3.61 \(\int (a+b x^2+c x^4)^2 (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6) \, dx\)

Optimal. Leaf size=259 \[ \frac{1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac{1}{3} a^2 x^3 (a f+3 b d)+\frac{3}{4} a^2 b e x^4+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{9} x^9 \left (6 a b c f+3 a c^2 d+3 b^2 c d+b^3 f\right )+\frac{3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac{3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac{3}{10} c e x^{10} \left (a c+b^2\right )+\frac{1}{8} b e x^8 \left (6 a c+b^2\right )+\frac{1}{2} a e x^6 \left (a c+b^2\right )+\frac{1}{13} c^2 x^{13} (3 b f+c d)+\frac{1}{4} b c^2 e x^{12}+\frac{1}{14} c^3 e x^{14}+\frac{1}{15} c^3 f x^{15} \]

[Out]

a^3*d*x + (a^3*e*x^2)/2 + (a^2*(3*b*d + a*f)*x^3)/3 + (3*a^2*b*e*x^4)/4 + (3*a*(b^2*d + a*c*d + a*b*f)*x^5)/5
+ (a*(b^2 + a*c)*e*x^6)/2 + ((b^3*d + 6*a*b*c*d + 3*a*b^2*f + 3*a^2*c*f)*x^7)/7 + (b*(b^2 + 6*a*c)*e*x^8)/8 +
((3*b^2*c*d + 3*a*c^2*d + b^3*f + 6*a*b*c*f)*x^9)/9 + (3*c*(b^2 + a*c)*e*x^10)/10 + (3*c*(b*c*d + b^2*f + a*c*
f)*x^11)/11 + (b*c^2*e*x^12)/4 + (c^2*(c*d + 3*b*f)*x^13)/13 + (c^3*e*x^14)/14 + (c^3*f*x^15)/15

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Rubi [A]  time = 0.332331, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 63, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.016, Rules used = {1671} \[ \frac{1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac{1}{3} a^2 x^3 (a f+3 b d)+\frac{3}{4} a^2 b e x^4+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{9} x^9 \left (6 a b c f+3 a c^2 d+3 b^2 c d+b^3 f\right )+\frac{3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac{3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac{3}{10} c e x^{10} \left (a c+b^2\right )+\frac{1}{8} b e x^8 \left (6 a c+b^2\right )+\frac{1}{2} a e x^6 \left (a c+b^2\right )+\frac{1}{13} c^2 x^{13} (3 b f+c d)+\frac{1}{4} b c^2 e x^{12}+\frac{1}{14} c^3 e x^{14}+\frac{1}{15} c^3 f x^{15} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]

[Out]

a^3*d*x + (a^3*e*x^2)/2 + (a^2*(3*b*d + a*f)*x^3)/3 + (3*a^2*b*e*x^4)/4 + (3*a*(b^2*d + a*c*d + a*b*f)*x^5)/5
+ (a*(b^2 + a*c)*e*x^6)/2 + ((b^3*d + 6*a*b*c*d + 3*a*b^2*f + 3*a^2*c*f)*x^7)/7 + (b*(b^2 + 6*a*c)*e*x^8)/8 +
((3*b^2*c*d + 3*a*c^2*d + b^3*f + 6*a*b*c*f)*x^9)/9 + (3*c*(b^2 + a*c)*e*x^10)/10 + (3*c*(b*c*d + b^2*f + a*c*
f)*x^11)/11 + (b*c^2*e*x^12)/4 + (c^2*(c*d + 3*b*f)*x^13)/13 + (c^3*e*x^14)/14 + (c^3*f*x^15)/15

Rule 1671

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2 + c*x^4)^
p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx &=\int \left (a^3 d+a^3 e x+a^2 (3 b d+a f) x^2+3 a^2 b e x^3+3 a \left (b^2 d+a c d+a b f\right ) x^4+3 a \left (b^2+a c\right ) e x^5+\left (b^3 d+6 a b c d+3 a b^2 f+3 a^2 c f\right ) x^6+b \left (b^2+6 a c\right ) e x^7+\left (3 b^2 c d+3 a c^2 d+b^3 f+6 a b c f\right ) x^8+3 c \left (b^2+a c\right ) e x^9+3 c \left (b c d+b^2 f+a c f\right ) x^{10}+3 b c^2 e x^{11}+c^2 (c d+3 b f) x^{12}+c^3 e x^{13}+c^3 f x^{14}\right ) \, dx\\ &=a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{3} a^2 (3 b d+a f) x^3+\frac{3}{4} a^2 b e x^4+\frac{3}{5} a \left (b^2 d+a c d+a b f\right ) x^5+\frac{1}{2} a \left (b^2+a c\right ) e x^6+\frac{1}{7} \left (b^3 d+6 a b c d+3 a b^2 f+3 a^2 c f\right ) x^7+\frac{1}{8} b \left (b^2+6 a c\right ) e x^8+\frac{1}{9} \left (3 b^2 c d+3 a c^2 d+b^3 f+6 a b c f\right ) x^9+\frac{3}{10} c \left (b^2+a c\right ) e x^{10}+\frac{3}{11} c \left (b c d+b^2 f+a c f\right ) x^{11}+\frac{1}{4} b c^2 e x^{12}+\frac{1}{13} c^2 (c d+3 b f) x^{13}+\frac{1}{14} c^3 e x^{14}+\frac{1}{15} c^3 f x^{15}\\ \end{align*}

Mathematica [A]  time = 0.054027, size = 259, normalized size = 1. \[ \frac{1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac{1}{3} a^2 x^3 (a f+3 b d)+\frac{3}{4} a^2 b e x^4+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{9} x^9 \left (6 a b c f+3 a c^2 d+3 b^2 c d+b^3 f\right )+\frac{3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac{3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac{3}{10} c e x^{10} \left (a c+b^2\right )+\frac{1}{8} b e x^8 \left (6 a c+b^2\right )+\frac{1}{2} a e x^6 \left (a c+b^2\right )+\frac{1}{13} c^2 x^{13} (3 b f+c d)+\frac{1}{4} b c^2 e x^{12}+\frac{1}{14} c^3 e x^{14}+\frac{1}{15} c^3 f x^{15} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6
),x]

[Out]

a^3*d*x + (a^3*e*x^2)/2 + (a^2*(3*b*d + a*f)*x^3)/3 + (3*a^2*b*e*x^4)/4 + (3*a*(b^2*d + a*c*d + a*b*f)*x^5)/5
+ (a*(b^2 + a*c)*e*x^6)/2 + ((b^3*d + 6*a*b*c*d + 3*a*b^2*f + 3*a^2*c*f)*x^7)/7 + (b*(b^2 + 6*a*c)*e*x^8)/8 +
((3*b^2*c*d + 3*a*c^2*d + b^3*f + 6*a*b*c*f)*x^9)/9 + (3*c*(b^2 + a*c)*e*x^10)/10 + (3*c*(b*c*d + b^2*f + a*c*
f)*x^11)/11 + (b*c^2*e*x^12)/4 + (c^2*(c*d + 3*b*f)*x^13)/13 + (c^3*e*x^14)/14 + (c^3*f*x^15)/15

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Maple [A]  time = 0.002, size = 354, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}f{x}^{15}}{15}}+{\frac{{c}^{3}e{x}^{14}}{14}}+{\frac{ \left ( 2\,b{c}^{2}f+{c}^{2} \left ( bf+cd \right ) \right ){x}^{13}}{13}}+{\frac{b{c}^{2}e{x}^{12}}{4}}+{\frac{ \left ( \left ( 2\,ac+{b}^{2} \right ) cf+2\,bc \left ( bf+cd \right ) +{c}^{2} \left ( af+bd \right ) \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 2\,ac+{b}^{2} \right ) ce+2\,{b}^{2}ce+a{c}^{2}e \right ){x}^{10}}{10}}+{\frac{ \left ( 2\,abcf+ \left ( 2\,ac+{b}^{2} \right ) \left ( bf+cd \right ) +2\,bc \left ( af+bd \right ) +a{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,abce+ \left ( 2\,ac+{b}^{2} \right ) be \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}cf+2\,ab \left ( bf+cd \right ) + \left ( 2\,ac+{b}^{2} \right ) \left ( af+bd \right ) +2\,abcd \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{2}ce+2\,a{b}^{2}e+ \left ( 2\,ac+{b}^{2} \right ) ae \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2} \left ( bf+cd \right ) +2\,ab \left ( af+bd \right ) + \left ( 2\,ac+{b}^{2} \right ) ad \right ){x}^{5}}{5}}+{\frac{3\,{a}^{2}be{x}^{4}}{4}}+{\frac{ \left ({a}^{2} \left ( af+bd \right ) +2\,{a}^{2}bd \right ){x}^{3}}{3}}+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{3}dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6),x)

[Out]

1/15*c^3*f*x^15+1/14*c^3*e*x^14+1/13*(2*b*c^2*f+c^2*(b*f+c*d))*x^13+1/4*b*c^2*e*x^12+1/11*((2*a*c+b^2)*c*f+2*b
*c*(b*f+c*d)+c^2*(a*f+b*d))*x^11+1/10*((2*a*c+b^2)*c*e+2*b^2*c*e+a*c^2*e)*x^10+1/9*(2*a*b*c*f+(2*a*c+b^2)*(b*f
+c*d)+2*b*c*(a*f+b*d)+a*c^2*d)*x^9+1/8*(4*a*b*c*e+(2*a*c+b^2)*b*e)*x^8+1/7*(a^2*c*f+2*a*b*(b*f+c*d)+(2*a*c+b^2
)*(a*f+b*d)+2*a*b*c*d)*x^7+1/6*(a^2*c*e+2*a*b^2*e+(2*a*c+b^2)*a*e)*x^6+1/5*(a^2*(b*f+c*d)+2*a*b*(a*f+b*d)+(2*a
*c+b^2)*a*d)*x^5+3/4*a^2*b*e*x^4+1/3*(a^2*(a*f+b*d)+2*a^2*b*d)*x^3+1/2*a^3*e*x^2+a^3*d*x

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Maxima [A]  time = 0.965589, size = 339, normalized size = 1.31 \begin{align*} \frac{1}{15} \, c^{3} f x^{15} + \frac{1}{14} \, c^{3} e x^{14} + \frac{1}{4} \, b c^{2} e x^{12} + \frac{1}{13} \,{\left (c^{3} d + 3 \, b c^{2} f\right )} x^{13} + \frac{3}{10} \,{\left (b^{2} c + a c^{2}\right )} e x^{10} + \frac{3}{11} \,{\left (b c^{2} d +{\left (b^{2} c + a c^{2}\right )} f\right )} x^{11} + \frac{1}{8} \,{\left (b^{3} + 6 \, a b c\right )} e x^{8} + \frac{1}{9} \,{\left (3 \,{\left (b^{2} c + a c^{2}\right )} d +{\left (b^{3} + 6 \, a b c\right )} f\right )} x^{9} + \frac{3}{4} \, a^{2} b e x^{4} + \frac{1}{2} \,{\left (a b^{2} + a^{2} c\right )} e x^{6} + \frac{1}{7} \,{\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \,{\left (a b^{2} + a^{2} c\right )} f\right )} x^{7} + \frac{1}{2} \, a^{3} e x^{2} + \frac{3}{5} \,{\left (a^{2} b f +{\left (a b^{2} + a^{2} c\right )} d\right )} x^{5} + a^{3} d x + \frac{1}{3} \,{\left (3 \, a^{2} b d + a^{3} f\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6),x, algorithm="maxi
ma")

[Out]

1/15*c^3*f*x^15 + 1/14*c^3*e*x^14 + 1/4*b*c^2*e*x^12 + 1/13*(c^3*d + 3*b*c^2*f)*x^13 + 3/10*(b^2*c + a*c^2)*e*
x^10 + 3/11*(b*c^2*d + (b^2*c + a*c^2)*f)*x^11 + 1/8*(b^3 + 6*a*b*c)*e*x^8 + 1/9*(3*(b^2*c + a*c^2)*d + (b^3 +
6*a*b*c)*f)*x^9 + 3/4*a^2*b*e*x^4 + 1/2*(a*b^2 + a^2*c)*e*x^6 + 1/7*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*f)
*x^7 + 1/2*a^3*e*x^2 + 3/5*(a^2*b*f + (a*b^2 + a^2*c)*d)*x^5 + a^3*d*x + 1/3*(3*a^2*b*d + a^3*f)*x^3

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Fricas [A]  time = 1.57373, size = 716, normalized size = 2.76 \begin{align*} \frac{1}{15} x^{15} f c^{3} + \frac{1}{14} x^{14} e c^{3} + \frac{1}{13} x^{13} d c^{3} + \frac{3}{13} x^{13} f c^{2} b + \frac{1}{4} x^{12} e c^{2} b + \frac{3}{11} x^{11} d c^{2} b + \frac{3}{11} x^{11} f c b^{2} + \frac{3}{11} x^{11} f c^{2} a + \frac{3}{10} x^{10} e c b^{2} + \frac{3}{10} x^{10} e c^{2} a + \frac{1}{3} x^{9} d c b^{2} + \frac{1}{9} x^{9} f b^{3} + \frac{1}{3} x^{9} d c^{2} a + \frac{2}{3} x^{9} f c b a + \frac{1}{8} x^{8} e b^{3} + \frac{3}{4} x^{8} e c b a + \frac{1}{7} x^{7} d b^{3} + \frac{6}{7} x^{7} d c b a + \frac{3}{7} x^{7} f b^{2} a + \frac{3}{7} x^{7} f c a^{2} + \frac{1}{2} x^{6} e b^{2} a + \frac{1}{2} x^{6} e c a^{2} + \frac{3}{5} x^{5} d b^{2} a + \frac{3}{5} x^{5} d c a^{2} + \frac{3}{5} x^{5} f b a^{2} + \frac{3}{4} x^{4} e b a^{2} + x^{3} d b a^{2} + \frac{1}{3} x^{3} f a^{3} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6),x, algorithm="fric
as")

[Out]

1/15*x^15*f*c^3 + 1/14*x^14*e*c^3 + 1/13*x^13*d*c^3 + 3/13*x^13*f*c^2*b + 1/4*x^12*e*c^2*b + 3/11*x^11*d*c^2*b
+ 3/11*x^11*f*c*b^2 + 3/11*x^11*f*c^2*a + 3/10*x^10*e*c*b^2 + 3/10*x^10*e*c^2*a + 1/3*x^9*d*c*b^2 + 1/9*x^9*f
*b^3 + 1/3*x^9*d*c^2*a + 2/3*x^9*f*c*b*a + 1/8*x^8*e*b^3 + 3/4*x^8*e*c*b*a + 1/7*x^7*d*b^3 + 6/7*x^7*d*c*b*a +
3/7*x^7*f*b^2*a + 3/7*x^7*f*c*a^2 + 1/2*x^6*e*b^2*a + 1/2*x^6*e*c*a^2 + 3/5*x^5*d*b^2*a + 3/5*x^5*d*c*a^2 + 3
/5*x^5*f*b*a^2 + 3/4*x^4*e*b*a^2 + x^3*d*b*a^2 + 1/3*x^3*f*a^3 + 1/2*x^2*e*a^3 + x*d*a^3

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Sympy [A]  time = 0.112955, size = 309, normalized size = 1.19 \begin{align*} a^{3} d x + \frac{a^{3} e x^{2}}{2} + \frac{3 a^{2} b e x^{4}}{4} + \frac{b c^{2} e x^{12}}{4} + \frac{c^{3} e x^{14}}{14} + \frac{c^{3} f x^{15}}{15} + x^{13} \left (\frac{3 b c^{2} f}{13} + \frac{c^{3} d}{13}\right ) + x^{11} \left (\frac{3 a c^{2} f}{11} + \frac{3 b^{2} c f}{11} + \frac{3 b c^{2} d}{11}\right ) + x^{10} \left (\frac{3 a c^{2} e}{10} + \frac{3 b^{2} c e}{10}\right ) + x^{9} \left (\frac{2 a b c f}{3} + \frac{a c^{2} d}{3} + \frac{b^{3} f}{9} + \frac{b^{2} c d}{3}\right ) + x^{8} \left (\frac{3 a b c e}{4} + \frac{b^{3} e}{8}\right ) + x^{7} \left (\frac{3 a^{2} c f}{7} + \frac{3 a b^{2} f}{7} + \frac{6 a b c d}{7} + \frac{b^{3} d}{7}\right ) + x^{6} \left (\frac{a^{2} c e}{2} + \frac{a b^{2} e}{2}\right ) + x^{5} \left (\frac{3 a^{2} b f}{5} + \frac{3 a^{2} c d}{5} + \frac{3 a b^{2} d}{5}\right ) + x^{3} \left (\frac{a^{3} f}{3} + a^{2} b d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2*(a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x**4+c*e*x**5+c*f*x**6),x)

[Out]

a**3*d*x + a**3*e*x**2/2 + 3*a**2*b*e*x**4/4 + b*c**2*e*x**12/4 + c**3*e*x**14/14 + c**3*f*x**15/15 + x**13*(3
*b*c**2*f/13 + c**3*d/13) + x**11*(3*a*c**2*f/11 + 3*b**2*c*f/11 + 3*b*c**2*d/11) + x**10*(3*a*c**2*e/10 + 3*b
**2*c*e/10) + x**9*(2*a*b*c*f/3 + a*c**2*d/3 + b**3*f/9 + b**2*c*d/3) + x**8*(3*a*b*c*e/4 + b**3*e/8) + x**7*(
3*a**2*c*f/7 + 3*a*b**2*f/7 + 6*a*b*c*d/7 + b**3*d/7) + x**6*(a**2*c*e/2 + a*b**2*e/2) + x**5*(3*a**2*b*f/5 +
3*a**2*c*d/5 + 3*a*b**2*d/5) + x**3*(a**3*f/3 + a**2*b*d)

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Giac [A]  time = 1.11794, size = 398, normalized size = 1.54 \begin{align*} \frac{1}{15} \, c^{3} f x^{15} + \frac{1}{14} \, c^{3} x^{14} e + \frac{1}{13} \, c^{3} d x^{13} + \frac{3}{13} \, b c^{2} f x^{13} + \frac{1}{4} \, b c^{2} x^{12} e + \frac{3}{11} \, b c^{2} d x^{11} + \frac{3}{11} \, b^{2} c f x^{11} + \frac{3}{11} \, a c^{2} f x^{11} + \frac{3}{10} \, b^{2} c x^{10} e + \frac{3}{10} \, a c^{2} x^{10} e + \frac{1}{3} \, b^{2} c d x^{9} + \frac{1}{3} \, a c^{2} d x^{9} + \frac{1}{9} \, b^{3} f x^{9} + \frac{2}{3} \, a b c f x^{9} + \frac{1}{8} \, b^{3} x^{8} e + \frac{3}{4} \, a b c x^{8} e + \frac{1}{7} \, b^{3} d x^{7} + \frac{6}{7} \, a b c d x^{7} + \frac{3}{7} \, a b^{2} f x^{7} + \frac{3}{7} \, a^{2} c f x^{7} + \frac{1}{2} \, a b^{2} x^{6} e + \frac{1}{2} \, a^{2} c x^{6} e + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{5} \, a^{2} c d x^{5} + \frac{3}{5} \, a^{2} b f x^{5} + \frac{3}{4} \, a^{2} b x^{4} e + a^{2} b d x^{3} + \frac{1}{3} \, a^{3} f x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6),x, algorithm="giac
")

[Out]

1/15*c^3*f*x^15 + 1/14*c^3*x^14*e + 1/13*c^3*d*x^13 + 3/13*b*c^2*f*x^13 + 1/4*b*c^2*x^12*e + 3/11*b*c^2*d*x^11
+ 3/11*b^2*c*f*x^11 + 3/11*a*c^2*f*x^11 + 3/10*b^2*c*x^10*e + 3/10*a*c^2*x^10*e + 1/3*b^2*c*d*x^9 + 1/3*a*c^2
*d*x^9 + 1/9*b^3*f*x^9 + 2/3*a*b*c*f*x^9 + 1/8*b^3*x^8*e + 3/4*a*b*c*x^8*e + 1/7*b^3*d*x^7 + 6/7*a*b*c*d*x^7 +
3/7*a*b^2*f*x^7 + 3/7*a^2*c*f*x^7 + 1/2*a*b^2*x^6*e + 1/2*a^2*c*x^6*e + 3/5*a*b^2*d*x^5 + 3/5*a^2*c*d*x^5 + 3
/5*a^2*b*f*x^5 + 3/4*a^2*b*x^4*e + a^2*b*d*x^3 + 1/3*a^3*f*x^3 + 1/2*a^3*x^2*e + a^3*d*x