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

Optimal. Leaf size=411 \[ \frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^8}+\frac {c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac {5 c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{8 e^8}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8}-\frac {7 c^3 (d+e x)^{10} (2 c d-b e)}{10 e^8}+\frac {2 c^4 (d+e x)^{11}}{11 e^8} \]

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Rubi [A]  time = 0.59, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \frac {(d+e x)^7 \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 )}{7 e^8}+\frac {c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac {5 c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{8 e^8}-\frac {(d+e x)^6 (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^8}+\frac {(d+e x)^5 \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 )}{5 e^8}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8}-\frac {7 c^3 (d+e x)^{10} (2 c d-b e)}{10 e^8}+\frac {2 c^4 (d+e x)^{11}}{11 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

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

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 562, normalized size = 1.37 \begin {gather*} a^3 b d^3 x+\frac {1}{2} a^2 d^2 x^2 \left (3 a b e+2 a c d+3 b^2 d\right )+a d x^3 \left (2 a^2 c d e+3 a b^2 d e+a b \left (a e^2+3 c d^2\right )+b^3 d^2\right )+\frac {1}{5} x^5 \left (2 a^2 c e \left (a e^2+9 c d^2\right )+b^3 \left (9 a d e^2+5 c d^3\right )+3 a b^2 e \left (a e^2+12 c d^2\right )+3 a b c d \left (9 a e^2+5 c d^2\right )+3 b^4 d^2 e\right )+\frac {1}{4} x^4 \left (a^2 b e \left (a e^2+27 c d^2\right )+6 a^2 c d \left (a e^2+c d^2\right )+9 a b^3 d^2 e+3 a b^2 d \left (3 a e^2+4 c d^2\right )+b^4 d^3\right )+\frac {1}{8} c x^8 \left (3 c^2 d e (6 a e+7 b d)+3 b c e^2 (5 a e+9 b d)+5 b^3 e^3+2 c^3 d^3\right )+\frac {1}{3} c^2 e x^9 \left (c e (2 a e+7 b d)+3 b^2 e^2+2 c^2 d^2\right )+\frac {1}{7} x^7 \left (3 b^2 c e \left (4 a e^2+9 c d^2\right )+b c^2 d \left (45 a e^2+7 c d^2\right )+6 a c^2 e \left (a e^2+3 c d^2\right )+b^4 e^3+15 b^3 c d e^2\right )+\frac {1}{2} x^6 \left (b^3 \left (a e^3+5 c d^2 e\right )+3 b^2 c d \left (4 a e^2+c d^2\right )+3 a b c e \left (a e^2+5 c d^2\right )+2 a c^2 d \left (3 a e^2+c d^2\right )+b^4 d e^2\right )+\frac {1}{10} c^3 e^2 x^{10} (7 b e+6 c d)+\frac {2}{11} c^4 e^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*b*d^3*x + (a^2*d^2*(3*b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + a*d*(b^3*d^2 + 3*a*b^2*d*e + 2*a^2*c*d*e + a*b*(
3*c*d^2 + a*e^2))*x^3 + ((b^4*d^3 + 9*a*b^3*d^2*e + 6*a^2*c*d*(c*d^2 + a*e^2) + a^2*b*e*(27*c*d^2 + a*e^2) + 3
*a*b^2*d*(4*c*d^2 + 3*a*e^2))*x^4)/4 + ((3*b^4*d^2*e + 2*a^2*c*e*(9*c*d^2 + a*e^2) + 3*a*b^2*e*(12*c*d^2 + a*e
^2) + 3*a*b*c*d*(5*c*d^2 + 9*a*e^2) + b^3*(5*c*d^3 + 9*a*d*e^2))*x^5)/5 + ((b^4*d*e^2 + 3*a*b*c*e*(5*c*d^2 + a
*e^2) + 2*a*c^2*d*(c*d^2 + 3*a*e^2) + 3*b^2*c*d*(c*d^2 + 4*a*e^2) + b^3*(5*c*d^2*e + a*e^3))*x^6)/2 + ((15*b^3
*c*d*e^2 + b^4*e^3 + 6*a*c^2*e*(3*c*d^2 + a*e^2) + 3*b^2*c*e*(9*c*d^2 + 4*a*e^2) + b*c^2*d*(7*c*d^2 + 45*a*e^2
))*x^7)/7 + (c*(2*c^3*d^3 + 5*b^3*e^3 + 3*b*c*e^2*(9*b*d + 5*a*e) + 3*c^2*d*e*(7*b*d + 6*a*e))*x^8)/8 + (c^2*e
*(2*c^2*d^2 + 3*b^2*e^2 + c*e*(7*b*d + 2*a*e))*x^9)/3 + (c^3*e^2*(6*c*d + 7*b*e)*x^10)/10 + (2*c^4*e^3*x^11)/1
1

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 719, normalized size = 1.75 \begin {gather*} \frac {2}{11} x^{11} e^{3} c^{4} + \frac {3}{5} x^{10} e^{2} d c^{4} + \frac {7}{10} x^{10} e^{3} c^{3} b + \frac {2}{3} x^{9} e d^{2} c^{4} + \frac {7}{3} x^{9} e^{2} d c^{3} b + x^{9} e^{3} c^{2} b^{2} + \frac {2}{3} x^{9} e^{3} c^{3} a + \frac {1}{4} x^{8} d^{3} c^{4} + \frac {21}{8} x^{8} e d^{2} c^{3} b + \frac {27}{8} x^{8} e^{2} d c^{2} b^{2} + \frac {5}{8} x^{8} e^{3} c b^{3} + \frac {9}{4} x^{8} e^{2} d c^{3} a + \frac {15}{8} x^{8} e^{3} c^{2} b a + x^{7} d^{3} c^{3} b + \frac {27}{7} x^{7} e d^{2} c^{2} b^{2} + \frac {15}{7} x^{7} e^{2} d c b^{3} + \frac {1}{7} x^{7} e^{3} b^{4} + \frac {18}{7} x^{7} e d^{2} c^{3} a + \frac {45}{7} x^{7} e^{2} d c^{2} b a + \frac {12}{7} x^{7} e^{3} c b^{2} a + \frac {6}{7} x^{7} e^{3} c^{2} a^{2} + \frac {3}{2} x^{6} d^{3} c^{2} b^{2} + \frac {5}{2} x^{6} e d^{2} c b^{3} + \frac {1}{2} x^{6} e^{2} d b^{4} + x^{6} d^{3} c^{3} a + \frac {15}{2} x^{6} e d^{2} c^{2} b a + 6 x^{6} e^{2} d c b^{2} a + \frac {1}{2} x^{6} e^{3} b^{3} a + 3 x^{6} e^{2} d c^{2} a^{2} + \frac {3}{2} x^{6} e^{3} c b a^{2} + x^{5} d^{3} c b^{3} + \frac {3}{5} x^{5} e d^{2} b^{4} + 3 x^{5} d^{3} c^{2} b a + \frac {36}{5} x^{5} e d^{2} c b^{2} a + \frac {9}{5} x^{5} e^{2} d b^{3} a + \frac {18}{5} x^{5} e d^{2} c^{2} a^{2} + \frac {27}{5} x^{5} e^{2} d c b a^{2} + \frac {3}{5} x^{5} e^{3} b^{2} a^{2} + \frac {2}{5} x^{5} e^{3} c a^{3} + \frac {1}{4} x^{4} d^{3} b^{4} + 3 x^{4} d^{3} c b^{2} a + \frac {9}{4} x^{4} e d^{2} b^{3} a + \frac {3}{2} x^{4} d^{3} c^{2} a^{2} + \frac {27}{4} x^{4} e d^{2} c b a^{2} + \frac {9}{4} x^{4} e^{2} d b^{2} a^{2} + \frac {3}{2} x^{4} e^{2} d c a^{3} + \frac {1}{4} x^{4} e^{3} b a^{3} + x^{3} d^{3} b^{3} a + 3 x^{3} d^{3} c b a^{2} + 3 x^{3} e d^{2} b^{2} a^{2} + 2 x^{3} e d^{2} c a^{3} + x^{3} e^{2} d b a^{3} + \frac {3}{2} x^{2} d^{3} b^{2} a^{2} + x^{2} d^{3} c a^{3} + \frac {3}{2} x^{2} e d^{2} b a^{3} + x d^{3} b a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*x^11*e^3*c^4 + 3/5*x^10*e^2*d*c^4 + 7/10*x^10*e^3*c^3*b + 2/3*x^9*e*d^2*c^4 + 7/3*x^9*e^2*d*c^3*b + x^9*e
^3*c^2*b^2 + 2/3*x^9*e^3*c^3*a + 1/4*x^8*d^3*c^4 + 21/8*x^8*e*d^2*c^3*b + 27/8*x^8*e^2*d*c^2*b^2 + 5/8*x^8*e^3
*c*b^3 + 9/4*x^8*e^2*d*c^3*a + 15/8*x^8*e^3*c^2*b*a + x^7*d^3*c^3*b + 27/7*x^7*e*d^2*c^2*b^2 + 15/7*x^7*e^2*d*
c*b^3 + 1/7*x^7*e^3*b^4 + 18/7*x^7*e*d^2*c^3*a + 45/7*x^7*e^2*d*c^2*b*a + 12/7*x^7*e^3*c*b^2*a + 6/7*x^7*e^3*c
^2*a^2 + 3/2*x^6*d^3*c^2*b^2 + 5/2*x^6*e*d^2*c*b^3 + 1/2*x^6*e^2*d*b^4 + x^6*d^3*c^3*a + 15/2*x^6*e*d^2*c^2*b*
a + 6*x^6*e^2*d*c*b^2*a + 1/2*x^6*e^3*b^3*a + 3*x^6*e^2*d*c^2*a^2 + 3/2*x^6*e^3*c*b*a^2 + x^5*d^3*c*b^3 + 3/5*
x^5*e*d^2*b^4 + 3*x^5*d^3*c^2*b*a + 36/5*x^5*e*d^2*c*b^2*a + 9/5*x^5*e^2*d*b^3*a + 18/5*x^5*e*d^2*c^2*a^2 + 27
/5*x^5*e^2*d*c*b*a^2 + 3/5*x^5*e^3*b^2*a^2 + 2/5*x^5*e^3*c*a^3 + 1/4*x^4*d^3*b^4 + 3*x^4*d^3*c*b^2*a + 9/4*x^4
*e*d^2*b^3*a + 3/2*x^4*d^3*c^2*a^2 + 27/4*x^4*e*d^2*c*b*a^2 + 9/4*x^4*e^2*d*b^2*a^2 + 3/2*x^4*e^2*d*c*a^3 + 1/
4*x^4*e^3*b*a^3 + x^3*d^3*b^3*a + 3*x^3*d^3*c*b*a^2 + 3*x^3*e*d^2*b^2*a^2 + 2*x^3*e*d^2*c*a^3 + x^3*e^2*d*b*a^
3 + 3/2*x^2*d^3*b^2*a^2 + x^2*d^3*c*a^3 + 3/2*x^2*e*d^2*b*a^3 + x*d^3*b*a^3

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giac [A]  time = 0.18, size = 705, normalized size = 1.72 \begin {gather*} \frac {2}{11} \, c^{4} x^{11} e^{3} + \frac {3}{5} \, c^{4} d x^{10} e^{2} + \frac {2}{3} \, c^{4} d^{2} x^{9} e + \frac {1}{4} \, c^{4} d^{3} x^{8} + \frac {7}{10} \, b c^{3} x^{10} e^{3} + \frac {7}{3} \, b c^{3} d x^{9} e^{2} + \frac {21}{8} \, b c^{3} d^{2} x^{8} e + b c^{3} d^{3} x^{7} + b^{2} c^{2} x^{9} e^{3} + \frac {2}{3} \, a c^{3} x^{9} e^{3} + \frac {27}{8} \, b^{2} c^{2} d x^{8} e^{2} + \frac {9}{4} \, a c^{3} d x^{8} e^{2} + \frac {27}{7} \, b^{2} c^{2} d^{2} x^{7} e + \frac {18}{7} \, a c^{3} d^{2} x^{7} e + \frac {3}{2} \, b^{2} c^{2} d^{3} x^{6} + a c^{3} d^{3} x^{6} + \frac {5}{8} \, b^{3} c x^{8} e^{3} + \frac {15}{8} \, a b c^{2} x^{8} e^{3} + \frac {15}{7} \, b^{3} c d x^{7} e^{2} + \frac {45}{7} \, a b c^{2} d x^{7} e^{2} + \frac {5}{2} \, b^{3} c d^{2} x^{6} e + \frac {15}{2} \, a b c^{2} d^{2} x^{6} e + b^{3} c d^{3} x^{5} + 3 \, a b c^{2} d^{3} x^{5} + \frac {1}{7} \, b^{4} x^{7} e^{3} + \frac {12}{7} \, a b^{2} c x^{7} e^{3} + \frac {6}{7} \, a^{2} c^{2} x^{7} e^{3} + \frac {1}{2} \, b^{4} d x^{6} e^{2} + 6 \, a b^{2} c d x^{6} e^{2} + 3 \, a^{2} c^{2} d x^{6} e^{2} + \frac {3}{5} \, b^{4} d^{2} x^{5} e + \frac {36}{5} \, a b^{2} c d^{2} x^{5} e + \frac {18}{5} \, a^{2} c^{2} d^{2} x^{5} e + \frac {1}{4} \, b^{4} d^{3} x^{4} + 3 \, a b^{2} c d^{3} x^{4} + \frac {3}{2} \, a^{2} c^{2} d^{3} x^{4} + \frac {1}{2} \, a b^{3} x^{6} e^{3} + \frac {3}{2} \, a^{2} b c x^{6} e^{3} + \frac {9}{5} \, a b^{3} d x^{5} e^{2} + \frac {27}{5} \, a^{2} b c d x^{5} e^{2} + \frac {9}{4} \, a b^{3} d^{2} x^{4} e + \frac {27}{4} \, a^{2} b c d^{2} x^{4} e + a b^{3} d^{3} x^{3} + 3 \, a^{2} b c d^{3} x^{3} + \frac {3}{5} \, a^{2} b^{2} x^{5} e^{3} + \frac {2}{5} \, a^{3} c x^{5} e^{3} + \frac {9}{4} \, a^{2} b^{2} d x^{4} e^{2} + \frac {3}{2} \, a^{3} c d x^{4} e^{2} + 3 \, a^{2} b^{2} d^{2} x^{3} e + 2 \, a^{3} c d^{2} x^{3} e + \frac {3}{2} \, a^{2} b^{2} d^{3} x^{2} + a^{3} c d^{3} x^{2} + \frac {1}{4} \, a^{3} b x^{4} e^{3} + a^{3} b d x^{3} e^{2} + \frac {3}{2} \, a^{3} b d^{2} x^{2} e + a^{3} b d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*c^4*x^11*e^3 + 3/5*c^4*d*x^10*e^2 + 2/3*c^4*d^2*x^9*e + 1/4*c^4*d^3*x^8 + 7/10*b*c^3*x^10*e^3 + 7/3*b*c^3
*d*x^9*e^2 + 21/8*b*c^3*d^2*x^8*e + b*c^3*d^3*x^7 + b^2*c^2*x^9*e^3 + 2/3*a*c^3*x^9*e^3 + 27/8*b^2*c^2*d*x^8*e
^2 + 9/4*a*c^3*d*x^8*e^2 + 27/7*b^2*c^2*d^2*x^7*e + 18/7*a*c^3*d^2*x^7*e + 3/2*b^2*c^2*d^3*x^6 + a*c^3*d^3*x^6
+ 5/8*b^3*c*x^8*e^3 + 15/8*a*b*c^2*x^8*e^3 + 15/7*b^3*c*d*x^7*e^2 + 45/7*a*b*c^2*d*x^7*e^2 + 5/2*b^3*c*d^2*x^
6*e + 15/2*a*b*c^2*d^2*x^6*e + b^3*c*d^3*x^5 + 3*a*b*c^2*d^3*x^5 + 1/7*b^4*x^7*e^3 + 12/7*a*b^2*c*x^7*e^3 + 6/
7*a^2*c^2*x^7*e^3 + 1/2*b^4*d*x^6*e^2 + 6*a*b^2*c*d*x^6*e^2 + 3*a^2*c^2*d*x^6*e^2 + 3/5*b^4*d^2*x^5*e + 36/5*a
*b^2*c*d^2*x^5*e + 18/5*a^2*c^2*d^2*x^5*e + 1/4*b^4*d^3*x^4 + 3*a*b^2*c*d^3*x^4 + 3/2*a^2*c^2*d^3*x^4 + 1/2*a*
b^3*x^6*e^3 + 3/2*a^2*b*c*x^6*e^3 + 9/5*a*b^3*d*x^5*e^2 + 27/5*a^2*b*c*d*x^5*e^2 + 9/4*a*b^3*d^2*x^4*e + 27/4*
a^2*b*c*d^2*x^4*e + a*b^3*d^3*x^3 + 3*a^2*b*c*d^3*x^3 + 3/5*a^2*b^2*x^5*e^3 + 2/5*a^3*c*x^5*e^3 + 9/4*a^2*b^2*
d*x^4*e^2 + 3/2*a^3*c*d*x^4*e^2 + 3*a^2*b^2*d^2*x^3*e + 2*a^3*c*d^2*x^3*e + 3/2*a^2*b^2*d^3*x^2 + a^3*c*d^3*x^
2 + 1/4*a^3*b*x^4*e^3 + a^3*b*d*x^3*e^2 + 3/2*a^3*b*d^2*x^2*e + a^3*b*d^3*x

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maple [B]  time = 0.04, size = 830, normalized size = 2.02 \begin {gather*} \frac {2 c^{4} e^{3} x^{11}}{11}+\frac {\left (6 b \,c^{3} e^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) c^{3}\right ) x^{10}}{10}+\frac {\left (2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c \,e^{3}+3 \left (b \,e^{3}+6 d \,e^{2} c \right ) b \,c^{2}+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) c^{3}\right ) x^{9}}{9}+a^{3} b \,d^{3} x +\frac {\left (2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c \,e^{3}+3 \left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b \,c^{2}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) c^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )\right ) x^{8}}{8}+\frac {\left (b \,c^{3} d^{3}+2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c \,e^{3}+3 \left (3 b \,d^{2} e +2 c \,d^{3}\right ) b \,c^{2}+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )+\left (b \,e^{3}+6 d \,e^{2} c \right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )\right ) x^{7}}{7}+\frac {\left (6 a^{2} b c \,e^{3}+3 b^{2} c^{2} d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )+\left (b \,e^{3}+6 d \,e^{2} c \right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{6}}{6}+\frac {\left (2 a^{3} c \,e^{3}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b \,d^{3}+3 \left (b \,e^{3}+6 d \,e^{2} c \right ) a^{2} b +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{5}}{5}+\frac {\left (\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b \,d^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) a^{3}+3 \left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a^{2} b +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{4}}{4}+\frac {\left (\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b \,d^{3}+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a^{3}+3 \left (3 b \,d^{2} e +2 c \,d^{3}\right ) a^{2} b \right ) x^{3}}{3}+\frac {\left (3 a^{2} b^{2} d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) a^{3}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 565, normalized size = 1.37 \begin {gather*} \frac {2}{11} \, c^{4} e^{3} x^{11} + \frac {1}{10} \, {\left (6 \, c^{4} d e^{2} + 7 \, b c^{3} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{4} d^{2} e + 7 \, b c^{3} d e^{2} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (2 \, c^{4} d^{3} + 21 \, b c^{3} d^{2} e + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x^{8} + a^{3} b d^{3} x + \frac {1}{7} \, {\left (7 \, b c^{3} d^{3} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e + 15 \, {\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^{7} + \frac {1}{2} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{2} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\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 + 9 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (a^{3} b e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 9 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e + 3 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{2}\right )} x^{4} + {\left (a^{3} b d e^{2} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{3} b d^{2} e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*c^4*e^3*x^11 + 1/10*(6*c^4*d*e^2 + 7*b*c^3*e^3)*x^10 + 1/3*(2*c^4*d^2*e + 7*b*c^3*d*e^2 + (3*b^2*c^2 + 2*
a*c^3)*e^3)*x^9 + 1/8*(2*c^4*d^3 + 21*b*c^3*d^2*e + 9*(3*b^2*c^2 + 2*a*c^3)*d*e^2 + 5*(b^3*c + 3*a*b*c^2)*e^3)
*x^8 + a^3*b*d^3*x + 1/7*(7*b*c^3*d^3 + 9*(3*b^2*c^2 + 2*a*c^3)*d^2*e + 15*(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^7 + 1/2*((3*b^2*c^2 + 2*a*c^3)*d^3 + 5*(b^3*c + 3*a*b*c^2)*d^2*e + (b^4 + 12*a*
b^2*c + 6*a^2*c^2)*d*e^2 + (a*b^3 + 3*a^2*b*c)*e^3)*x^6 + 1/5*(5*(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 + 9*(a*b^3 + 3*a^2*b*c)*d*e^2 + (3*a^2*b^2 + 2*a^3*c)*e^3)*x^5 + 1/4*(a^3*b*e^3 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^3 + 9*(a*b^3 + 3*a^2*b*c)*d^2*e + 3*(3*a^2*b^2 + 2*a^3*c)*d*e^2)*x^4 + (a^3*b*d*e^2 +
(a*b^3 + 3*a^2*b*c)*d^3 + (3*a^2*b^2 + 2*a^3*c)*d^2*e)*x^3 + 1/2*(3*a^3*b*d^2*e + (3*a^2*b^2 + 2*a^3*c)*d^3)*
x^2

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mupad [B]  time = 0.18, size = 589, normalized size = 1.43 \begin {gather*} x^4\,\left (\frac {a^3\,b\,e^3}{4}+\frac {3\,a^3\,c\,d\,e^2}{2}+\frac {9\,a^2\,b^2\,d\,e^2}{4}+\frac {27\,a^2\,b\,c\,d^2\,e}{4}+\frac {3\,a^2\,c^2\,d^3}{2}+\frac {9\,a\,b^3\,d^2\,e}{4}+3\,a\,b^2\,c\,d^3+\frac {b^4\,d^3}{4}\right )+x^7\,\left (\frac {6\,a^2\,c^2\,e^3}{7}+\frac {12\,a\,b^2\,c\,e^3}{7}+\frac {45\,a\,b\,c^2\,d\,e^2}{7}+\frac {18\,a\,c^3\,d^2\,e}{7}+\frac {b^4\,e^3}{7}+\frac {15\,b^3\,c\,d\,e^2}{7}+\frac {27\,b^2\,c^2\,d^2\,e}{7}+b\,c^3\,d^3\right )+x^5\,\left (\frac {2\,a^3\,c\,e^3}{5}+\frac {3\,a^2\,b^2\,e^3}{5}+\frac {27\,a^2\,b\,c\,d\,e^2}{5}+\frac {18\,a^2\,c^2\,d^2\,e}{5}+\frac {9\,a\,b^3\,d\,e^2}{5}+\frac {36\,a\,b^2\,c\,d^2\,e}{5}+3\,a\,b\,c^2\,d^3+\frac {3\,b^4\,d^2\,e}{5}+b^3\,c\,d^3\right )+x^6\,\left (\frac {3\,a^2\,b\,c\,e^3}{2}+3\,a^2\,c^2\,d\,e^2+\frac {a\,b^3\,e^3}{2}+6\,a\,b^2\,c\,d\,e^2+\frac {15\,a\,b\,c^2\,d^2\,e}{2}+a\,c^3\,d^3+\frac {b^4\,d\,e^2}{2}+\frac {5\,b^3\,c\,d^2\,e}{2}+\frac {3\,b^2\,c^2\,d^3}{2}\right )+x^8\,\left (\frac {5\,b^3\,c\,e^3}{8}+\frac {27\,b^2\,c^2\,d\,e^2}{8}+\frac {21\,b\,c^3\,d^2\,e}{8}+\frac {15\,a\,b\,c^2\,e^3}{8}+\frac {c^4\,d^3}{4}+\frac {9\,a\,c^3\,d\,e^2}{4}\right )+x^3\,\left (a^3\,b\,d\,e^2+2\,c\,a^3\,d^2\,e+3\,a^2\,b^2\,d^2\,e+3\,c\,a^2\,b\,d^3+a\,b^3\,d^3\right )+\frac {2\,c^4\,e^3\,x^{11}}{11}+\frac {c^2\,e\,x^9\,\left (3\,b^2\,e^2+7\,b\,c\,d\,e+2\,c^2\,d^2+2\,a\,c\,e^2\right )}{3}+\frac {c^3\,e^2\,x^{10}\,\left (7\,b\,e+6\,c\,d\right )}{10}+\frac {a^2\,d^2\,x^2\,\left (3\,d\,b^2+3\,a\,e\,b+2\,a\,c\,d\right )}{2}+a^3\,b\,d^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((b^4*d^3)/4 + (a^3*b*e^3)/4 + (3*a^2*c^2*d^3)/2 + (9*a^2*b^2*d*e^2)/4 + 3*a*b^2*c*d^3 + (9*a*b^3*d^2*e)/4
+ (3*a^3*c*d*e^2)/2 + (27*a^2*b*c*d^2*e)/4) + x^7*((b^4*e^3)/7 + b*c^3*d^3 + (6*a^2*c^2*e^3)/7 + (27*b^2*c^2*
d^2*e)/7 + (12*a*b^2*c*e^3)/7 + (18*a*c^3*d^2*e)/7 + (15*b^3*c*d*e^2)/7 + (45*a*b*c^2*d*e^2)/7) + x^5*((2*a^3*
c*e^3)/5 + b^3*c*d^3 + (3*b^4*d^2*e)/5 + (3*a^2*b^2*e^3)/5 + (18*a^2*c^2*d^2*e)/5 + 3*a*b*c^2*d^3 + (9*a*b^3*d
*e^2)/5 + (36*a*b^2*c*d^2*e)/5 + (27*a^2*b*c*d*e^2)/5) + x^6*((a*b^3*e^3)/2 + a*c^3*d^3 + (b^4*d*e^2)/2 + (3*b
^2*c^2*d^3)/2 + 3*a^2*c^2*d*e^2 + (3*a^2*b*c*e^3)/2 + (5*b^3*c*d^2*e)/2 + (15*a*b*c^2*d^2*e)/2 + 6*a*b^2*c*d*e
^2) + x^8*((c^4*d^3)/4 + (5*b^3*c*e^3)/8 + (27*b^2*c^2*d*e^2)/8 + (15*a*b*c^2*e^3)/8 + (9*a*c^3*d*e^2)/4 + (21
*b*c^3*d^2*e)/8) + x^3*(a*b^3*d^3 + 3*a^2*b^2*d^2*e + 3*a^2*b*c*d^3 + a^3*b*d*e^2 + 2*a^3*c*d^2*e) + (2*c^4*e^
3*x^11)/11 + (c^2*e*x^9*(3*b^2*e^2 + 2*c^2*d^2 + 2*a*c*e^2 + 7*b*c*d*e))/3 + (c^3*e^2*x^10*(7*b*e + 6*c*d))/10
+ (a^2*d^2*x^2*(3*b^2*d + 3*a*b*e + 2*a*c*d))/2 + a^3*b*d^3*x

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sympy [A]  time = 0.18, size = 726, normalized size = 1.77 \begin {gather*} a^{3} b d^{3} x + \frac {2 c^{4} e^{3} x^{11}}{11} + x^{10} \left (\frac {7 b c^{3} e^{3}}{10} + \frac {3 c^{4} d e^{2}}{5}\right ) + x^{9} \left (\frac {2 a c^{3} e^{3}}{3} + b^{2} c^{2} e^{3} + \frac {7 b c^{3} d e^{2}}{3} + \frac {2 c^{4} d^{2} e}{3}\right ) + x^{8} \left (\frac {15 a b c^{2} e^{3}}{8} + \frac {9 a c^{3} d e^{2}}{4} + \frac {5 b^{3} c e^{3}}{8} + \frac {27 b^{2} c^{2} d e^{2}}{8} + \frac {21 b c^{3} d^{2} e}{8} + \frac {c^{4} d^{3}}{4}\right ) + x^{7} \left (\frac {6 a^{2} c^{2} e^{3}}{7} + \frac {12 a b^{2} c e^{3}}{7} + \frac {45 a b c^{2} d e^{2}}{7} + \frac {18 a c^{3} d^{2} e}{7} + \frac {b^{4} e^{3}}{7} + \frac {15 b^{3} c d e^{2}}{7} + \frac {27 b^{2} c^{2} d^{2} e}{7} + b c^{3} d^{3}\right ) + x^{6} \left (\frac {3 a^{2} b c e^{3}}{2} + 3 a^{2} c^{2} d e^{2} + \frac {a b^{3} e^{3}}{2} + 6 a b^{2} c d e^{2} + \frac {15 a b c^{2} d^{2} e}{2} + a c^{3} d^{3} + \frac {b^{4} d e^{2}}{2} + \frac {5 b^{3} c d^{2} e}{2} + \frac {3 b^{2} c^{2} d^{3}}{2}\right ) + x^{5} \left (\frac {2 a^{3} c e^{3}}{5} + \frac {3 a^{2} b^{2} e^{3}}{5} + \frac {27 a^{2} b c d e^{2}}{5} + \frac {18 a^{2} c^{2} d^{2} e}{5} + \frac {9 a b^{3} d e^{2}}{5} + \frac {36 a b^{2} c d^{2} e}{5} + 3 a b c^{2} d^{3} + \frac {3 b^{4} d^{2} e}{5} + b^{3} c d^{3}\right ) + x^{4} \left (\frac {a^{3} b e^{3}}{4} + \frac {3 a^{3} c d e^{2}}{2} + \frac {9 a^{2} b^{2} d e^{2}}{4} + \frac {27 a^{2} b c d^{2} e}{4} + \frac {3 a^{2} c^{2} d^{3}}{2} + \frac {9 a b^{3} d^{2} e}{4} + 3 a b^{2} c d^{3} + \frac {b^{4} d^{3}}{4}\right ) + x^{3} \left (a^{3} b d e^{2} + 2 a^{3} c d^{2} e + 3 a^{2} b^{2} d^{2} e + 3 a^{2} b c d^{3} + a b^{3} d^{3}\right ) + x^{2} \left (\frac {3 a^{3} b d^{2} e}{2} + a^{3} c d^{3} + \frac {3 a^{2} b^{2} d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*b*d**3*x + 2*c**4*e**3*x**11/11 + x**10*(7*b*c**3*e**3/10 + 3*c**4*d*e**2/5) + x**9*(2*a*c**3*e**3/3 + b*
*2*c**2*e**3 + 7*b*c**3*d*e**2/3 + 2*c**4*d**2*e/3) + x**8*(15*a*b*c**2*e**3/8 + 9*a*c**3*d*e**2/4 + 5*b**3*c*
e**3/8 + 27*b**2*c**2*d*e**2/8 + 21*b*c**3*d**2*e/8 + c**4*d**3/4) + x**7*(6*a**2*c**2*e**3/7 + 12*a*b**2*c*e*
*3/7 + 45*a*b*c**2*d*e**2/7 + 18*a*c**3*d**2*e/7 + b**4*e**3/7 + 15*b**3*c*d*e**2/7 + 27*b**2*c**2*d**2*e/7 +
b*c**3*d**3) + x**6*(3*a**2*b*c*e**3/2 + 3*a**2*c**2*d*e**2 + a*b**3*e**3/2 + 6*a*b**2*c*d*e**2 + 15*a*b*c**2*
d**2*e/2 + a*c**3*d**3 + b**4*d*e**2/2 + 5*b**3*c*d**2*e/2 + 3*b**2*c**2*d**3/2) + x**5*(2*a**3*c*e**3/5 + 3*a
**2*b**2*e**3/5 + 27*a**2*b*c*d*e**2/5 + 18*a**2*c**2*d**2*e/5 + 9*a*b**3*d*e**2/5 + 36*a*b**2*c*d**2*e/5 + 3*
a*b*c**2*d**3 + 3*b**4*d**2*e/5 + b**3*c*d**3) + x**4*(a**3*b*e**3/4 + 3*a**3*c*d*e**2/2 + 9*a**2*b**2*d*e**2/
4 + 27*a**2*b*c*d**2*e/4 + 3*a**2*c**2*d**3/2 + 9*a*b**3*d**2*e/4 + 3*a*b**2*c*d**3 + b**4*d**3/4) + x**3*(a**
3*b*d*e**2 + 2*a**3*c*d**2*e + 3*a**2*b**2*d**2*e + 3*a**2*b*c*d**3 + a*b**3*d**3) + x**2*(3*a**3*b*d**2*e/2 +
a**3*c*d**3 + 3*a**2*b**2*d**3/2)

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