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

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

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Rubi [A]  time = 0.87, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.25, size = 766, normalized size = 1.73 \begin {gather*} a^4 d^4 x+2 a^3 d^3 x^2 (a e+b d)+\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 )+\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 )+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 {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}{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 )+16 a b^3 d^3 e+12 a b^2 d^2 \left (3 a e^2+c d^2\right )+b^4 d^4\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}{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 {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 {1}{3} c^3 e^3 x^{12} (b e+c d)+\frac {1}{13} c^4 e^4 x^{13} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.35, size = 1001, normalized size = 2.26 \begin {gather*} \frac {1}{13} x^{13} e^{4} c^{4} + \frac {1}{3} x^{12} e^{3} d c^{4} + \frac {1}{3} x^{12} e^{4} c^{3} b + \frac {6}{11} x^{11} e^{2} d^{2} c^{4} + \frac {16}{11} x^{11} e^{3} d c^{3} b + \frac {6}{11} x^{11} e^{4} c^{2} b^{2} + \frac {4}{11} x^{11} e^{4} c^{3} a + \frac {2}{5} x^{10} e d^{3} c^{4} + \frac {12}{5} x^{10} e^{2} d^{2} c^{3} b + \frac {12}{5} x^{10} e^{3} d c^{2} b^{2} + \frac {2}{5} x^{10} e^{4} c b^{3} + \frac {8}{5} x^{10} e^{3} d c^{3} a + \frac {6}{5} x^{10} e^{4} c^{2} b a + \frac {1}{9} x^{9} d^{4} c^{4} + \frac {16}{9} x^{9} e d^{3} c^{3} b + 4 x^{9} e^{2} d^{2} c^{2} b^{2} + \frac {16}{9} x^{9} e^{3} d c b^{3} + \frac {1}{9} x^{9} e^{4} b^{4} + \frac {8}{3} x^{9} e^{2} d^{2} c^{3} a + \frac {16}{3} x^{9} e^{3} d c^{2} b a + \frac {4}{3} x^{9} e^{4} c b^{2} a + \frac {2}{3} x^{9} e^{4} c^{2} a^{2} + \frac {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} + \frac {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 + \frac {1}{2} x^{8} e^{4} b^{3} a + 3 x^{8} e^{3} d c^{2} a^{2} + \frac {3}{2} x^{8} e^{4} c b a^{2} + \frac {6}{7} x^{7} d^{4} c^{2} b^{2} + \frac {16}{7} x^{7} e d^{3} c b^{3} + \frac {6}{7} x^{7} e^{2} d^{2} b^{4} + \frac {4}{7} x^{7} d^{4} c^{3} a + \frac {48}{7} x^{7} e d^{3} c^{2} b a + \frac {72}{7} x^{7} e^{2} d^{2} c b^{2} a + \frac {16}{7} x^{7} e^{3} d b^{3} a + \frac {36}{7} x^{7} e^{2} d^{2} c^{2} a^{2} + \frac {48}{7} x^{7} e^{3} d c b a^{2} + \frac {6}{7} x^{7} e^{4} b^{2} a^{2} + \frac {4}{7} x^{7} e^{4} c a^{3} + \frac {2}{3} x^{6} d^{4} c b^{3} + \frac {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} + \frac {8}{3} x^{6} e^{3} d c a^{3} + \frac {2}{3} x^{6} e^{4} b a^{3} + \frac {1}{5} x^{5} d^{4} b^{4} + \frac {12}{5} x^{5} d^{4} c b^{2} a + \frac {16}{5} x^{5} e d^{3} b^{3} a + \frac {6}{5} x^{5} d^{4} c^{2} a^{2} + \frac {48}{5} x^{5} e d^{3} c b a^{2} + \frac {36}{5} x^{5} e^{2} d^{2} b^{2} a^{2} + \frac {24}{5} x^{5} e^{2} d^{2} c a^{3} + \frac {16}{5} x^{5} e^{3} d b a^{3} + \frac {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} + \frac {4}{3} x^{3} d^{4} c a^{3} + \frac {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} \end {gather*}

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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giac [B]  time = 0.17, size = 971, normalized size = 2.19 \begin {gather*} \frac {1}{13} \, c^{4} x^{13} e^{4} + \frac {1}{3} \, c^{4} d x^{12} e^{3} + \frac {6}{11} \, c^{4} d^{2} x^{11} e^{2} + \frac {2}{5} \, c^{4} d^{3} x^{10} e + \frac {1}{9} \, c^{4} d^{4} x^{9} + \frac {1}{3} \, b c^{3} x^{12} e^{4} + \frac {16}{11} \, b c^{3} d x^{11} e^{3} + \frac {12}{5} \, b c^{3} d^{2} x^{10} e^{2} + \frac {16}{9} \, b c^{3} d^{3} x^{9} e + \frac {1}{2} \, b c^{3} d^{4} x^{8} + \frac {6}{11} \, b^{2} c^{2} x^{11} e^{4} + \frac {4}{11} \, a c^{3} x^{11} e^{4} + \frac {12}{5} \, b^{2} c^{2} d x^{10} e^{3} + \frac {8}{5} \, a c^{3} d x^{10} e^{3} + 4 \, b^{2} c^{2} d^{2} x^{9} e^{2} + \frac {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 + \frac {6}{7} \, b^{2} c^{2} d^{4} x^{7} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + \frac {2}{5} \, b^{3} c x^{10} e^{4} + \frac {6}{5} \, a b c^{2} x^{10} e^{4} + \frac {16}{9} \, b^{3} c d x^{9} e^{3} + \frac {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} + \frac {16}{7} \, b^{3} c d^{3} x^{7} e + \frac {48}{7} \, a b c^{2} d^{3} x^{7} e + \frac {2}{3} \, b^{3} c d^{4} x^{6} + 2 \, a b c^{2} d^{4} x^{6} + \frac {1}{9} \, b^{4} x^{9} e^{4} + \frac {4}{3} \, a b^{2} c x^{9} e^{4} + \frac {2}{3} \, a^{2} c^{2} x^{9} e^{4} + \frac {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} + \frac {6}{7} \, b^{4} d^{2} x^{7} e^{2} + \frac {72}{7} \, a b^{2} c d^{2} x^{7} e^{2} + \frac {36}{7} \, a^{2} c^{2} d^{2} x^{7} e^{2} + \frac {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 + \frac {1}{5} \, b^{4} d^{4} x^{5} + \frac {12}{5} \, a b^{2} c d^{4} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{4} x^{5} + \frac {1}{2} \, a b^{3} x^{8} e^{4} + \frac {3}{2} \, a^{2} b c x^{8} e^{4} + \frac {16}{7} \, a b^{3} d x^{7} e^{3} + \frac {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} + \frac {16}{5} \, a b^{3} d^{3} x^{5} e + \frac {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} + \frac {6}{7} \, a^{2} b^{2} x^{7} e^{4} + \frac {4}{7} \, a^{3} c x^{7} e^{4} + 4 \, a^{2} b^{2} d x^{6} e^{3} + \frac {8}{3} \, a^{3} c d x^{6} e^{3} + \frac {36}{5} \, a^{2} b^{2} d^{2} x^{5} e^{2} + \frac {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} + \frac {4}{3} \, a^{3} c d^{4} x^{3} + \frac {2}{3} \, a^{3} b x^{6} e^{4} + \frac {16}{5} \, a^{3} b d x^{5} e^{3} + 6 \, a^{3} b d^{2} x^{4} e^{2} + \frac {16}{3} \, a^{3} b d^{3} x^{3} e + 2 \, a^{3} b d^{4} x^{2} + \frac {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 \end {gather*}

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

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maple [B]  time = 0.04, size = 949, normalized size = 2.14 \begin {gather*} \frac {c^{4} e^{4} x^{13}}{13}+\frac {\left (4 e^{4} b \,c^{3}+4 d \,e^{3} c^{4}\right ) x^{12}}{12}+\frac {\left (16 b \,c^{3} d \,e^{3}+6 c^{4} d^{2} e^{2}+\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) e^{4}\right ) x^{11}}{11}+\frac {\left (24 b \,c^{3} d^{2} e^{2}+4 c^{4} d^{3} e +4 \left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d \,e^{3}+\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) e^{4}\right ) x^{10}}{10}+a^{4} d^{4} x +\frac {\left (16 b \,c^{3} d^{3} e +c^{4} d^{4}+6 \left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d^{2} e^{2}+4 \left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d \,e^{3}+\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) e^{4}\right ) x^{9}}{9}+\frac {\left (4 b \,c^{3} d^{4}+4 \left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d^{3} e +6 \left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d^{2} e^{2}+4 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d \,e^{3}+\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) e^{4}\right ) x^{8}}{8}+\frac {\left (\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d^{4}+4 \left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d^{3} e +6 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d^{2} e^{2}+4 \left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d \,e^{3}+\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) e^{4}\right ) x^{7}}{7}+\frac {\left (4 a^{3} b \,e^{4}+\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d^{4}+4 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d^{3} e +6 \left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d^{2} e^{2}+4 \left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d \,e^{3}\right ) x^{6}}{6}+\frac {\left (a^{4} e^{4}+16 a^{3} b d \,e^{3}+\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d^{4}+4 \left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d^{3} e +6 \left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d^{2} e^{2}\right ) x^{5}}{5}+\frac {\left (4 a^{4} d \,e^{3}+24 a^{3} b \,d^{2} e^{2}+\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d^{4}+4 \left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d^{3} e \right ) x^{4}}{4}+\frac {\left (6 a^{4} d^{2} e^{2}+16 a^{3} b \,d^{3} e +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{4}+4 d^{4} a^{3} b \right ) x^{2}}{2} \end {gather*}

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*b^2*c+(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*b^2*c+(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*b^2*c+(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*b^2*c+(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*b^2*c+(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 = 1.25, size = 762, normalized size = 1.72 \begin {gather*} \frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, {\left (c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{12} + \frac {2}{11} \, {\left (3 \, c^{4} d^{2} e^{2} + 8 \, b c^{3} d e^{3} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{11} + \frac {2}{5} \, {\left (c^{4} d^{3} e + 6 \, b c^{3} d^{2} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{4} + 16 \, b c^{3} d^{3} e + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} + 16 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b c^{3} d^{4} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e + 6 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{3} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} + 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{2} + 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{3} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (a^{3} b e^{4} + {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (16 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 16 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e + 12 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{2}\right )} x^{5} + {\left (6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e\right )} x^{4} + \frac {2}{3} \, {\left (8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \end {gather*}

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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mupad [B]  time = 0.85, size = 817, normalized size = 1.84 \begin {gather*} x^6\,\left (\frac {2\,a^3\,b\,e^4}{3}+\frac {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+\frac {2\,b^4\,d^3\,e}{3}+\frac {2\,b^3\,c\,d^4}{3}\right )+x^8\,\left (\frac {3\,a^2\,b\,c\,e^4}{2}+3\,a^2\,c^2\,d\,e^3+\frac {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+\frac {b^4\,d\,e^3}{2}+3\,b^3\,c\,d^2\,e^2+3\,b^2\,c^2\,d^3\,e+\frac {b\,c^3\,d^4}{2}\right )+x^7\,\left (\frac {4\,a^3\,c\,e^4}{7}+\frac {6\,a^2\,b^2\,e^4}{7}+\frac {48\,a^2\,b\,c\,d\,e^3}{7}+\frac {36\,a^2\,c^2\,d^2\,e^2}{7}+\frac {16\,a\,b^3\,d\,e^3}{7}+\frac {72\,a\,b^2\,c\,d^2\,e^2}{7}+\frac {48\,a\,b\,c^2\,d^3\,e}{7}+\frac {4\,a\,c^3\,d^4}{7}+\frac {6\,b^4\,d^2\,e^2}{7}+\frac {16\,b^3\,c\,d^3\,e}{7}+\frac {6\,b^2\,c^2\,d^4}{7}\right )+x^5\,\left (\frac {a^4\,e^4}{5}+\frac {16\,a^3\,b\,d\,e^3}{5}+\frac {24\,a^3\,c\,d^2\,e^2}{5}+\frac {36\,a^2\,b^2\,d^2\,e^2}{5}+\frac {48\,a^2\,b\,c\,d^3\,e}{5}+\frac {6\,a^2\,c^2\,d^4}{5}+\frac {16\,a\,b^3\,d^3\,e}{5}+\frac {12\,a\,b^2\,c\,d^4}{5}+\frac {b^4\,d^4}{5}\right )+x^4\,\left (a^4\,d\,e^3+6\,a^3\,b\,d^2\,e^2+4\,c\,a^3\,d^3\,e+6\,a^2\,b^2\,d^3\,e+3\,c\,a^2\,b\,d^4+a\,b^3\,d^4\right )+x^9\,\left (\frac {2\,a^2\,c^2\,e^4}{3}+\frac {4\,a\,b^2\,c\,e^4}{3}+\frac {16\,a\,b\,c^2\,d\,e^3}{3}+\frac {8\,a\,c^3\,d^2\,e^2}{3}+\frac {b^4\,e^4}{9}+\frac {16\,b^3\,c\,d\,e^3}{9}+4\,b^2\,c^2\,d^2\,e^2+\frac {16\,b\,c^3\,d^3\,e}{9}+\frac {c^4\,d^4}{9}\right )+x^{10}\,\left (\frac {2\,b^3\,c\,e^4}{5}+\frac {12\,b^2\,c^2\,d\,e^3}{5}+\frac {12\,b\,c^3\,d^2\,e^2}{5}+\frac {6\,a\,b\,c^2\,e^4}{5}+\frac {2\,c^4\,d^3\,e}{5}+\frac {8\,a\,c^3\,d\,e^3}{5}\right )+a^4\,d^4\,x+\frac {c^4\,e^4\,x^{13}}{13}+2\,a^3\,d^3\,x^2\,\left (a\,e+b\,d\right )+\frac {c^3\,e^3\,x^{12}\,\left (b\,e+c\,d\right )}{3}+\frac {2\,a^2\,d^2\,x^3\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+2\,c\,a\,d^2+3\,b^2\,d^2\right )}{3}+\frac {2\,c^2\,e^2\,x^{11}\,\left (3\,b^2\,e^2+8\,b\,c\,d\,e+3\,c^2\,d^2+2\,a\,c\,e^2\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.21, size = 998, normalized size = 2.25 \begin {gather*} a^{4} d^{4} x + \frac {c^{4} e^{4} x^{13}}{13} + x^{12} \left (\frac {b c^{3} e^{4}}{3} + \frac {c^{4} d e^{3}}{3}\right ) + x^{11} \left (\frac {4 a c^{3} e^{4}}{11} + \frac {6 b^{2} c^{2} e^{4}}{11} + \frac {16 b c^{3} d e^{3}}{11} + \frac {6 c^{4} d^{2} e^{2}}{11}\right ) + x^{10} \left (\frac {6 a b c^{2} e^{4}}{5} + \frac {8 a c^{3} d e^{3}}{5} + \frac {2 b^{3} c e^{4}}{5} + \frac {12 b^{2} c^{2} d e^{3}}{5} + \frac {12 b c^{3} d^{2} e^{2}}{5} + \frac {2 c^{4} d^{3} e}{5}\right ) + x^{9} \left (\frac {2 a^{2} c^{2} e^{4}}{3} + \frac {4 a b^{2} c e^{4}}{3} + \frac {16 a b c^{2} d e^{3}}{3} + \frac {8 a c^{3} d^{2} e^{2}}{3} + \frac {b^{4} e^{4}}{9} + \frac {16 b^{3} c d e^{3}}{9} + 4 b^{2} c^{2} d^{2} e^{2} + \frac {16 b c^{3} d^{3} e}{9} + \frac {c^{4} d^{4}}{9}\right ) + x^{8} \left (\frac {3 a^{2} b c e^{4}}{2} + 3 a^{2} c^{2} d e^{3} + \frac {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 + \frac {b^{4} d e^{3}}{2} + 3 b^{3} c d^{2} e^{2} + 3 b^{2} c^{2} d^{3} e + \frac {b c^{3} d^{4}}{2}\right ) + x^{7} \left (\frac {4 a^{3} c e^{4}}{7} + \frac {6 a^{2} b^{2} e^{4}}{7} + \frac {48 a^{2} b c d e^{3}}{7} + \frac {36 a^{2} c^{2} d^{2} e^{2}}{7} + \frac {16 a b^{3} d e^{3}}{7} + \frac {72 a b^{2} c d^{2} e^{2}}{7} + \frac {48 a b c^{2} d^{3} e}{7} + \frac {4 a c^{3} d^{4}}{7} + \frac {6 b^{4} d^{2} e^{2}}{7} + \frac {16 b^{3} c d^{3} e}{7} + \frac {6 b^{2} c^{2} d^{4}}{7}\right ) + x^{6} \left (\frac {2 a^{3} b e^{4}}{3} + \frac {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} + \frac {2 b^{4} d^{3} e}{3} + \frac {2 b^{3} c d^{4}}{3}\right ) + x^{5} \left (\frac {a^{4} e^{4}}{5} + \frac {16 a^{3} b d e^{3}}{5} + \frac {24 a^{3} c d^{2} e^{2}}{5} + \frac {36 a^{2} b^{2} d^{2} e^{2}}{5} + \frac {48 a^{2} b c d^{3} e}{5} + \frac {6 a^{2} c^{2} d^{4}}{5} + \frac {16 a b^{3} d^{3} e}{5} + \frac {12 a b^{2} c d^{4}}{5} + \frac {b^{4} d^{4}}{5}\right ) + x^{4} \left (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}\right ) + x^{3} \left (2 a^{4} d^{2} e^{2} + \frac {16 a^{3} b d^{3} e}{3} + \frac {4 a^{3} c d^{4}}{3} + 2 a^{2} b^{2} d^{4}\right ) + x^{2} \left (2 a^{4} d^{3} e + 2 a^{3} b d^{4}\right ) \end {gather*}

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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