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]

1/4*(a*e^2-b*d*e+c*d^2)^4*(e*x+d)^4/e^9-4/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^9+1/3*(a*e^2-b*d*e+
c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^6/e^9-4/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d
^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^7/e^9+1/8*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(
-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^8/e^9-4/9*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c
*e*(-3*a*e+7*b*d))*(e*x+d)^9/e^9+1/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^10/e^9-4/11*c^3*(-b
*e+2*c*d)*(e*x+d)^11/e^9+1/12*c^4*(e*x+d)^12/e^9

________________________________________________________________________________________

Rubi [A]  time = 0.67, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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 = 0.19, size = 611, normalized size = 1.38 \[ a^4 d^3 x+\frac {1}{2} a^3 d^2 x^2 (3 a e+4 b d)+\frac {1}{3} a^2 d x^3 \left (12 a b d e+a \left (3 a e^2+4 c d^2\right )+6 b^2 d^2\right )+\frac {1}{4} a x^4 \left (a^2 e \left (a e^2+12 c d^2\right )+18 a b^2 d^2 e+12 a b d \left (a e^2+c d^2\right )+4 b^3 d^3\right )+\frac {1}{6} x^6 \left (2 a^2 c e \left (2 a e^2+9 c d^2\right )+4 b^3 \left (3 a d e^2+c d^3\right )+6 a b^2 e \left (a e^2+6 c d^2\right )+12 a b c d \left (3 a e^2+c d^2\right )+3 b^4 d^2 e\right )+\frac {1}{5} x^5 \left (4 a^2 b e \left (a e^2+9 c d^2\right )+6 a^2 c d \left (2 a e^2+c d^2\right )+12 a b^3 d^2 e+6 a b^2 d \left (3 a e^2+2 c d^2\right )+b^4 d^3\right )+\frac {1}{9} c x^9 \left (12 c^2 d e (a e+b d)+6 b c e^2 (2 a e+3 b d)+4 b^3 e^3+c^3 d^3\right )+\frac {1}{10} c^2 e x^{10} \left (4 c e (a e+3 b d)+6 b^2 e^2+3 c^2 d^2\right )+\frac {1}{8} x^8 \left (6 b^2 c e \left (2 a e^2+3 c d^2\right )+4 b c^2 d \left (9 a e^2+c d^2\right )+6 a c^2 e \left (a e^2+2 c d^2\right )+b^4 e^3+12 b^3 c d e^2\right )+\frac {1}{7} x^7 \left (4 b^3 \left (a e^3+3 c d^2 e\right )+6 b^2 c d \left (6 a e^2+c d^2\right )+12 a b c e \left (a e^2+3 c d^2\right )+2 a c^2 d \left (9 a e^2+2 c d^2\right )+3 b^4 d e^2\right )+\frac {1}{11} c^3 e^2 x^{11} (4 b e+3 c d)+\frac {1}{12} c^4 e^3 x^{12} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 769, normalized size = 1.74 \[ \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} \]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 754, normalized size = 1.70 \[ \frac {1}{12} \, c^{4} x^{12} e^{3} + \frac {3}{11} \, c^{4} d x^{11} e^{2} + \frac {3}{10} \, c^{4} d^{2} x^{10} e + \frac {1}{9} \, c^{4} d^{3} x^{9} + \frac {4}{11} \, b c^{3} x^{11} e^{3} + \frac {6}{5} \, b c^{3} d x^{10} e^{2} + \frac {4}{3} \, b c^{3} d^{2} x^{9} e + \frac {1}{2} \, b c^{3} d^{3} x^{8} + \frac {3}{5} \, b^{2} c^{2} x^{10} e^{3} + \frac {2}{5} \, a c^{3} x^{10} e^{3} + 2 \, b^{2} c^{2} d x^{9} e^{2} + \frac {4}{3} \, a c^{3} d x^{9} e^{2} + \frac {9}{4} \, b^{2} c^{2} d^{2} x^{8} e + \frac {3}{2} \, a c^{3} d^{2} x^{8} e + \frac {6}{7} \, b^{2} c^{2} d^{3} x^{7} + \frac {4}{7} \, a c^{3} d^{3} x^{7} + \frac {4}{9} \, b^{3} c x^{9} e^{3} + \frac {4}{3} \, a b c^{2} x^{9} e^{3} + \frac {3}{2} \, b^{3} c d x^{8} e^{2} + \frac {9}{2} \, a b c^{2} d x^{8} e^{2} + \frac {12}{7} \, b^{3} c d^{2} x^{7} e + \frac {36}{7} \, a b c^{2} d^{2} x^{7} e + \frac {2}{3} \, b^{3} c d^{3} x^{6} + 2 \, a b c^{2} d^{3} x^{6} + \frac {1}{8} \, b^{4} x^{8} e^{3} + \frac {3}{2} \, a b^{2} c x^{8} e^{3} + \frac {3}{4} \, a^{2} c^{2} x^{8} e^{3} + \frac {3}{7} \, b^{4} d x^{7} e^{2} + \frac {36}{7} \, a b^{2} c d x^{7} e^{2} + \frac {18}{7} \, a^{2} c^{2} d x^{7} e^{2} + \frac {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 + \frac {1}{5} \, b^{4} d^{3} x^{5} + \frac {12}{5} \, a b^{2} c d^{3} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {4}{7} \, a b^{3} x^{7} e^{3} + \frac {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} + \frac {12}{5} \, a b^{3} d^{2} x^{5} e + \frac {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} + \frac {2}{3} \, a^{3} c x^{6} e^{3} + \frac {18}{5} \, a^{2} b^{2} d x^{5} e^{2} + \frac {12}{5} \, a^{3} c d x^{5} e^{2} + \frac {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} + \frac {4}{3} \, a^{3} c d^{3} x^{3} + \frac {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} + \frac {1}{4} \, a^{4} x^{4} e^{3} + a^{4} d x^{3} e^{2} + \frac {3}{2} \, a^{4} d^{2} x^{2} e + a^{4} d^{3} x \]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 747, normalized size = 1.69 \[ \frac {c^{4} e^{3} x^{12}}{12}+\frac {\left (4 e^{3} b \,c^{3}+3 d \,e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (12 b \,c^{3} d \,e^{2}+3 c^{4} d^{2} e +\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) e^{3}\right ) x^{10}}{10}+\frac {\left (12 b \,c^{3} d^{2} e +c^{4} d^{3}+3 \left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d \,e^{2}+\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) e^{3}\right ) x^{9}}{9}+a^{4} d^{3} x +\frac {\left (4 b \,c^{3} d^{3}+3 \left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d^{2} e +3 \left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d \,e^{2}+\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) e^{3}\right ) x^{8}}{8}+\frac {\left (\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d^{3}+3 \left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d^{2} e +3 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d \,e^{2}+\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) e^{3}\right ) x^{7}}{7}+\frac {\left (\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d^{3}+3 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d^{2} e +3 \left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d \,e^{2}+\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) e^{3}\right ) x^{6}}{6}+\frac {\left (4 a^{3} b \,e^{3}+\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d^{3}+3 \left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d^{2} e +3 \left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{4} e^{3}+12 a^{3} b d \,e^{2}+\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d^{3}+3 \left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d^{2} e \right ) x^{4}}{4}+\frac {\left (3 a^{4} d \,e^{2}+12 a^{3} b \,d^{2} e +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{4}+4 d^{3} a^{3} b \right ) x^{2}}{2} \]

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*(4*b^2*c^2+2*(2*a*c
+b^2)*c^2))*x^10+1/9*(d^3*c^4+12*d^2*e*b*c^3+3*d*e^2*(4*b^2*c^2+2*(2*a*c+b^2)*c^2)+e^3*(4*a*b*c^2+4*(2*a*c+b^2
)*b*c))*x^9+1/8*(4*d^3*b*c^3+3*d^2*e*(4*b^2*c^2+2*(2*a*c+b^2)*c^2)+3*d*e^2*(4*a*b*c^2+4*(2*a*c+b^2)*b*c)+e^3*(
2*a^2*c^2+8*a*b^2*c+(2*a*c+b^2)^2))*x^8+1/7*(d^3*(4*b^2*c^2+2*(2*a*c+b^2)*c^2)+3*d^2*e*(4*a*b*c^2+4*(2*a*c+b^2
)*b*c)+3*d*e^2*(2*a^2*c^2+8*a*b^2*c+(2*a*c+b^2)^2)+e^3*(4*a^2*b*c+4*(2*a*c+b^2)*a*b))*x^7+1/6*(d^3*(4*a*b*c^2+
4*(2*a*c+b^2)*b*c)+3*d^2*e*(2*a^2*c^2+8*a*b^2*c+(2*a*c+b^2)^2)+3*d*e^2*(4*a^2*b*c+4*(2*a*c+b^2)*a*b)+e^3*(4*a^
2*b^2+2*(2*a*c+b^2)*a^2))*x^6+1/5*(d^3*(2*a^2*c^2+8*a*b^2*c+(2*a*c+b^2)^2)+3*d^2*e*(4*a^2*b*c+4*(2*a*c+b^2)*a*
b)+3*d*e^2*(4*a^2*b^2+2*(2*a*c+b^2)*a^2)+4*a^3*b*e^3)*x^5+1/4*(d^3*(4*a^2*b*c+4*(2*a*c+b^2)*a*b)+3*d^2*e*(4*a^
2*b^2+2*(2*a*c+b^2)*a^2)+12*d*e^2*a^3*b+e^3*a^4)*x^4+1/3*(d^3*(4*a^2*b^2+2*(2*a*c+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 = 1.14, size = 614, normalized size = 1.39 \[ \frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {1}{11} \, {\left (3 \, c^{4} d e^{2} + 4 \, b c^{3} e^{3}\right )} x^{11} + \frac {1}{10} \, {\left (3 \, c^{4} d^{2} e + 12 \, b c^{3} d e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{3} + 12 \, b c^{3} d^{2} e + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, b c^{3} d^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e + 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{2} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, a^{3} b e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (12 \, a^{3} b d e^{2} + a^{4} e^{3} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (12 \, a^{3} b d^{2} e + 3 \, a^{4} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 630, normalized size = 1.42 \[ x^5\,\left (\frac {4\,a^3\,b\,e^3}{5}+\frac {12\,a^3\,c\,d\,e^2}{5}+\frac {18\,a^2\,b^2\,d\,e^2}{5}+\frac {36\,a^2\,b\,c\,d^2\,e}{5}+\frac {6\,a^2\,c^2\,d^3}{5}+\frac {12\,a\,b^3\,d^2\,e}{5}+\frac {12\,a\,b^2\,c\,d^3}{5}+\frac {b^4\,d^3}{5}\right )+x^8\,\left (\frac {3\,a^2\,c^2\,e^3}{4}+\frac {3\,a\,b^2\,c\,e^3}{2}+\frac {9\,a\,b\,c^2\,d\,e^2}{2}+\frac {3\,a\,c^3\,d^2\,e}{2}+\frac {b^4\,e^3}{8}+\frac {3\,b^3\,c\,d\,e^2}{2}+\frac {9\,b^2\,c^2\,d^2\,e}{4}+\frac {b\,c^3\,d^3}{2}\right )+x^6\,\left (\frac {2\,a^3\,c\,e^3}{3}+a^2\,b^2\,e^3+6\,a^2\,b\,c\,d\,e^2+3\,a^2\,c^2\,d^2\,e+2\,a\,b^3\,d\,e^2+6\,a\,b^2\,c\,d^2\,e+2\,a\,b\,c^2\,d^3+\frac {b^4\,d^2\,e}{2}+\frac {2\,b^3\,c\,d^3}{3}\right )+x^7\,\left (\frac {12\,a^2\,b\,c\,e^3}{7}+\frac {18\,a^2\,c^2\,d\,e^2}{7}+\frac {4\,a\,b^3\,e^3}{7}+\frac {36\,a\,b^2\,c\,d\,e^2}{7}+\frac {36\,a\,b\,c^2\,d^2\,e}{7}+\frac {4\,a\,c^3\,d^3}{7}+\frac {3\,b^4\,d\,e^2}{7}+\frac {12\,b^3\,c\,d^2\,e}{7}+\frac {6\,b^2\,c^2\,d^3}{7}\right )+x^4\,\left (\frac {a^4\,e^3}{4}+3\,a^3\,b\,d\,e^2+3\,c\,a^3\,d^2\,e+\frac {9\,a^2\,b^2\,d^2\,e}{2}+3\,c\,a^2\,b\,d^3+a\,b^3\,d^3\right )+x^9\,\left (\frac {4\,b^3\,c\,e^3}{9}+2\,b^2\,c^2\,d\,e^2+\frac {4\,b\,c^3\,d^2\,e}{3}+\frac {4\,a\,b\,c^2\,e^3}{3}+\frac {c^4\,d^3}{9}+\frac {4\,a\,c^3\,d\,e^2}{3}\right )+a^4\,d^3\,x+\frac {c^4\,e^3\,x^{12}}{12}+\frac {a^2\,d\,x^3\,\left (3\,a^2\,e^2+12\,a\,b\,d\,e+4\,c\,a\,d^2+6\,b^2\,d^2\right )}{3}+\frac {c^2\,e\,x^{10}\,\left (6\,b^2\,e^2+12\,b\,c\,d\,e+3\,c^2\,d^2+4\,a\,c\,e^2\right )}{10}+\frac {a^3\,d^2\,x^2\,\left (3\,a\,e+4\,b\,d\right )}{2}+\frac {c^3\,e^2\,x^{11}\,\left (4\,b\,e+3\,c\,d\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 777, normalized size = 1.75 \[ a^{4} d^{3} x + \frac {c^{4} e^{3} x^{12}}{12} + x^{11} \left (\frac {4 b c^{3} e^{3}}{11} + \frac {3 c^{4} d e^{2}}{11}\right ) + x^{10} \left (\frac {2 a c^{3} e^{3}}{5} + \frac {3 b^{2} c^{2} e^{3}}{5} + \frac {6 b c^{3} d e^{2}}{5} + \frac {3 c^{4} d^{2} e}{10}\right ) + x^{9} \left (\frac {4 a b c^{2} e^{3}}{3} + \frac {4 a c^{3} d e^{2}}{3} + \frac {4 b^{3} c e^{3}}{9} + 2 b^{2} c^{2} d e^{2} + \frac {4 b c^{3} d^{2} e}{3} + \frac {c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac {3 a^{2} c^{2} e^{3}}{4} + \frac {3 a b^{2} c e^{3}}{2} + \frac {9 a b c^{2} d e^{2}}{2} + \frac {3 a c^{3} d^{2} e}{2} + \frac {b^{4} e^{3}}{8} + \frac {3 b^{3} c d e^{2}}{2} + \frac {9 b^{2} c^{2} d^{2} e}{4} + \frac {b c^{3} d^{3}}{2}\right ) + x^{7} \left (\frac {12 a^{2} b c e^{3}}{7} + \frac {18 a^{2} c^{2} d e^{2}}{7} + \frac {4 a b^{3} e^{3}}{7} + \frac {36 a b^{2} c d e^{2}}{7} + \frac {36 a b c^{2} d^{2} e}{7} + \frac {4 a c^{3} d^{3}}{7} + \frac {3 b^{4} d e^{2}}{7} + \frac {12 b^{3} c d^{2} e}{7} + \frac {6 b^{2} c^{2} d^{3}}{7}\right ) + x^{6} \left (\frac {2 a^{3} c e^{3}}{3} + a^{2} b^{2} e^{3} + 6 a^{2} b c d e^{2} + 3 a^{2} c^{2} d^{2} e + 2 a b^{3} d e^{2} + 6 a b^{2} c d^{2} e + 2 a b c^{2} d^{3} + \frac {b^{4} d^{2} e}{2} + \frac {2 b^{3} c d^{3}}{3}\right ) + x^{5} \left (\frac {4 a^{3} b e^{3}}{5} + \frac {12 a^{3} c d e^{2}}{5} + \frac {18 a^{2} b^{2} d e^{2}}{5} + \frac {36 a^{2} b c d^{2} e}{5} + \frac {6 a^{2} c^{2} d^{3}}{5} + \frac {12 a b^{3} d^{2} e}{5} + \frac {12 a b^{2} c d^{3}}{5} + \frac {b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + 3 a^{3} c d^{2} e + \frac {9 a^{2} b^{2} d^{2} e}{2} + 3 a^{2} b c d^{3} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + \frac {4 a^{3} c d^{3}}{3} + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \left (\frac {3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \]

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)

________________________________________________________________________________________