^3 + 3*a^3*b*d*x^4*e^2 + 4*a^3*b*d^2*x^3*e + 2*a^3*b*d^3*x^2 + 1/4*a^4*x^4*e
^3 + a^4*d*x^3*e^2 + 3/2*a^4*d^2*x^2*e + a^4*d^3*x