3.140 \(\int (a+b x+c x^2)^4 (A+C x^2) \, dx\)

Optimal. Leaf size=254 \[ a^4 A x+2 a^3 A b x^2+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+\frac {1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac {1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]

[Out]

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

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Rubi [A]  time = 0.33, antiderivative size = 254, 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 = {1657} \[ \frac {1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac {1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4*(A + C*x^2),x]

[Out]

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx &=\int \left (a^4 A+4 a^3 A b x+a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^2+4 a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^3+\left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^4+4 b \left (b^2+3 a c\right ) (A c+a C) x^5+\left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^6+4 b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^7+c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^8+4 b c^3 C x^9+c^4 C x^{10}\right ) \, dx\\ &=a^4 A x+2 a^3 A b x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^4+\frac {1}{5} \left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^5+\frac {2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac {1}{7} \left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^7+\frac {1}{2} b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^8+\frac {1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 256, normalized size = 1.01 \[ a^4 A x+2 a^3 A b x^2+a b x^4 \left (a^2 C+3 a A c+A b^2\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+\frac {1}{7} x^7 \left (6 a^2 c^2 C+4 a A c^3+12 a b^2 c C+6 A b^2 c^2+b^4 C\right )+\frac {1}{5} x^5 \left (4 a^3 c C+6 a^2 A c^2+6 a^2 b^2 C+12 a A b^2 c+A b^4\right )+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (3 a c C+A c^2+b^2 C\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4*(A + C*x^2),x]

[Out]

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

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fricas [A]  time = 0.71, size = 308, normalized size = 1.21 \[ \frac {1}{11} x^{11} c^{4} C + \frac {2}{5} x^{10} c^{3} b C + \frac {2}{3} x^{9} c^{2} b^{2} C + \frac {4}{9} x^{9} c^{3} a C + \frac {1}{9} x^{9} c^{4} A + \frac {1}{2} x^{8} c b^{3} C + \frac {3}{2} x^{8} c^{2} b a C + \frac {1}{2} x^{8} c^{3} b A + \frac {1}{7} x^{7} b^{4} C + \frac {12}{7} x^{7} c b^{2} a C + \frac {6}{7} x^{7} c^{2} a^{2} C + \frac {6}{7} x^{7} c^{2} b^{2} A + \frac {4}{7} x^{7} c^{3} a A + \frac {2}{3} x^{6} b^{3} a C + 2 x^{6} c b a^{2} C + \frac {2}{3} x^{6} c b^{3} A + 2 x^{6} c^{2} b a A + \frac {6}{5} x^{5} b^{2} a^{2} C + \frac {4}{5} x^{5} c a^{3} C + \frac {1}{5} x^{5} b^{4} A + \frac {12}{5} x^{5} c b^{2} a A + \frac {6}{5} x^{5} c^{2} a^{2} A + x^{4} b a^{3} C + x^{4} b^{3} a A + 3 x^{4} c b a^{2} A + \frac {1}{3} x^{3} a^{4} C + 2 x^{3} b^{2} a^{2} A + \frac {4}{3} x^{3} c a^{3} A + 2 x^{2} b a^{3} A + x a^{4} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*c^4*C + 2/5*x^10*c^3*b*C + 2/3*x^9*c^2*b^2*C + 4/9*x^9*c^3*a*C + 1/9*x^9*c^4*A + 1/2*x^8*c*b^3*C + 3
/2*x^8*c^2*b*a*C + 1/2*x^8*c^3*b*A + 1/7*x^7*b^4*C + 12/7*x^7*c*b^2*a*C + 6/7*x^7*c^2*a^2*C + 6/7*x^7*c^2*b^2*
A + 4/7*x^7*c^3*a*A + 2/3*x^6*b^3*a*C + 2*x^6*c*b*a^2*C + 2/3*x^6*c*b^3*A + 2*x^6*c^2*b*a*A + 6/5*x^5*b^2*a^2*
C + 4/5*x^5*c*a^3*C + 1/5*x^5*b^4*A + 12/5*x^5*c*b^2*a*A + 6/5*x^5*c^2*a^2*A + x^4*b*a^3*C + x^4*b^3*a*A + 3*x
^4*c*b*a^2*A + 1/3*x^3*a^4*C + 2*x^3*b^2*a^2*A + 4/3*x^3*c*a^3*A + 2*x^2*b*a^3*A + x*a^4*A

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giac [A]  time = 0.15, size = 308, normalized size = 1.21 \[ \frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {2}{3} \, C b^{2} c^{2} x^{9} + \frac {4}{9} \, C a c^{3} x^{9} + \frac {1}{9} \, A c^{4} x^{9} + \frac {1}{2} \, C b^{3} c x^{8} + \frac {3}{2} \, C a b c^{2} x^{8} + \frac {1}{2} \, A b c^{3} x^{8} + \frac {1}{7} \, C b^{4} x^{7} + \frac {12}{7} \, C a b^{2} c x^{7} + \frac {6}{7} \, C a^{2} c^{2} x^{7} + \frac {6}{7} \, A b^{2} c^{2} x^{7} + \frac {4}{7} \, A a c^{3} x^{7} + \frac {2}{3} \, C a b^{3} x^{6} + 2 \, C a^{2} b c x^{6} + \frac {2}{3} \, A b^{3} c x^{6} + 2 \, A a b c^{2} x^{6} + \frac {6}{5} \, C a^{2} b^{2} x^{5} + \frac {1}{5} \, A b^{4} x^{5} + \frac {4}{5} \, C a^{3} c x^{5} + \frac {12}{5} \, A a b^{2} c x^{5} + \frac {6}{5} \, A a^{2} c^{2} x^{5} + C a^{3} b x^{4} + A a b^{3} x^{4} + 3 \, A a^{2} b c x^{4} + \frac {1}{3} \, C a^{4} x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac {4}{3} \, A a^{3} c x^{3} + 2 \, A a^{3} b x^{2} + A a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*C*c^4*x^11 + 2/5*C*b*c^3*x^10 + 2/3*C*b^2*c^2*x^9 + 4/9*C*a*c^3*x^9 + 1/9*A*c^4*x^9 + 1/2*C*b^3*c*x^8 + 3
/2*C*a*b*c^2*x^8 + 1/2*A*b*c^3*x^8 + 1/7*C*b^4*x^7 + 12/7*C*a*b^2*c*x^7 + 6/7*C*a^2*c^2*x^7 + 6/7*A*b^2*c^2*x^
7 + 4/7*A*a*c^3*x^7 + 2/3*C*a*b^3*x^6 + 2*C*a^2*b*c*x^6 + 2/3*A*b^3*c*x^6 + 2*A*a*b*c^2*x^6 + 6/5*C*a^2*b^2*x^
5 + 1/5*A*b^4*x^5 + 4/5*C*a^3*c*x^5 + 12/5*A*a*b^2*c*x^5 + 6/5*A*a^2*c^2*x^5 + C*a^3*b*x^4 + A*a*b^3*x^4 + 3*A
*a^2*b*c*x^4 + 1/3*C*a^4*x^3 + 2*A*a^2*b^2*x^3 + 4/3*A*a^3*c*x^3 + 2*A*a^3*b*x^2 + A*a^4*x

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maple [A]  time = 0.00, size = 343, normalized size = 1.35 \[ \frac {C \,c^{4} x^{11}}{11}+\frac {2 C b \,c^{3} x^{10}}{5}+\frac {\left (A \,c^{4}+\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) C \right ) x^{9}}{9}+\frac {\left (4 A b \,c^{3}+\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) C \right ) x^{8}}{8}+2 A \,a^{3} b \,x^{2}+\frac {\left (\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) A +\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) C \right ) x^{7}}{7}+A \,a^{4} x +\frac {\left (\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) A +\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) C \right ) x^{6}}{6}+\frac {\left (\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) A +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) C \right ) x^{5}}{5}+\frac {\left (4 C \,a^{3} b +\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) A \right ) x^{4}}{4}+\frac {\left (C \,a^{4}+\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) A \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^4*(C*x^2+A),x)

[Out]

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

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maxima [A]  time = 0.44, size = 263, normalized size = 1.04 \[ \frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {1}{9} \, {\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac {1}{2} \, {\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac {2}{3} \, {\left (C a b^{3} + 3 \, A a b c^{2} + {\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} + {\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.13, size = 244, normalized size = 0.96 \[ x^5\,\left (\frac {4\,C\,a^3\,c}{5}+\frac {6\,C\,a^2\,b^2}{5}+\frac {6\,A\,a^2\,c^2}{5}+\frac {12\,A\,a\,b^2\,c}{5}+\frac {A\,b^4}{5}\right )+x^7\,\left (\frac {6\,C\,a^2\,c^2}{7}+\frac {12\,C\,a\,b^2\,c}{7}+\frac {4\,A\,a\,c^3}{7}+\frac {C\,b^4}{7}+\frac {6\,A\,b^2\,c^2}{7}\right )+x^3\,\left (\frac {C\,a^4}{3}+\frac {4\,A\,c\,a^3}{3}+2\,A\,a^2\,b^2\right )+x^9\,\left (\frac {2\,C\,b^2\,c^2}{3}+\frac {A\,c^4}{9}+\frac {4\,C\,a\,c^3}{9}\right )+\frac {C\,c^4\,x^{11}}{11}+A\,a^4\,x+\frac {2\,b\,x^6\,\left (b^2+3\,a\,c\right )\,\left (A\,c+C\,a\right )}{3}+a\,b\,x^4\,\left (C\,a^2+3\,A\,c\,a+A\,b^2\right )+\frac {b\,c\,x^8\,\left (C\,b^2+A\,c^2+3\,C\,a\,c\right )}{2}+2\,A\,a^3\,b\,x^2+\frac {2\,C\,b\,c^3\,x^{10}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + C*x^2)*(a + b*x + c*x^2)^4,x)

[Out]

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

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sympy [A]  time = 0.14, size = 320, normalized size = 1.26 \[ A a^{4} x + 2 A a^{3} b x^{2} + \frac {2 C b c^{3} x^{10}}{5} + \frac {C c^{4} x^{11}}{11} + x^{9} \left (\frac {A c^{4}}{9} + \frac {4 C a c^{3}}{9} + \frac {2 C b^{2} c^{2}}{3}\right ) + x^{8} \left (\frac {A b c^{3}}{2} + \frac {3 C a b c^{2}}{2} + \frac {C b^{3} c}{2}\right ) + x^{7} \left (\frac {4 A a c^{3}}{7} + \frac {6 A b^{2} c^{2}}{7} + \frac {6 C a^{2} c^{2}}{7} + \frac {12 C a b^{2} c}{7} + \frac {C b^{4}}{7}\right ) + x^{6} \left (2 A a b c^{2} + \frac {2 A b^{3} c}{3} + 2 C a^{2} b c + \frac {2 C a b^{3}}{3}\right ) + x^{5} \left (\frac {6 A a^{2} c^{2}}{5} + \frac {12 A a b^{2} c}{5} + \frac {A b^{4}}{5} + \frac {4 C a^{3} c}{5} + \frac {6 C a^{2} b^{2}}{5}\right ) + x^{4} \left (3 A a^{2} b c + A a b^{3} + C a^{3} b\right ) + x^{3} \left (\frac {4 A a^{3} c}{3} + 2 A a^{2} b^{2} + \frac {C a^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4*(C*x**2+A),x)

[Out]

A*a**4*x + 2*A*a**3*b*x**2 + 2*C*b*c**3*x**10/5 + C*c**4*x**11/11 + x**9*(A*c**4/9 + 4*C*a*c**3/9 + 2*C*b**2*c
**2/3) + x**8*(A*b*c**3/2 + 3*C*a*b*c**2/2 + C*b**3*c/2) + x**7*(4*A*a*c**3/7 + 6*A*b**2*c**2/7 + 6*C*a**2*c**
2/7 + 12*C*a*b**2*c/7 + C*b**4/7) + x**6*(2*A*a*b*c**2 + 2*A*b**3*c/3 + 2*C*a**2*b*c + 2*C*a*b**3/3) + x**5*(6
*A*a**2*c**2/5 + 12*A*a*b**2*c/5 + A*b**4/5 + 4*C*a**3*c/5 + 6*C*a**2*b**2/5) + x**4*(3*A*a**2*b*c + A*a*b**3
+ C*a**3*b) + x**3*(4*A*a**3*c/3 + 2*A*a**2*b**2 + C*a**4/3)

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