3.1.94 \(\int x (b+2 c x^2) (a+b x^2+c x^4)^{13} \, dx\) [94]

Optimal. Leaf size=18 \[ \frac {1}{28} \left (a+b x^2+c x^4\right )^{14} \]

[Out]

1/28*(c*x^4+b*x^2+a)^14

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Rubi [A]
time = 0.21, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1261, 643} \begin {gather*} \frac {1}{28} \left (a+b x^2+c x^4\right )^{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^13,x]

[Out]

(a + b*x^2 + c*x^4)^14/28

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int x \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{13} \, dx &=\frac {1}{2} \text {Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^{13} \, dx,x,x^2\right )\\ &=\frac {1}{28} \left (a+b x^2+c x^4\right )^{14}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(18)=36\).
time = 0.12, size = 233, normalized size = 12.94 \begin {gather*} \frac {1}{28} x^2 \left (b+c x^2\right ) \left (14 a^{13}+91 a^{12} x^2 \left (b+c x^2\right )+364 a^{11} x^4 \left (b+c x^2\right )^2+1001 a^{10} x^6 \left (b+c x^2\right )^3+2002 a^9 x^8 \left (b+c x^2\right )^4+3003 a^8 x^{10} \left (b+c x^2\right )^5+3432 a^7 x^{12} \left (b+c x^2\right )^6+3003 a^6 x^{14} \left (b+c x^2\right )^7+2002 a^5 x^{16} \left (b+c x^2\right )^8+1001 a^4 x^{18} \left (b+c x^2\right )^9+364 a^3 x^{20} \left (b+c x^2\right )^{10}+91 a^2 x^{22} \left (b+c x^2\right )^{11}+14 a x^{24} \left (b+c x^2\right )^{12}+x^{26} \left (b+c x^2\right )^{13}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^13,x]

[Out]

(x^2*(b + c*x^2)*(14*a^13 + 91*a^12*x^2*(b + c*x^2) + 364*a^11*x^4*(b + c*x^2)^2 + 1001*a^10*x^6*(b + c*x^2)^3
+ 2002*a^9*x^8*(b + c*x^2)^4 + 3003*a^8*x^10*(b + c*x^2)^5 + 3432*a^7*x^12*(b + c*x^2)^6 + 3003*a^6*x^14*(b +
c*x^2)^7 + 2002*a^5*x^16*(b + c*x^2)^8 + 1001*a^4*x^18*(b + c*x^2)^9 + 364*a^3*x^20*(b + c*x^2)^10 + 91*a^2*x
^22*(b + c*x^2)^11 + 14*a*x^24*(b + c*x^2)^12 + x^26*(b + c*x^2)^13))/28

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Maple [A]
time = 0.09, size = 17, normalized size = 0.94

method	result	size
default	\(\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{14}}{28}\)	\(17\)
gosper	\(\text {Expression too large to display}\)	\(1455\)
risch	\(\text {Expression too large to display}\)	\(1460\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*c*x^2+b)*(c*x^4+b*x^2+a)^13,x,method=_RETURNVERBOSE)

[Out]

1/28*(c*x^4+b*x^2+a)^14

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1240 vs. \(2 (16) = 32\).
time = 0.30, size = 1240, normalized size = 68.89 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {1}{4} \, {\left (13 \, b^{2} c^{12} + 2 \, a c^{13}\right )} x^{52} + \frac {13}{2} \, {\left (2 \, b^{3} c^{11} + a b c^{12}\right )} x^{50} + \frac {13}{4} \, {\left (11 \, b^{4} c^{10} + 12 \, a b^{2} c^{11} + a^{2} c^{12}\right )} x^{48} + \frac {13}{2} \, {\left (11 \, b^{5} c^{9} + 22 \, a b^{3} c^{10} + 6 \, a^{2} b c^{11}\right )} x^{46} + \frac {13}{4} \, {\left (33 \, b^{6} c^{8} + 110 \, a b^{4} c^{9} + 66 \, a^{2} b^{2} c^{10} + 4 \, a^{3} c^{11}\right )} x^{44} + \frac {143}{14} \, {\left (12 \, b^{7} c^{7} + 63 \, a b^{5} c^{8} + 70 \, a^{2} b^{3} c^{9} + 14 \, a^{3} b c^{10}\right )} x^{42} + \frac {143}{4} \, {\left (3 \, b^{8} c^{6} + 24 \, a b^{6} c^{7} + 45 \, a^{2} b^{4} c^{8} + 20 \, a^{3} b^{2} c^{9} + a^{4} c^{10}\right )} x^{40} + \frac {143}{2} \, {\left (b^{9} c^{5} + 12 \, a b^{7} c^{6} + 36 \, a^{2} b^{5} c^{7} + 30 \, a^{3} b^{3} c^{8} + 5 \, a^{4} b c^{9}\right )} x^{38} + \frac {143}{4} \, {\left (b^{10} c^{4} + 18 \, a b^{8} c^{5} + 84 \, a^{2} b^{6} c^{6} + 120 \, a^{3} b^{4} c^{7} + 45 \, a^{4} b^{2} c^{8} + 2 \, a^{5} c^{9}\right )} x^{36} + \frac {13}{2} \, {\left (2 \, b^{11} c^{3} + 55 \, a b^{9} c^{4} + 396 \, a^{2} b^{7} c^{5} + 924 \, a^{3} b^{5} c^{6} + 660 \, a^{4} b^{3} c^{7} + 99 \, a^{5} b c^{8}\right )} x^{34} + \frac {13}{4} \, {\left (b^{12} c^{2} + 44 \, a b^{10} c^{3} + 495 \, a^{2} b^{8} c^{4} + 1848 \, a^{3} b^{6} c^{5} + 2310 \, a^{4} b^{4} c^{6} + 792 \, a^{5} b^{2} c^{7} + 33 \, a^{6} c^{8}\right )} x^{32} + \frac {1}{2} \, {\left (b^{13} c + 78 \, a b^{11} c^{2} + 1430 \, a^{2} b^{9} c^{3} + 8580 \, a^{3} b^{7} c^{4} + 18018 \, a^{4} b^{5} c^{5} + 12012 \, a^{5} b^{3} c^{6} + 1716 \, a^{6} b c^{7}\right )} x^{30} + \frac {1}{28} \, {\left (b^{14} + 182 \, a b^{12} c + 6006 \, a^{2} b^{10} c^{2} + 60060 \, a^{3} b^{8} c^{3} + 210210 \, a^{4} b^{6} c^{4} + 252252 \, a^{5} b^{4} c^{5} + 84084 \, a^{6} b^{2} c^{6} + 3432 \, a^{7} c^{7}\right )} x^{28} + \frac {1}{2} \, {\left (a b^{13} + 78 \, a^{2} b^{11} c + 1430 \, a^{3} b^{9} c^{2} + 8580 \, a^{4} b^{7} c^{3} + 18018 \, a^{5} b^{5} c^{4} + 12012 \, a^{6} b^{3} c^{5} + 1716 \, a^{7} b c^{6}\right )} x^{26} + \frac {13}{4} \, {\left (a^{2} b^{12} + 44 \, a^{3} b^{10} c + 495 \, a^{4} b^{8} c^{2} + 1848 \, a^{5} b^{6} c^{3} + 2310 \, a^{6} b^{4} c^{4} + 792 \, a^{7} b^{2} c^{5} + 33 \, a^{8} c^{6}\right )} x^{24} + \frac {13}{2} \, {\left (2 \, a^{3} b^{11} + 55 \, a^{4} b^{9} c + 396 \, a^{5} b^{7} c^{2} + 924 \, a^{6} b^{5} c^{3} + 660 \, a^{7} b^{3} c^{4} + 99 \, a^{8} b c^{5}\right )} x^{22} + \frac {143}{4} \, {\left (a^{4} b^{10} + 18 \, a^{5} b^{8} c + 84 \, a^{6} b^{6} c^{2} + 120 \, a^{7} b^{4} c^{3} + 45 \, a^{8} b^{2} c^{4} + 2 \, a^{9} c^{5}\right )} x^{20} + \frac {143}{2} \, {\left (a^{5} b^{9} + 12 \, a^{6} b^{7} c + 36 \, a^{7} b^{5} c^{2} + 30 \, a^{8} b^{3} c^{3} + 5 \, a^{9} b c^{4}\right )} x^{18} + \frac {143}{4} \, {\left (3 \, a^{6} b^{8} + 24 \, a^{7} b^{6} c + 45 \, a^{8} b^{4} c^{2} + 20 \, a^{9} b^{2} c^{3} + a^{10} c^{4}\right )} x^{16} + \frac {1}{2} \, a^{13} b x^{2} + \frac {143}{14} \, {\left (12 \, a^{7} b^{7} + 63 \, a^{8} b^{5} c + 70 \, a^{9} b^{3} c^{2} + 14 \, a^{10} b c^{3}\right )} x^{14} + \frac {13}{4} \, {\left (33 \, a^{8} b^{6} + 110 \, a^{9} b^{4} c + 66 \, a^{10} b^{2} c^{2} + 4 \, a^{11} c^{3}\right )} x^{12} + \frac {13}{2} \, {\left (11 \, a^{9} b^{5} + 22 \, a^{10} b^{3} c + 6 \, a^{11} b c^{2}\right )} x^{10} + \frac {13}{4} \, {\left (11 \, a^{10} b^{4} + 12 \, a^{11} b^{2} c + a^{12} c^{2}\right )} x^{8} + \frac {13}{2} \, {\left (2 \, a^{11} b^{3} + a^{12} b c\right )} x^{6} + \frac {1}{4} \, {\left (13 \, a^{12} b^{2} + 2 \, a^{13} c\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2+a)^13,x, algorithm="maxima")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 1/4*(13*b^2*c^12 + 2*a*c^13)*x^52 + 13/2*(2*b^3*c^11 + a*b*c^12)*x^50 + 13/
4*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c^12)*x^48 + 13/2*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^46 + 13/4
*(33*b^6*c^8 + 110*a*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^44 + 143/14*(12*b^7*c^7 + 63*a*b^5*c^8 + 70*a^2
*b^3*c^9 + 14*a^3*b*c^10)*x^42 + 143/4*(3*b^8*c^6 + 24*a*b^6*c^7 + 45*a^2*b^4*c^8 + 20*a^3*b^2*c^9 + a^4*c^10)
*x^40 + 143/2*(b^9*c^5 + 12*a*b^7*c^6 + 36*a^2*b^5*c^7 + 30*a^3*b^3*c^8 + 5*a^4*b*c^9)*x^38 + 143/4*(b^10*c^4
+ 18*a*b^8*c^5 + 84*a^2*b^6*c^6 + 120*a^3*b^4*c^7 + 45*a^4*b^2*c^8 + 2*a^5*c^9)*x^36 + 13/2*(2*b^11*c^3 + 55*a
*b^9*c^4 + 396*a^2*b^7*c^5 + 924*a^3*b^5*c^6 + 660*a^4*b^3*c^7 + 99*a^5*b*c^8)*x^34 + 13/4*(b^12*c^2 + 44*a*b^
10*c^3 + 495*a^2*b^8*c^4 + 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6 + 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^32 + 1/2*(b^1
3*c + 78*a*b^11*c^2 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 + 12012*a^5*b^3*c^6 + 1716*a^6*b
*c^7)*x^30 + 1/28*(b^14 + 182*a*b^12*c + 6006*a^2*b^10*c^2 + 60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 + 252252*a
^5*b^4*c^5 + 84084*a^6*b^2*c^6 + 3432*a^7*c^7)*x^28 + 1/2*(a*b^13 + 78*a^2*b^11*c + 1430*a^3*b^9*c^2 + 8580*a^
4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*a^6*b^3*c^5 + 1716*a^7*b*c^6)*x^26 + 13/4*(a^2*b^12 + 44*a^3*b^10*c + 49
5*a^4*b^8*c^2 + 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 + 792*a^7*b^2*c^5 + 33*a^8*c^6)*x^24 + 13/2*(2*a^3*b^11 +
55*a^4*b^9*c + 396*a^5*b^7*c^2 + 924*a^6*b^5*c^3 + 660*a^7*b^3*c^4 + 99*a^8*b*c^5)*x^22 + 143/4*(a^4*b^10 + 18
*a^5*b^8*c + 84*a^6*b^6*c^2 + 120*a^7*b^4*c^3 + 45*a^8*b^2*c^4 + 2*a^9*c^5)*x^20 + 143/2*(a^5*b^9 + 12*a^6*b^7
*c + 36*a^7*b^5*c^2 + 30*a^8*b^3*c^3 + 5*a^9*b*c^4)*x^18 + 143/4*(3*a^6*b^8 + 24*a^7*b^6*c + 45*a^8*b^4*c^2 +
20*a^9*b^2*c^3 + a^10*c^4)*x^16 + 1/2*a^13*b*x^2 + 143/14*(12*a^7*b^7 + 63*a^8*b^5*c + 70*a^9*b^3*c^2 + 14*a^1
0*b*c^3)*x^14 + 13/4*(33*a^8*b^6 + 110*a^9*b^4*c + 66*a^10*b^2*c^2 + 4*a^11*c^3)*x^12 + 13/2*(11*a^9*b^5 + 22*
a^10*b^3*c + 6*a^11*b*c^2)*x^10 + 13/4*(11*a^10*b^4 + 12*a^11*b^2*c + a^12*c^2)*x^8 + 13/2*(2*a^11*b^3 + a^12*
b*c)*x^6 + 1/4*(13*a^12*b^2 + 2*a^13*c)*x^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1240 vs. \(2 (16) = 32\).
time = 0.35, size = 1240, normalized size = 68.89 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {1}{4} \, {\left (13 \, b^{2} c^{12} + 2 \, a c^{13}\right )} x^{52} + \frac {13}{2} \, {\left (2 \, b^{3} c^{11} + a b c^{12}\right )} x^{50} + \frac {13}{4} \, {\left (11 \, b^{4} c^{10} + 12 \, a b^{2} c^{11} + a^{2} c^{12}\right )} x^{48} + \frac {13}{2} \, {\left (11 \, b^{5} c^{9} + 22 \, a b^{3} c^{10} + 6 \, a^{2} b c^{11}\right )} x^{46} + \frac {13}{4} \, {\left (33 \, b^{6} c^{8} + 110 \, a b^{4} c^{9} + 66 \, a^{2} b^{2} c^{10} + 4 \, a^{3} c^{11}\right )} x^{44} + \frac {143}{14} \, {\left (12 \, b^{7} c^{7} + 63 \, a b^{5} c^{8} + 70 \, a^{2} b^{3} c^{9} + 14 \, a^{3} b c^{10}\right )} x^{42} + \frac {143}{4} \, {\left (3 \, b^{8} c^{6} + 24 \, a b^{6} c^{7} + 45 \, a^{2} b^{4} c^{8} + 20 \, a^{3} b^{2} c^{9} + a^{4} c^{10}\right )} x^{40} + \frac {143}{2} \, {\left (b^{9} c^{5} + 12 \, a b^{7} c^{6} + 36 \, a^{2} b^{5} c^{7} + 30 \, a^{3} b^{3} c^{8} + 5 \, a^{4} b c^{9}\right )} x^{38} + \frac {143}{4} \, {\left (b^{10} c^{4} + 18 \, a b^{8} c^{5} + 84 \, a^{2} b^{6} c^{6} + 120 \, a^{3} b^{4} c^{7} + 45 \, a^{4} b^{2} c^{8} + 2 \, a^{5} c^{9}\right )} x^{36} + \frac {13}{2} \, {\left (2 \, b^{11} c^{3} + 55 \, a b^{9} c^{4} + 396 \, a^{2} b^{7} c^{5} + 924 \, a^{3} b^{5} c^{6} + 660 \, a^{4} b^{3} c^{7} + 99 \, a^{5} b c^{8}\right )} x^{34} + \frac {13}{4} \, {\left (b^{12} c^{2} + 44 \, a b^{10} c^{3} + 495 \, a^{2} b^{8} c^{4} + 1848 \, a^{3} b^{6} c^{5} + 2310 \, a^{4} b^{4} c^{6} + 792 \, a^{5} b^{2} c^{7} + 33 \, a^{6} c^{8}\right )} x^{32} + \frac {1}{2} \, {\left (b^{13} c + 78 \, a b^{11} c^{2} + 1430 \, a^{2} b^{9} c^{3} + 8580 \, a^{3} b^{7} c^{4} + 18018 \, a^{4} b^{5} c^{5} + 12012 \, a^{5} b^{3} c^{6} + 1716 \, a^{6} b c^{7}\right )} x^{30} + \frac {1}{28} \, {\left (b^{14} + 182 \, a b^{12} c + 6006 \, a^{2} b^{10} c^{2} + 60060 \, a^{3} b^{8} c^{3} + 210210 \, a^{4} b^{6} c^{4} + 252252 \, a^{5} b^{4} c^{5} + 84084 \, a^{6} b^{2} c^{6} + 3432 \, a^{7} c^{7}\right )} x^{28} + \frac {1}{2} \, {\left (a b^{13} + 78 \, a^{2} b^{11} c + 1430 \, a^{3} b^{9} c^{2} + 8580 \, a^{4} b^{7} c^{3} + 18018 \, a^{5} b^{5} c^{4} + 12012 \, a^{6} b^{3} c^{5} + 1716 \, a^{7} b c^{6}\right )} x^{26} + \frac {13}{4} \, {\left (a^{2} b^{12} + 44 \, a^{3} b^{10} c + 495 \, a^{4} b^{8} c^{2} + 1848 \, a^{5} b^{6} c^{3} + 2310 \, a^{6} b^{4} c^{4} + 792 \, a^{7} b^{2} c^{5} + 33 \, a^{8} c^{6}\right )} x^{24} + \frac {13}{2} \, {\left (2 \, a^{3} b^{11} + 55 \, a^{4} b^{9} c + 396 \, a^{5} b^{7} c^{2} + 924 \, a^{6} b^{5} c^{3} + 660 \, a^{7} b^{3} c^{4} + 99 \, a^{8} b c^{5}\right )} x^{22} + \frac {143}{4} \, {\left (a^{4} b^{10} + 18 \, a^{5} b^{8} c + 84 \, a^{6} b^{6} c^{2} + 120 \, a^{7} b^{4} c^{3} + 45 \, a^{8} b^{2} c^{4} + 2 \, a^{9} c^{5}\right )} x^{20} + \frac {143}{2} \, {\left (a^{5} b^{9} + 12 \, a^{6} b^{7} c + 36 \, a^{7} b^{5} c^{2} + 30 \, a^{8} b^{3} c^{3} + 5 \, a^{9} b c^{4}\right )} x^{18} + \frac {143}{4} \, {\left (3 \, a^{6} b^{8} + 24 \, a^{7} b^{6} c + 45 \, a^{8} b^{4} c^{2} + 20 \, a^{9} b^{2} c^{3} + a^{10} c^{4}\right )} x^{16} + \frac {1}{2} \, a^{13} b x^{2} + \frac {143}{14} \, {\left (12 \, a^{7} b^{7} + 63 \, a^{8} b^{5} c + 70 \, a^{9} b^{3} c^{2} + 14 \, a^{10} b c^{3}\right )} x^{14} + \frac {13}{4} \, {\left (33 \, a^{8} b^{6} + 110 \, a^{9} b^{4} c + 66 \, a^{10} b^{2} c^{2} + 4 \, a^{11} c^{3}\right )} x^{12} + \frac {13}{2} \, {\left (11 \, a^{9} b^{5} + 22 \, a^{10} b^{3} c + 6 \, a^{11} b c^{2}\right )} x^{10} + \frac {13}{4} \, {\left (11 \, a^{10} b^{4} + 12 \, a^{11} b^{2} c + a^{12} c^{2}\right )} x^{8} + \frac {13}{2} \, {\left (2 \, a^{11} b^{3} + a^{12} b c\right )} x^{6} + \frac {1}{4} \, {\left (13 \, a^{12} b^{2} + 2 \, a^{13} c\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2+a)^13,x, algorithm="fricas")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 1/4*(13*b^2*c^12 + 2*a*c^13)*x^52 + 13/2*(2*b^3*c^11 + a*b*c^12)*x^50 + 13/
4*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c^12)*x^48 + 13/2*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^46 + 13/4
*(33*b^6*c^8 + 110*a*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^44 + 143/14*(12*b^7*c^7 + 63*a*b^5*c^8 + 70*a^2
*b^3*c^9 + 14*a^3*b*c^10)*x^42 + 143/4*(3*b^8*c^6 + 24*a*b^6*c^7 + 45*a^2*b^4*c^8 + 20*a^3*b^2*c^9 + a^4*c^10)
*x^40 + 143/2*(b^9*c^5 + 12*a*b^7*c^6 + 36*a^2*b^5*c^7 + 30*a^3*b^3*c^8 + 5*a^4*b*c^9)*x^38 + 143/4*(b^10*c^4
+ 18*a*b^8*c^5 + 84*a^2*b^6*c^6 + 120*a^3*b^4*c^7 + 45*a^4*b^2*c^8 + 2*a^5*c^9)*x^36 + 13/2*(2*b^11*c^3 + 55*a
*b^9*c^4 + 396*a^2*b^7*c^5 + 924*a^3*b^5*c^6 + 660*a^4*b^3*c^7 + 99*a^5*b*c^8)*x^34 + 13/4*(b^12*c^2 + 44*a*b^
10*c^3 + 495*a^2*b^8*c^4 + 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6 + 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^32 + 1/2*(b^1
3*c + 78*a*b^11*c^2 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 + 12012*a^5*b^3*c^6 + 1716*a^6*b
*c^7)*x^30 + 1/28*(b^14 + 182*a*b^12*c + 6006*a^2*b^10*c^2 + 60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 + 252252*a
^5*b^4*c^5 + 84084*a^6*b^2*c^6 + 3432*a^7*c^7)*x^28 + 1/2*(a*b^13 + 78*a^2*b^11*c + 1430*a^3*b^9*c^2 + 8580*a^
4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*a^6*b^3*c^5 + 1716*a^7*b*c^6)*x^26 + 13/4*(a^2*b^12 + 44*a^3*b^10*c + 49
5*a^4*b^8*c^2 + 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 + 792*a^7*b^2*c^5 + 33*a^8*c^6)*x^24 + 13/2*(2*a^3*b^11 +
55*a^4*b^9*c + 396*a^5*b^7*c^2 + 924*a^6*b^5*c^3 + 660*a^7*b^3*c^4 + 99*a^8*b*c^5)*x^22 + 143/4*(a^4*b^10 + 18
*a^5*b^8*c + 84*a^6*b^6*c^2 + 120*a^7*b^4*c^3 + 45*a^8*b^2*c^4 + 2*a^9*c^5)*x^20 + 143/2*(a^5*b^9 + 12*a^6*b^7
*c + 36*a^7*b^5*c^2 + 30*a^8*b^3*c^3 + 5*a^9*b*c^4)*x^18 + 143/4*(3*a^6*b^8 + 24*a^7*b^6*c + 45*a^8*b^4*c^2 +
20*a^9*b^2*c^3 + a^10*c^4)*x^16 + 1/2*a^13*b*x^2 + 143/14*(12*a^7*b^7 + 63*a^8*b^5*c + 70*a^9*b^3*c^2 + 14*a^1
0*b*c^3)*x^14 + 13/4*(33*a^8*b^6 + 110*a^9*b^4*c + 66*a^10*b^2*c^2 + 4*a^11*c^3)*x^12 + 13/2*(11*a^9*b^5 + 22*
a^10*b^3*c + 6*a^11*b*c^2)*x^10 + 13/4*(11*a^10*b^4 + 12*a^11*b^2*c + a^12*c^2)*x^8 + 13/2*(2*a^11*b^3 + a^12*
b*c)*x^6 + 1/4*(13*a^12*b^2 + 2*a^13*c)*x^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1384 vs. \(2 (14) = 28\).
time = 0.17, size = 1384, normalized size = 76.89 \begin {gather*} \frac {a^{13} b x^{2}}{2} + \frac {b c^{13} x^{54}}{2} + \frac {c^{14} x^{56}}{28} + x^{52} \left (\frac {a c^{13}}{2} + \frac {13 b^{2} c^{12}}{4}\right ) + x^{50} \cdot \left (\frac {13 a b c^{12}}{2} + 13 b^{3} c^{11}\right ) + x^{48} \cdot \left (\frac {13 a^{2} c^{12}}{4} + 39 a b^{2} c^{11} + \frac {143 b^{4} c^{10}}{4}\right ) + x^{46} \cdot \left (39 a^{2} b c^{11} + 143 a b^{3} c^{10} + \frac {143 b^{5} c^{9}}{2}\right ) + x^{44} \cdot \left (13 a^{3} c^{11} + \frac {429 a^{2} b^{2} c^{10}}{2} + \frac {715 a b^{4} c^{9}}{2} + \frac {429 b^{6} c^{8}}{4}\right ) + x^{42} \cdot \left (143 a^{3} b c^{10} + 715 a^{2} b^{3} c^{9} + \frac {1287 a b^{5} c^{8}}{2} + \frac {858 b^{7} c^{7}}{7}\right ) + x^{40} \cdot \left (\frac {143 a^{4} c^{10}}{4} + 715 a^{3} b^{2} c^{9} + \frac {6435 a^{2} b^{4} c^{8}}{4} + 858 a b^{6} c^{7} + \frac {429 b^{8} c^{6}}{4}\right ) + x^{38} \cdot \left (\frac {715 a^{4} b c^{9}}{2} + 2145 a^{3} b^{3} c^{8} + 2574 a^{2} b^{5} c^{7} + 858 a b^{7} c^{6} + \frac {143 b^{9} c^{5}}{2}\right ) + x^{36} \cdot \left (\frac {143 a^{5} c^{9}}{2} + \frac {6435 a^{4} b^{2} c^{8}}{4} + 4290 a^{3} b^{4} c^{7} + 3003 a^{2} b^{6} c^{6} + \frac {1287 a b^{8} c^{5}}{2} + \frac {143 b^{10} c^{4}}{4}\right ) + x^{34} \cdot \left (\frac {1287 a^{5} b c^{8}}{2} + 4290 a^{4} b^{3} c^{7} + 6006 a^{3} b^{5} c^{6} + 2574 a^{2} b^{7} c^{5} + \frac {715 a b^{9} c^{4}}{2} + 13 b^{11} c^{3}\right ) + x^{32} \cdot \left (\frac {429 a^{6} c^{8}}{4} + 2574 a^{5} b^{2} c^{7} + \frac {15015 a^{4} b^{4} c^{6}}{2} + 6006 a^{3} b^{6} c^{5} + \frac {6435 a^{2} b^{8} c^{4}}{4} + 143 a b^{10} c^{3} + \frac {13 b^{12} c^{2}}{4}\right ) + x^{30} \cdot \left (858 a^{6} b c^{7} + 6006 a^{5} b^{3} c^{6} + 9009 a^{4} b^{5} c^{5} + 4290 a^{3} b^{7} c^{4} + 715 a^{2} b^{9} c^{3} + 39 a b^{11} c^{2} + \frac {b^{13} c}{2}\right ) + x^{28} \cdot \left (\frac {858 a^{7} c^{7}}{7} + 3003 a^{6} b^{2} c^{6} + 9009 a^{5} b^{4} c^{5} + \frac {15015 a^{4} b^{6} c^{4}}{2} + 2145 a^{3} b^{8} c^{3} + \frac {429 a^{2} b^{10} c^{2}}{2} + \frac {13 a b^{12} c}{2} + \frac {b^{14}}{28}\right ) + x^{26} \cdot \left (858 a^{7} b c^{6} + 6006 a^{6} b^{3} c^{5} + 9009 a^{5} b^{5} c^{4} + 4290 a^{4} b^{7} c^{3} + 715 a^{3} b^{9} c^{2} + 39 a^{2} b^{11} c + \frac {a b^{13}}{2}\right ) + x^{24} \cdot \left (\frac {429 a^{8} c^{6}}{4} + 2574 a^{7} b^{2} c^{5} + \frac {15015 a^{6} b^{4} c^{4}}{2} + 6006 a^{5} b^{6} c^{3} + \frac {6435 a^{4} b^{8} c^{2}}{4} + 143 a^{3} b^{10} c + \frac {13 a^{2} b^{12}}{4}\right ) + x^{22} \cdot \left (\frac {1287 a^{8} b c^{5}}{2} + 4290 a^{7} b^{3} c^{4} + 6006 a^{6} b^{5} c^{3} + 2574 a^{5} b^{7} c^{2} + \frac {715 a^{4} b^{9} c}{2} + 13 a^{3} b^{11}\right ) + x^{20} \cdot \left (\frac {143 a^{9} c^{5}}{2} + \frac {6435 a^{8} b^{2} c^{4}}{4} + 4290 a^{7} b^{4} c^{3} + 3003 a^{6} b^{6} c^{2} + \frac {1287 a^{5} b^{8} c}{2} + \frac {143 a^{4} b^{10}}{4}\right ) + x^{18} \cdot \left (\frac {715 a^{9} b c^{4}}{2} + 2145 a^{8} b^{3} c^{3} + 2574 a^{7} b^{5} c^{2} + 858 a^{6} b^{7} c + \frac {143 a^{5} b^{9}}{2}\right ) + x^{16} \cdot \left (\frac {143 a^{10} c^{4}}{4} + 715 a^{9} b^{2} c^{3} + \frac {6435 a^{8} b^{4} c^{2}}{4} + 858 a^{7} b^{6} c + \frac {429 a^{6} b^{8}}{4}\right ) + x^{14} \cdot \left (143 a^{10} b c^{3} + 715 a^{9} b^{3} c^{2} + \frac {1287 a^{8} b^{5} c}{2} + \frac {858 a^{7} b^{7}}{7}\right ) + x^{12} \cdot \left (13 a^{11} c^{3} + \frac {429 a^{10} b^{2} c^{2}}{2} + \frac {715 a^{9} b^{4} c}{2} + \frac {429 a^{8} b^{6}}{4}\right ) + x^{10} \cdot \left (39 a^{11} b c^{2} + 143 a^{10} b^{3} c + \frac {143 a^{9} b^{5}}{2}\right ) + x^{8} \cdot \left (\frac {13 a^{12} c^{2}}{4} + 39 a^{11} b^{2} c + \frac {143 a^{10} b^{4}}{4}\right ) + x^{6} \cdot \left (\frac {13 a^{12} b c}{2} + 13 a^{11} b^{3}\right ) + x^{4} \left (\frac {a^{13} c}{2} + \frac {13 a^{12} b^{2}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2+a)**13,x)

[Out]

a**13*b*x**2/2 + b*c**13*x**54/2 + c**14*x**56/28 + x**52*(a*c**13/2 + 13*b**2*c**12/4) + x**50*(13*a*b*c**12/
2 + 13*b**3*c**11) + x**48*(13*a**2*c**12/4 + 39*a*b**2*c**11 + 143*b**4*c**10/4) + x**46*(39*a**2*b*c**11 + 1
43*a*b**3*c**10 + 143*b**5*c**9/2) + x**44*(13*a**3*c**11 + 429*a**2*b**2*c**10/2 + 715*a*b**4*c**9/2 + 429*b*
*6*c**8/4) + x**42*(143*a**3*b*c**10 + 715*a**2*b**3*c**9 + 1287*a*b**5*c**8/2 + 858*b**7*c**7/7) + x**40*(143
*a**4*c**10/4 + 715*a**3*b**2*c**9 + 6435*a**2*b**4*c**8/4 + 858*a*b**6*c**7 + 429*b**8*c**6/4) + x**38*(715*a
**4*b*c**9/2 + 2145*a**3*b**3*c**8 + 2574*a**2*b**5*c**7 + 858*a*b**7*c**6 + 143*b**9*c**5/2) + x**36*(143*a**
5*c**9/2 + 6435*a**4*b**2*c**8/4 + 4290*a**3*b**4*c**7 + 3003*a**2*b**6*c**6 + 1287*a*b**8*c**5/2 + 143*b**10*
c**4/4) + x**34*(1287*a**5*b*c**8/2 + 4290*a**4*b**3*c**7 + 6006*a**3*b**5*c**6 + 2574*a**2*b**7*c**5 + 715*a*
b**9*c**4/2 + 13*b**11*c**3) + x**32*(429*a**6*c**8/4 + 2574*a**5*b**2*c**7 + 15015*a**4*b**4*c**6/2 + 6006*a*
*3*b**6*c**5 + 6435*a**2*b**8*c**4/4 + 143*a*b**10*c**3 + 13*b**12*c**2/4) + x**30*(858*a**6*b*c**7 + 6006*a**
5*b**3*c**6 + 9009*a**4*b**5*c**5 + 4290*a**3*b**7*c**4 + 715*a**2*b**9*c**3 + 39*a*b**11*c**2 + b**13*c/2) +
x**28*(858*a**7*c**7/7 + 3003*a**6*b**2*c**6 + 9009*a**5*b**4*c**5 + 15015*a**4*b**6*c**4/2 + 2145*a**3*b**8*c
**3 + 429*a**2*b**10*c**2/2 + 13*a*b**12*c/2 + b**14/28) + x**26*(858*a**7*b*c**6 + 6006*a**6*b**3*c**5 + 9009
*a**5*b**5*c**4 + 4290*a**4*b**7*c**3 + 715*a**3*b**9*c**2 + 39*a**2*b**11*c + a*b**13/2) + x**24*(429*a**8*c*
*6/4 + 2574*a**7*b**2*c**5 + 15015*a**6*b**4*c**4/2 + 6006*a**5*b**6*c**3 + 6435*a**4*b**8*c**2/4 + 143*a**3*b
**10*c + 13*a**2*b**12/4) + x**22*(1287*a**8*b*c**5/2 + 4290*a**7*b**3*c**4 + 6006*a**6*b**5*c**3 + 2574*a**5*
b**7*c**2 + 715*a**4*b**9*c/2 + 13*a**3*b**11) + x**20*(143*a**9*c**5/2 + 6435*a**8*b**2*c**4/4 + 4290*a**7*b*
*4*c**3 + 3003*a**6*b**6*c**2 + 1287*a**5*b**8*c/2 + 143*a**4*b**10/4) + x**18*(715*a**9*b*c**4/2 + 2145*a**8*
b**3*c**3 + 2574*a**7*b**5*c**2 + 858*a**6*b**7*c + 143*a**5*b**9/2) + x**16*(143*a**10*c**4/4 + 715*a**9*b**2
*c**3 + 6435*a**8*b**4*c**2/4 + 858*a**7*b**6*c + 429*a**6*b**8/4) + x**14*(143*a**10*b*c**3 + 715*a**9*b**3*c
**2 + 1287*a**8*b**5*c/2 + 858*a**7*b**7/7) + x**12*(13*a**11*c**3 + 429*a**10*b**2*c**2/2 + 715*a**9*b**4*c/2
+ 429*a**8*b**6/4) + x**10*(39*a**11*b*c**2 + 143*a**10*b**3*c + 143*a**9*b**5/2) + x**8*(13*a**12*c**2/4 + 3
9*a**11*b**2*c + 143*a**10*b**4/4) + x**6*(13*a**12*b*c/2 + 13*a**11*b**3) + x**4*(a**13*c/2 + 13*a**12*b**2/4
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (16) = 32\).
time = 3.60, size = 246, normalized size = 13.67 \begin {gather*} \frac {1}{28} \, {\left (c x^{4} + b x^{2}\right )}^{14} + \frac {1}{2} \, {\left (c x^{4} + b x^{2}\right )}^{13} a + \frac {13}{4} \, {\left (c x^{4} + b x^{2}\right )}^{12} a^{2} + 13 \, {\left (c x^{4} + b x^{2}\right )}^{11} a^{3} + \frac {143}{4} \, {\left (c x^{4} + b x^{2}\right )}^{10} a^{4} + \frac {143}{2} \, {\left (c x^{4} + b x^{2}\right )}^{9} a^{5} + \frac {429}{4} \, {\left (c x^{4} + b x^{2}\right )}^{8} a^{6} + \frac {858}{7} \, {\left (c x^{4} + b x^{2}\right )}^{7} a^{7} + \frac {429}{4} \, {\left (c x^{4} + b x^{2}\right )}^{6} a^{8} + \frac {143}{2} \, {\left (c x^{4} + b x^{2}\right )}^{5} a^{9} + \frac {143}{4} \, {\left (c x^{4} + b x^{2}\right )}^{4} a^{10} + 13 \, {\left (c x^{4} + b x^{2}\right )}^{3} a^{11} + \frac {13}{4} \, {\left (c x^{4} + b x^{2}\right )}^{2} a^{12} + \frac {1}{2} \, {\left (c x^{4} + b x^{2}\right )} a^{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2+a)^13,x, algorithm="giac")

[Out]

1/28*(c*x^4 + b*x^2)^14 + 1/2*(c*x^4 + b*x^2)^13*a + 13/4*(c*x^4 + b*x^2)^12*a^2 + 13*(c*x^4 + b*x^2)^11*a^3 +
143/4*(c*x^4 + b*x^2)^10*a^4 + 143/2*(c*x^4 + b*x^2)^9*a^5 + 429/4*(c*x^4 + b*x^2)^8*a^6 + 858/7*(c*x^4 + b*x
^2)^7*a^7 + 429/4*(c*x^4 + b*x^2)^6*a^8 + 143/2*(c*x^4 + b*x^2)^5*a^9 + 143/4*(c*x^4 + b*x^2)^4*a^10 + 13*(c*x
^4 + b*x^2)^3*a^11 + 13/4*(c*x^4 + b*x^2)^2*a^12 + 1/2*(c*x^4 + b*x^2)*a^13

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Mupad [B]
time = 3.23, size = 1210, normalized size = 67.22 \begin {gather*} x^{24}\,\left (\frac {429\,a^8\,c^6}{4}+2574\,a^7\,b^2\,c^5+\frac {15015\,a^6\,b^4\,c^4}{2}+6006\,a^5\,b^6\,c^3+\frac {6435\,a^4\,b^8\,c^2}{4}+143\,a^3\,b^{10}\,c+\frac {13\,a^2\,b^{12}}{4}\right )+x^{32}\,\left (\frac {429\,a^6\,c^8}{4}+2574\,a^5\,b^2\,c^7+\frac {15015\,a^4\,b^4\,c^6}{2}+6006\,a^3\,b^6\,c^5+\frac {6435\,a^2\,b^8\,c^4}{4}+143\,a\,b^{10}\,c^3+\frac {13\,b^{12}\,c^2}{4}\right )+x^{26}\,\left (858\,a^7\,b\,c^6+6006\,a^6\,b^3\,c^5+9009\,a^5\,b^5\,c^4+4290\,a^4\,b^7\,c^3+715\,a^3\,b^9\,c^2+39\,a^2\,b^{11}\,c+\frac {a\,b^{13}}{2}\right )+x^{30}\,\left (858\,a^6\,b\,c^7+6006\,a^5\,b^3\,c^6+9009\,a^4\,b^5\,c^5+4290\,a^3\,b^7\,c^4+715\,a^2\,b^9\,c^3+39\,a\,b^{11}\,c^2+\frac {b^{13}\,c}{2}\right )+x^{12}\,\left (13\,a^{11}\,c^3+\frac {429\,a^{10}\,b^2\,c^2}{2}+\frac {715\,a^9\,b^4\,c}{2}+\frac {429\,a^8\,b^6}{4}\right )+x^{44}\,\left (13\,a^3\,c^{11}+\frac {429\,a^2\,b^2\,c^{10}}{2}+\frac {715\,a\,b^4\,c^9}{2}+\frac {429\,b^6\,c^8}{4}\right )+x^{20}\,\left (\frac {143\,a^9\,c^5}{2}+\frac {6435\,a^8\,b^2\,c^4}{4}+4290\,a^7\,b^4\,c^3+3003\,a^6\,b^6\,c^2+\frac {1287\,a^5\,b^8\,c}{2}+\frac {143\,a^4\,b^{10}}{4}\right )+x^{36}\,\left (\frac {143\,a^5\,c^9}{2}+\frac {6435\,a^4\,b^2\,c^8}{4}+4290\,a^3\,b^4\,c^7+3003\,a^2\,b^6\,c^6+\frac {1287\,a\,b^8\,c^5}{2}+\frac {143\,b^{10}\,c^4}{4}\right )+x^{28}\,\left (\frac {858\,a^7\,c^7}{7}+3003\,a^6\,b^2\,c^6+9009\,a^5\,b^4\,c^5+\frac {15015\,a^4\,b^6\,c^4}{2}+2145\,a^3\,b^8\,c^3+\frac {429\,a^2\,b^{10}\,c^2}{2}+\frac {13\,a\,b^{12}\,c}{2}+\frac {b^{14}}{28}\right )+x^{16}\,\left (\frac {143\,a^{10}\,c^4}{4}+715\,a^9\,b^2\,c^3+\frac {6435\,a^8\,b^4\,c^2}{4}+858\,a^7\,b^6\,c+\frac {429\,a^6\,b^8}{4}\right )+x^{40}\,\left (\frac {143\,a^4\,c^{10}}{4}+715\,a^3\,b^2\,c^9+\frac {6435\,a^2\,b^4\,c^8}{4}+858\,a\,b^6\,c^7+\frac {429\,b^8\,c^6}{4}\right )+\frac {c^{14}\,x^{56}}{28}+x^4\,\left (\frac {c\,a^{13}}{2}+\frac {13\,a^{12}\,b^2}{4}\right )+\frac {13\,a^{10}\,x^8\,\left (a^2\,c^2+12\,a\,b^2\,c+11\,b^4\right )}{4}+\frac {13\,c^{10}\,x^{48}\,\left (a^2\,c^2+12\,a\,b^2\,c+11\,b^4\right )}{4}+\frac {a^{13}\,b\,x^2}{2}+\frac {b\,c^{13}\,x^{54}}{2}+\frac {c^{12}\,x^{52}\,\left (13\,b^2+2\,a\,c\right )}{4}+\frac {143\,a^7\,b\,x^{14}\,\left (14\,a^3\,c^3+70\,a^2\,b^2\,c^2+63\,a\,b^4\,c+12\,b^6\right )}{14}+\frac {143\,b\,c^7\,x^{42}\,\left (14\,a^3\,c^3+70\,a^2\,b^2\,c^2+63\,a\,b^4\,c+12\,b^6\right )}{14}+\frac {143\,a^5\,b\,x^{18}\,\left (5\,a^4\,c^4+30\,a^3\,b^2\,c^3+36\,a^2\,b^4\,c^2+12\,a\,b^6\,c+b^8\right )}{2}+\frac {143\,b\,c^5\,x^{38}\,\left (5\,a^4\,c^4+30\,a^3\,b^2\,c^3+36\,a^2\,b^4\,c^2+12\,a\,b^6\,c+b^8\right )}{2}+\frac {13\,a^3\,b\,x^{22}\,\left (99\,a^5\,c^5+660\,a^4\,b^2\,c^4+924\,a^3\,b^4\,c^3+396\,a^2\,b^6\,c^2+55\,a\,b^8\,c+2\,b^{10}\right )}{2}+\frac {13\,b\,c^3\,x^{34}\,\left (99\,a^5\,c^5+660\,a^4\,b^2\,c^4+924\,a^3\,b^4\,c^3+396\,a^2\,b^6\,c^2+55\,a\,b^8\,c+2\,b^{10}\right )}{2}+\frac {13\,a^9\,b\,x^{10}\,\left (6\,a^2\,c^2+22\,a\,b^2\,c+11\,b^4\right )}{2}+\frac {13\,b\,c^9\,x^{46}\,\left (6\,a^2\,c^2+22\,a\,b^2\,c+11\,b^4\right )}{2}+\frac {13\,a^{11}\,b\,x^6\,\left (2\,b^2+a\,c\right )}{2}+\frac {13\,b\,c^{11}\,x^{50}\,\left (2\,b^2+a\,c\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^13,x)

[Out]

x^24*((13*a^2*b^12)/4 + (429*a^8*c^6)/4 + 143*a^3*b^10*c + (6435*a^4*b^8*c^2)/4 + 6006*a^5*b^6*c^3 + (15015*a^
6*b^4*c^4)/2 + 2574*a^7*b^2*c^5) + x^32*((429*a^6*c^8)/4 + (13*b^12*c^2)/4 + 143*a*b^10*c^3 + (6435*a^2*b^8*c^
4)/4 + 6006*a^3*b^6*c^5 + (15015*a^4*b^4*c^6)/2 + 2574*a^5*b^2*c^7) + x^26*((a*b^13)/2 + 39*a^2*b^11*c + 858*a
^7*b*c^6 + 715*a^3*b^9*c^2 + 4290*a^4*b^7*c^3 + 9009*a^5*b^5*c^4 + 6006*a^6*b^3*c^5) + x^30*((b^13*c)/2 + 39*a
*b^11*c^2 + 858*a^6*b*c^7 + 715*a^2*b^9*c^3 + 4290*a^3*b^7*c^4 + 9009*a^4*b^5*c^5 + 6006*a^5*b^3*c^6) + x^12*(
(429*a^8*b^6)/4 + 13*a^11*c^3 + (715*a^9*b^4*c)/2 + (429*a^10*b^2*c^2)/2) + x^44*(13*a^3*c^11 + (429*b^6*c^8)/
4 + (715*a*b^4*c^9)/2 + (429*a^2*b^2*c^10)/2) + x^20*((143*a^4*b^10)/4 + (143*a^9*c^5)/2 + (1287*a^5*b^8*c)/2
+ 3003*a^6*b^6*c^2 + 4290*a^7*b^4*c^3 + (6435*a^8*b^2*c^4)/4) + x^36*((143*a^5*c^9)/2 + (143*b^10*c^4)/4 + (12
87*a*b^8*c^5)/2 + 3003*a^2*b^6*c^6 + 4290*a^3*b^4*c^7 + (6435*a^4*b^2*c^8)/4) + x^28*(b^14/28 + (858*a^7*c^7)/
7 + (429*a^2*b^10*c^2)/2 + 2145*a^3*b^8*c^3 + (15015*a^4*b^6*c^4)/2 + 9009*a^5*b^4*c^5 + 3003*a^6*b^2*c^6 + (1
3*a*b^12*c)/2) + x^16*((429*a^6*b^8)/4 + (143*a^10*c^4)/4 + 858*a^7*b^6*c + (6435*a^8*b^4*c^2)/4 + 715*a^9*b^2
*c^3) + x^40*((143*a^4*c^10)/4 + (429*b^8*c^6)/4 + 858*a*b^6*c^7 + (6435*a^2*b^4*c^8)/4 + 715*a^3*b^2*c^9) + (
c^14*x^56)/28 + x^4*((a^13*c)/2 + (13*a^12*b^2)/4) + (13*a^10*x^8*(11*b^4 + a^2*c^2 + 12*a*b^2*c))/4 + (13*c^1
0*x^48*(11*b^4 + a^2*c^2 + 12*a*b^2*c))/4 + (a^13*b*x^2)/2 + (b*c^13*x^54)/2 + (c^12*x^52*(2*a*c + 13*b^2))/4
+ (143*a^7*b*x^14*(12*b^6 + 14*a^3*c^3 + 70*a^2*b^2*c^2 + 63*a*b^4*c))/14 + (143*b*c^7*x^42*(12*b^6 + 14*a^3*c
^3 + 70*a^2*b^2*c^2 + 63*a*b^4*c))/14 + (143*a^5*b*x^18*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 + 30*a^3*b^2*c^3 + 1
2*a*b^6*c))/2 + (143*b*c^5*x^38*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 + 30*a^3*b^2*c^3 + 12*a*b^6*c))/2 + (13*a^3*
b*x^22*(2*b^10 + 99*a^5*c^5 + 396*a^2*b^6*c^2 + 924*a^3*b^4*c^3 + 660*a^4*b^2*c^4 + 55*a*b^8*c))/2 + (13*b*c^3
*x^34*(2*b^10 + 99*a^5*c^5 + 396*a^2*b^6*c^2 + 924*a^3*b^4*c^3 + 660*a^4*b^2*c^4 + 55*a*b^8*c))/2 + (13*a^9*b*
x^10*(11*b^4 + 6*a^2*c^2 + 22*a*b^2*c))/2 + (13*b*c^9*x^46*(11*b^4 + 6*a^2*c^2 + 22*a*b^2*c))/2 + (13*a^11*b*x
^6*(a*c + 2*b^2))/2 + (13*b*c^11*x^50*(a*c + 2*b^2))/2

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