\(\int (a+b x^2+c x^4)^3 (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6) \, dx\) [60]

Optimal result

Integrand size = 63, antiderivative size = 416 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {1}{3} a^3 (4 b d+a f) x^3+a^3 b e x^4+\frac {2}{5} a^2 \left (3 b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{3} a^2 \left (3 b^2+2 a c\right ) e x^6+\frac {2}{7} a \left (2 b^3 d+6 a b c d+3 a b^2 f+2 a^2 c f\right ) x^7+\frac {1}{2} a b \left (b^2+3 a c\right ) e x^8+\frac {1}{9} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 f+12 a^2 b c f\right ) x^9+\frac {1}{10} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) e x^{10}+\frac {1}{11} \left (4 b^3 c d+12 a b c^2 d+b^4 f+12 a b^2 c f+6 a^2 c^2 f\right ) x^{11}+\frac {1}{3} b c \left (b^2+3 a c\right ) e x^{12}+\frac {2}{13} c \left (3 b^2 c d+2 a c^2 d+2 b^3 f+6 a b c f\right ) x^{13}+\frac {1}{7} c^2 \left (3 b^2+2 a c\right ) e x^{14}+\frac {2}{15} c^2 \left (2 b c d+3 b^2 f+2 a c f\right ) x^{15}+\frac {1}{4} b c^3 e x^{16}+\frac {1}{17} c^3 (c d+4 b f) x^{17}+\frac {1}{18} c^4 e x^{18}+\frac {1}{19} c^4 f x^{19} \]

[Out]

a^4*d*x+1/2*a^4*e*x^2+1/3*a^3*(a*f+4*b*d)*x^3+a^3*b*e*x^4+2/5*a^2*(2*a*b*f+2*a*c*d+3*b^2*d)*x^5+1/3*a^2*(2*a*c
+3*b^2)*e*x^6+2/7*a*(2*a^2*c*f+3*a*b^2*f+6*a*b*c*d+2*b^3*d)*x^7+1/2*a*b*(3*a*c+b^2)*e*x^8+1/9*(12*a^2*b*c*f+6*
a^2*c^2*d+4*a*b^3*f+12*a*b^2*c*d+b^4*d)*x^9+1/10*(6*a^2*c^2+12*a*b^2*c+b^4)*e*x^10+1/11*(6*a^2*c^2*f+12*a*b^2*
c*f+12*a*b*c^2*d+b^4*f+4*b^3*c*d)*x^11+1/3*b*c*(3*a*c+b^2)*e*x^12+2/13*c*(6*a*b*c*f+2*a*c^2*d+2*b^3*f+3*b^2*c*
d)*x^13+1/7*c^2*(2*a*c+3*b^2)*e*x^14+2/15*c^2*(2*a*c*f+3*b^2*f+2*b*c*d)*x^15+1/4*b*c^3*e*x^16+1/17*c^3*(4*b*f+
c*d)*x^17+1/18*c^4*e*x^18+1/19*c^4*f*x^19

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {1685} \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac {2}{5} a^2 x^5 \left (2 a b f+2 a c d+3 b^2 d\right )+\frac {1}{3} a^2 e x^6 \left (2 a c+3 b^2\right )+\frac {1}{10} e x^{10} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac {2}{7} a x^7 \left (2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right )+\frac {1}{11} x^{11} \left (6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right )+\frac {1}{9} x^9 \left (12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right )+\frac {2}{15} c^2 x^{15} \left (2 a c f+3 b^2 f+2 b c d\right )+\frac {1}{7} c^2 e x^{14} \left (2 a c+3 b^2\right )+\frac {1}{3} b c e x^{12} \left (3 a c+b^2\right )+\frac {1}{2} a b e x^8 \left (3 a c+b^2\right )+\frac {2}{13} c x^{13} \left (6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right )+\frac {1}{17} c^3 x^{17} (4 b f+c d)+\frac {1}{4} b c^3 e x^{16}+\frac {1}{18} c^4 e x^{18}+\frac {1}{19} c^4 f x^{19} \]

[In]

Int[(a + b*x^2 + c*x^4)^3*(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^4*d*x + (a^4*e*x^2)/2 + (a^3*(4*b*d + a*f)*x^3)/3 + a^3*b*e*x^4 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/
5 + (a^2*(3*b^2 + 2*a*c)*e*x^6)/3 + (2*a*(2*b^3*d + 6*a*b*c*d + 3*a*b^2*f + 2*a^2*c*f)*x^7)/7 + (a*b*(b^2 + 3*
a*c)*e*x^8)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*f + 12*a^2*b*c*f)*x^9)/9 + ((b^4 + 12*a*b^2*c +
6*a^2*c^2)*e*x^10)/10 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*f + 12*a*b^2*c*f + 6*a^2*c^2*f)*x^11)/11 + (b*c*(b^2
+ 3*a*c)*e*x^12)/3 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*f + 6*a*b*c*f)*x^13)/13 + (c^2*(3*b^2 + 2*a*c)*e*x^1
4)/7 + (2*c^2*(2*b*c*d + 3*b^2*f + 2*a*c*f)*x^15)/15 + (b*c^3*e*x^16)/4 + (c^3*(c*d + 4*b*f)*x^17)/17 + (c^4*e
*x^18)/18 + (c^4*f*x^19)/19

Rule 1685

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*} \text {integral}& = \int \left (a^4 d+a^4 e x+a^3 (4 b d+a f) x^2+4 a^3 b e x^3+2 a^2 \left (3 b^2 d+2 a c d+2 a b f\right ) x^4+2 a^2 \left (3 b^2+2 a c\right ) e x^5+2 a \left (2 b^3 d+6 a b c d+3 a b^2 f+2 a^2 c f\right ) x^6+4 a b \left (b^2+3 a c\right ) e x^7+\left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 f+12 a^2 b c f\right ) x^8+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) e x^9+\left (4 b^3 c d+12 a b c^2 d+b^4 f+12 a b^2 c f+6 a^2 c^2 f\right ) x^{10}+4 b c \left (b^2+3 a c\right ) e x^{11}+2 c \left (3 b^2 c d+2 a c^2 d+2 b^3 f+6 a b c f\right ) x^{12}+2 c^2 \left (3 b^2+2 a c\right ) e x^{13}+2 c^2 \left (2 b c d+3 b^2 f+2 a c f\right ) x^{14}+4 b c^3 e x^{15}+c^3 (c d+4 b f) x^{16}+c^4 e x^{17}+c^4 f x^{18}\right ) \, dx \\ & = a^4 d x+\frac {1}{2} a^4 e x^2+\frac {1}{3} a^3 (4 b d+a f) x^3+a^3 b e x^4+\frac {2}{5} a^2 \left (3 b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{3} a^2 \left (3 b^2+2 a c\right ) e x^6+\frac {2}{7} a \left (2 b^3 d+6 a b c d+3 a b^2 f+2 a^2 c f\right ) x^7+\frac {1}{2} a b \left (b^2+3 a c\right ) e x^8+\frac {1}{9} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 f+12 a^2 b c f\right ) x^9+\frac {1}{10} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) e x^{10}+\frac {1}{11} \left (4 b^3 c d+12 a b c^2 d+b^4 f+12 a b^2 c f+6 a^2 c^2 f\right ) x^{11}+\frac {1}{3} b c \left (b^2+3 a c\right ) e x^{12}+\frac {2}{13} c \left (3 b^2 c d+2 a c^2 d+2 b^3 f+6 a b c f\right ) x^{13}+\frac {1}{7} c^2 \left (3 b^2+2 a c\right ) e x^{14}+\frac {2}{15} c^2 \left (2 b c d+3 b^2 f+2 a c f\right ) x^{15}+\frac {1}{4} b c^3 e x^{16}+\frac {1}{17} c^3 (c d+4 b f) x^{17}+\frac {1}{18} c^4 e x^{18}+\frac {1}{19} c^4 f x^{19} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {1}{3} a^3 (4 b d+a f) x^3+a^3 b e x^4+\frac {2}{5} a^2 \left (3 b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{3} a^2 \left (3 b^2+2 a c\right ) e x^6+\frac {2}{7} a \left (2 b^3 d+6 a b c d+3 a b^2 f+2 a^2 c f\right ) x^7+\frac {1}{2} a b \left (b^2+3 a c\right ) e x^8+\frac {1}{9} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 f+12 a^2 b c f\right ) x^9+\frac {1}{10} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) e x^{10}+\frac {1}{11} \left (4 b^3 c d+12 a b c^2 d+b^4 f+12 a b^2 c f+6 a^2 c^2 f\right ) x^{11}+\frac {1}{3} b c \left (b^2+3 a c\right ) e x^{12}+\frac {2}{13} c \left (3 b^2 c d+2 a c^2 d+2 b^3 f+6 a b c f\right ) x^{13}+\frac {1}{7} c^2 \left (3 b^2+2 a c\right ) e x^{14}+\frac {2}{15} c^2 \left (2 b c d+3 b^2 f+2 a c f\right ) x^{15}+\frac {1}{4} b c^3 e x^{16}+\frac {1}{17} c^3 (c d+4 b f) x^{17}+\frac {1}{18} c^4 e x^{18}+\frac {1}{19} c^4 f x^{19} \]

[In]

Integrate[(a + b*x^2 + c*x^4)^3*(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^4*d*x + (a^4*e*x^2)/2 + (a^3*(4*b*d + a*f)*x^3)/3 + a^3*b*e*x^4 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/
5 + (a^2*(3*b^2 + 2*a*c)*e*x^6)/3 + (2*a*(2*b^3*d + 6*a*b*c*d + 3*a*b^2*f + 2*a^2*c*f)*x^7)/7 + (a*b*(b^2 + 3*
a*c)*e*x^8)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*f + 12*a^2*b*c*f)*x^9)/9 + ((b^4 + 12*a*b^2*c +
6*a^2*c^2)*e*x^10)/10 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*f + 12*a*b^2*c*f + 6*a^2*c^2*f)*x^11)/11 + (b*c*(b^2
+ 3*a*c)*e*x^12)/3 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*f + 6*a*b*c*f)*x^13)/13 + (c^2*(3*b^2 + 2*a*c)*e*x^1
4)/7 + (2*c^2*(2*b*c*d + 3*b^2*f + 2*a*c*f)*x^15)/15 + (b*c^3*e*x^16)/4 + (c^3*(c*d + 4*b*f)*x^17)/17 + (c^4*e
*x^18)/18 + (c^4*f*x^19)/19

Maple [A] (verified)

Time = 28.81 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.99

method	result	size
norman	\(\left (\frac {1}{3} f \,a^{4}+\frac {4}{3} d \,a^{3} b \right ) x^{3}+\left (\frac {2}{7} a \,c^{3} e +\frac {3}{7} b^{2} c^{2} e \right ) x^{14}+\left (\frac {2}{3} a^{3} c e +a^{2} b^{2} e \right ) x^{6}+\left (\frac {4}{17} b \,c^{3} f +\frac {1}{17} c^{4} d \right ) x^{17}+\left (a b \,c^{2} e +\frac {1}{3} b^{3} c e \right ) x^{12}+\left (\frac {3}{2} a^{2} b c e +\frac {1}{2} a \,b^{3} e \right ) x^{8}+\left (\frac {4}{15} a \,c^{3} f +\frac {2}{5} b^{2} c^{2} f +\frac {4}{15} b \,c^{3} d \right ) x^{15}+\left (\frac {3}{5} a^{2} c^{2} e +\frac {6}{5} a \,b^{2} c e +\frac {1}{10} b^{4} e \right ) x^{10}+\left (\frac {4}{5} f \,a^{3} b +\frac {4}{5} a^{3} c d +\frac {6}{5} a^{2} b^{2} d \right ) x^{5}+\left (\frac {4}{7} a^{3} c f +\frac {6}{7} a^{2} b^{2} f +\frac {12}{7} a^{2} b c d +\frac {4}{7} a \,b^{3} d \right ) x^{7}+\left (\frac {12}{13} a b \,c^{2} f +\frac {4}{13} a \,c^{3} d +\frac {4}{13} b^{3} c f +\frac {6}{13} b^{2} c^{2} d \right ) x^{13}+\left (\frac {6}{11} a^{2} c^{2} f +\frac {12}{11} a \,b^{2} c f +\frac {12}{11} a b \,c^{2} d +\frac {1}{11} b^{4} f +\frac {4}{11} b^{3} c d \right ) x^{11}+\left (\frac {4}{3} a^{2} b c f +\frac {2}{3} a^{2} c^{2} d +\frac {4}{9} a \,b^{3} f +\frac {4}{3} a \,b^{2} c d +\frac {1}{9} d \,b^{4}\right ) x^{9}+a^{4} d x +a^{3} b e \,x^{4}+\frac {a^{4} e \,x^{2}}{2}+\frac {c^{4} e \,x^{18}}{18}+\frac {c^{4} f \,x^{19}}{19}+\frac {b \,c^{3} e \,x^{16}}{4}\)	\(412\)
risch	\(\frac {1}{2} a^{4} e \,x^{2}+\frac {1}{18} c^{4} e \,x^{18}+\frac {1}{19} c^{4} f \,x^{19}+\frac {2}{3} x^{6} a^{3} c e +x^{6} a^{2} b^{2} e +\frac {4}{5} x^{5} f \,a^{3} b +\frac {4}{5} x^{5} a^{3} c d +\frac {6}{5} x^{5} a^{2} b^{2} d +\frac {4}{3} x^{3} d \,a^{3} b +a^{4} d x +a^{3} b e \,x^{4}+\frac {1}{4} b \,c^{3} e \,x^{16}+\frac {6}{5} x^{10} a \,b^{2} c e +\frac {3}{2} x^{8} a^{2} b c e +\frac {12}{7} x^{7} a^{2} b c d +\frac {1}{3} x^{3} f \,a^{4}+\frac {4}{17} x^{17} b \,c^{3} f +\frac {4}{15} x^{15} a \,c^{3} f +\frac {2}{5} x^{15} b^{2} c^{2} f +\frac {4}{15} x^{15} b \,c^{3} d +\frac {2}{7} x^{14} a \,c^{3} e +\frac {3}{7} x^{14} b^{2} c^{2} e +\frac {1}{9} x^{9} d \,b^{4}+\frac {1}{11} x^{11} b^{4} f +\frac {1}{10} x^{10} b^{4} e +\frac {1}{17} x^{17} c^{4} d +\frac {12}{13} x^{13} a b \,c^{2} f +x^{12} a b \,c^{2} e +\frac {12}{11} x^{11} a \,b^{2} c f +\frac {12}{11} x^{11} a b \,c^{2} d +\frac {4}{3} x^{9} a^{2} b c f +\frac {4}{3} x^{9} a \,b^{2} c d +\frac {4}{13} x^{13} a \,c^{3} d +\frac {4}{13} x^{13} b^{3} c f +\frac {6}{13} x^{13} b^{2} c^{2} d +\frac {1}{3} x^{12} b^{3} c e +\frac {6}{11} x^{11} a^{2} c^{2} f +\frac {4}{11} x^{11} b^{3} c d +\frac {3}{5} x^{10} a^{2} c^{2} e +\frac {2}{3} x^{9} a^{2} c^{2} d +\frac {4}{9} x^{9} a \,b^{3} f +\frac {1}{2} x^{8} a \,b^{3} e +\frac {4}{7} x^{7} a^{3} c f +\frac {6}{7} x^{7} a^{2} b^{2} f +\frac {4}{7} x^{7} a \,b^{3} d\)	\(464\)
parallelrisch	\(\frac {1}{2} a^{4} e \,x^{2}+\frac {1}{18} c^{4} e \,x^{18}+\frac {1}{19} c^{4} f \,x^{19}+\frac {2}{3} x^{6} a^{3} c e +x^{6} a^{2} b^{2} e +\frac {4}{5} x^{5} f \,a^{3} b +\frac {4}{5} x^{5} a^{3} c d +\frac {6}{5} x^{5} a^{2} b^{2} d +\frac {4}{3} x^{3} d \,a^{3} b +a^{4} d x +a^{3} b e \,x^{4}+\frac {1}{4} b \,c^{3} e \,x^{16}+\frac {6}{5} x^{10} a \,b^{2} c e +\frac {3}{2} x^{8} a^{2} b c e +\frac {12}{7} x^{7} a^{2} b c d +\frac {1}{3} x^{3} f \,a^{4}+\frac {4}{17} x^{17} b \,c^{3} f +\frac {4}{15} x^{15} a \,c^{3} f +\frac {2}{5} x^{15} b^{2} c^{2} f +\frac {4}{15} x^{15} b \,c^{3} d +\frac {2}{7} x^{14} a \,c^{3} e +\frac {3}{7} x^{14} b^{2} c^{2} e +\frac {1}{9} x^{9} d \,b^{4}+\frac {1}{11} x^{11} b^{4} f +\frac {1}{10} x^{10} b^{4} e +\frac {1}{17} x^{17} c^{4} d +\frac {12}{13} x^{13} a b \,c^{2} f +x^{12} a b \,c^{2} e +\frac {12}{11} x^{11} a \,b^{2} c f +\frac {12}{11} x^{11} a b \,c^{2} d +\frac {4}{3} x^{9} a^{2} b c f +\frac {4}{3} x^{9} a \,b^{2} c d +\frac {4}{13} x^{13} a \,c^{3} d +\frac {4}{13} x^{13} b^{3} c f +\frac {6}{13} x^{13} b^{2} c^{2} d +\frac {1}{3} x^{12} b^{3} c e +\frac {6}{11} x^{11} a^{2} c^{2} f +\frac {4}{11} x^{11} b^{3} c d +\frac {3}{5} x^{10} a^{2} c^{2} e +\frac {2}{3} x^{9} a^{2} c^{2} d +\frac {4}{9} x^{9} a \,b^{3} f +\frac {1}{2} x^{8} a \,b^{3} e +\frac {4}{7} x^{7} a^{3} c f +\frac {6}{7} x^{7} a^{2} b^{2} f +\frac {4}{7} x^{7} a \,b^{3} d\)	\(464\)
gosper	\(\frac {x \left (3063060 f \,c^{4} x^{18}+3233230 c^{4} e \,x^{17}[phone] b \,c^{3} f \,x^{16}+3423420 c^{4} d \,x^{16}[phone] b \,c^{3} e \,x^{15}[phone] a \,c^{3} f \,x^{14}[phone] b^{2} c^{2} f \,x^{14}[phone] b \,c^{3} d \,x^{14}[phone] a \,c^{3} e \,x^{13}[phone] b^{2} c^{2} e \,x^{13}[phone] a b \,c^{2} f \,x^{12}[phone] a \,c^{3} d \,x^{12}[phone] b^{3} c f \,x^{12}[phone] b^{2} c^{2} d \,x^{12}[phone] a b \,c^{2} e \,x^{11}[phone] b^{3} c e \,x^{11}[phone] a^{2} c^{2} f \,x^{10}[phone] a \,b^{2} c f \,x^{10}[phone] a b \,c^{2} d \,x^{10}+5290740 b^{4} f \,x^{10}[phone] b^{3} c d \,x^{10}[phone] a^{2} c^{2} e \,x^{9}[phone] a \,b^{2} c e \,x^{9}+5819814 b^{4} e \,x^{9}[phone] a^{2} b c f \,x^{8}[phone] a^{2} c^{2} d \,x^{8}[phone] a \,b^{3} f \,x^{8}[phone] a \,b^{2} c d \,x^{8}+6466460 b^{4} d \,x^{8}[phone] a^{2} b c e \,x^{7}[phone] a \,b^{3} e \,x^{7}[phone] a^{3} c f \,x^{6}[phone] a^{2} b^{2} f \,x^{6}[phone] a^{2} b c d \,x^{6}[phone] a \,b^{3} d \,x^{6}[phone] a^{3} c e \,x^{5}[phone] a^{2} b^{2} e \,x^{5}[phone] a^{3} b f \,x^{4}[phone] a^{3} c d \,x^{4}[phone] a^{2} b^{2} d \,x^{4}[phone] e \,a^{3} b \,x^{3}[phone] a^{4} f \,x^{2}[phone] a^{3} b d \,x^{2}[phone] e \,a^{4} x [phone] a^{4} d \right )}{58198140}\)	\(468\)
default	\(\frac {c^{4} f \,x^{19}}{19}+\frac {c^{4} e \,x^{18}}{18}+\frac {\left (3 b \,c^{3} f +c^{3} \left (b f +c d \right )\right ) x^{17}}{17}+\frac {b \,c^{3} e \,x^{16}}{4}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) c f +3 b \,c^{2} \left (b f +c d \right )+c^{3} \left (a f +b d \right )\right ) x^{15}}{15}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) e c +3 b^{2} c^{2} e +a \,c^{3} e \right ) x^{14}}{14}+\frac {\left (\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) c f +\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) \left (b f +c d \right )+3 b \,c^{2} \left (a f +b d \right )+a \,c^{3} d \right ) x^{13}}{13}+\frac {\left (\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) e c +\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) b e +3 a b \,c^{2} e \right ) x^{12}}{12}+\frac {\left (\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) c f +\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) \left (b f +c d \right )+\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) \left (a f +b d \right )+3 a b \,c^{2} d \right ) x^{11}}{11}+\frac {\left (\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) e c +\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) b e +\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) a e \right ) x^{10}}{10}+\frac {\left (3 a^{2} b c f +\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) \left (b f +c d \right )+\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) \left (a f +b d \right )+\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) d a \right ) x^{9}}{9}+\frac {\left (3 a^{2} b c e +\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) b e +\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) a e \right ) x^{8}}{8}+\frac {\left (a^{3} c f +3 a^{2} b \left (b f +c d \right )+\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) \left (a f +b d \right )+\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) d a \right ) x^{7}}{7}+\frac {\left (a^{3} c e +3 a^{2} b^{2} e +\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) a e \right ) x^{6}}{6}+\frac {\left (a^{3} \left (b f +c d \right )+3 a^{2} b \left (a f +b d \right )+\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) d a \right ) x^{5}}{5}+a^{3} b e \,x^{4}+\frac {\left (a^{3} \left (a f +b d \right )+3 d \,a^{3} b \right ) x^{3}}{3}+\frac {a^{4} e \,x^{2}}{2}+a^{4} d x\)	\(829\)

[In]

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

[Out]

(1/3*f*a^4+4/3*d*a^3*b)*x^3+(2/7*a*c^3*e+3/7*b^2*c^2*e)*x^14+(2/3*a^3*c*e+a^2*b^2*e)*x^6+(4/17*b*c^3*f+1/17*c^
4*d)*x^17+(a*b*c^2*e+1/3*b^3*c*e)*x^12+(3/2*a^2*b*c*e+1/2*a*b^3*e)*x^8+(4/15*a*c^3*f+2/5*b^2*c^2*f+4/15*b*c^3*
d)*x^15+(3/5*a^2*c^2*e+6/5*a*b^2*c*e+1/10*b^4*e)*x^10+(4/5*f*a^3*b+4/5*a^3*c*d+6/5*a^2*b^2*d)*x^5+(4/7*a^3*c*f
+6/7*a^2*b^2*f+12/7*a^2*b*c*d+4/7*a*b^3*d)*x^7+(12/13*a*b*c^2*f+4/13*a*c^3*d+4/13*b^3*c*f+6/13*b^2*c^2*d)*x^13
+(6/11*a^2*c^2*f+12/11*a*b^2*c*f+12/11*a*b*c^2*d+1/11*b^4*f+4/11*b^3*c*d)*x^11+(4/3*a^2*b*c*f+2/3*a^2*c^2*d+4/
9*a*b^3*f+4/3*a*b^2*c*d+1/9*d*b^4)*x^9+a^4*d*x+a^3*b*e*x^4+1/2*a^4*e*x^2+1/18*c^4*e*x^18+1/19*c^4*f*x^19+1/4*b
*c^3*e*x^16

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{19} \, c^{4} f x^{19} + \frac {1}{18} \, c^{4} e x^{18} + \frac {1}{4} \, b c^{3} e x^{16} + \frac {1}{17} \, {\left (c^{4} d + 4 \, b c^{3} f\right )} x^{17} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e x^{14} + \frac {2}{15} \, {\left (2 \, b c^{3} d + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} f\right )} x^{15} + \frac {1}{3} \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e x^{12} + \frac {2}{13} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} f\right )} x^{13} + \frac {1}{10} \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e x^{10} + \frac {1}{11} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} f\right )} x^{11} + \frac {1}{2} \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e x^{8} + \frac {1}{9} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} f\right )} x^{9} + a^{3} b e x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e x^{6} + \frac {2}{7} \, {\left (2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} f\right )} x^{7} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x + \frac {2}{5} \, {\left (2 \, a^{3} b f + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} b d + a^{4} f\right )} x^{3} \]

[In]

integrate((c*x^4+b*x^2+a)^3*(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]

1/19*c^4*f*x^19 + 1/18*c^4*e*x^18 + 1/4*b*c^3*e*x^16 + 1/17*(c^4*d + 4*b*c^3*f)*x^17 + 1/7*(3*b^2*c^2 + 2*a*c^
3)*e*x^14 + 2/15*(2*b*c^3*d + (3*b^2*c^2 + 2*a*c^3)*f)*x^15 + 1/3*(b^3*c + 3*a*b*c^2)*e*x^12 + 2/13*((3*b^2*c^
2 + 2*a*c^3)*d + 2*(b^3*c + 3*a*b*c^2)*f)*x^13 + 1/10*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e*x^10 + 1/11*(4*(b^3*c +
3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*f)*x^11 + 1/2*(a*b^3 + 3*a^2*b*c)*e*x^8 + 1/9*((b^4 + 12*a*b^2*
c + 6*a^2*c^2)*d + 4*(a*b^3 + 3*a^2*b*c)*f)*x^9 + a^3*b*e*x^4 + 1/3*(3*a^2*b^2 + 2*a^3*c)*e*x^6 + 2/7*(2*(a*b^
3 + 3*a^2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*f)*x^7 + 1/2*a^4*e*x^2 + a^4*d*x + 2/5*(2*a^3*b*f + (3*a^2*b^2 + 2*a^
3*c)*d)*x^5 + 1/3*(4*a^3*b*d + a^4*f)*x^3

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.21 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^{4} d x + \frac {a^{4} e x^{2}}{2} + a^{3} b e x^{4} + \frac {b c^{3} e x^{16}}{4} + \frac {c^{4} e x^{18}}{18} + \frac {c^{4} f x^{19}}{19} + x^{17} \cdot \left (\frac {4 b c^{3} f}{17} + \frac {c^{4} d}{17}\right ) + x^{15} \cdot \left (\frac {4 a c^{3} f}{15} + \frac {2 b^{2} c^{2} f}{5} + \frac {4 b c^{3} d}{15}\right ) + x^{14} \cdot \left (\frac {2 a c^{3} e}{7} + \frac {3 b^{2} c^{2} e}{7}\right ) + x^{13} \cdot \left (\frac {12 a b c^{2} f}{13} + \frac {4 a c^{3} d}{13} + \frac {4 b^{3} c f}{13} + \frac {6 b^{2} c^{2} d}{13}\right ) + x^{12} \left (a b c^{2} e + \frac {b^{3} c e}{3}\right ) + x^{11} \cdot \left (\frac {6 a^{2} c^{2} f}{11} + \frac {12 a b^{2} c f}{11} + \frac {12 a b c^{2} d}{11} + \frac {b^{4} f}{11} + \frac {4 b^{3} c d}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} c^{2} e}{5} + \frac {6 a b^{2} c e}{5} + \frac {b^{4} e}{10}\right ) + x^{9} \cdot \left (\frac {4 a^{2} b c f}{3} + \frac {2 a^{2} c^{2} d}{3} + \frac {4 a b^{3} f}{9} + \frac {4 a b^{2} c d}{3} + \frac {b^{4} d}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} b c e}{2} + \frac {a b^{3} e}{2}\right ) + x^{7} \cdot \left (\frac {4 a^{3} c f}{7} + \frac {6 a^{2} b^{2} f}{7} + \frac {12 a^{2} b c d}{7} + \frac {4 a b^{3} d}{7}\right ) + x^{6} \cdot \left (\frac {2 a^{3} c e}{3} + a^{2} b^{2} e\right ) + x^{5} \cdot \left (\frac {4 a^{3} b f}{5} + \frac {4 a^{3} c d}{5} + \frac {6 a^{2} b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{4} f}{3} + \frac {4 a^{3} b d}{3}\right ) \]

[In]

integrate((c*x**4+b*x**2+a)**3*(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**4*d*x + a**4*e*x**2/2 + a**3*b*e*x**4 + b*c**3*e*x**16/4 + c**4*e*x**18/18 + c**4*f*x**19/19 + x**17*(4*b*c
**3*f/17 + c**4*d/17) + x**15*(4*a*c**3*f/15 + 2*b**2*c**2*f/5 + 4*b*c**3*d/15) + x**14*(2*a*c**3*e/7 + 3*b**2
*c**2*e/7) + x**13*(12*a*b*c**2*f/13 + 4*a*c**3*d/13 + 4*b**3*c*f/13 + 6*b**2*c**2*d/13) + x**12*(a*b*c**2*e +
b**3*c*e/3) + x**11*(6*a**2*c**2*f/11 + 12*a*b**2*c*f/11 + 12*a*b*c**2*d/11 + b**4*f/11 + 4*b**3*c*d/11) + x*
*10*(3*a**2*c**2*e/5 + 6*a*b**2*c*e/5 + b**4*e/10) + x**9*(4*a**2*b*c*f/3 + 2*a**2*c**2*d/3 + 4*a*b**3*f/9 + 4
*a*b**2*c*d/3 + b**4*d/9) + x**8*(3*a**2*b*c*e/2 + a*b**3*e/2) + x**7*(4*a**3*c*f/7 + 6*a**2*b**2*f/7 + 12*a**
2*b*c*d/7 + 4*a*b**3*d/7) + x**6*(2*a**3*c*e/3 + a**2*b**2*e) + x**5*(4*a**3*b*f/5 + 4*a**3*c*d/5 + 6*a**2*b**
2*d/5) + x**3*(a**4*f/3 + 4*a**3*b*d/3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{19} \, c^{4} f x^{19} + \frac {1}{18} \, c^{4} e x^{18} + \frac {1}{4} \, b c^{3} e x^{16} + \frac {1}{17} \, {\left (c^{4} d + 4 \, b c^{3} f\right )} x^{17} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e x^{14} + \frac {2}{15} \, {\left (2 \, b c^{3} d + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} f\right )} x^{15} + \frac {1}{3} \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e x^{12} + \frac {2}{13} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} f\right )} x^{13} + \frac {1}{10} \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e x^{10} + \frac {1}{11} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} f\right )} x^{11} + \frac {1}{2} \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e x^{8} + \frac {1}{9} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} f\right )} x^{9} + a^{3} b e x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e x^{6} + \frac {2}{7} \, {\left (2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} f\right )} x^{7} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x + \frac {2}{5} \, {\left (2 \, a^{3} b f + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} b d + a^{4} f\right )} x^{3} \]

[In]

integrate((c*x^4+b*x^2+a)^3*(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]

1/19*c^4*f*x^19 + 1/18*c^4*e*x^18 + 1/4*b*c^3*e*x^16 + 1/17*(c^4*d + 4*b*c^3*f)*x^17 + 1/7*(3*b^2*c^2 + 2*a*c^
3)*e*x^14 + 2/15*(2*b*c^3*d + (3*b^2*c^2 + 2*a*c^3)*f)*x^15 + 1/3*(b^3*c + 3*a*b*c^2)*e*x^12 + 2/13*((3*b^2*c^
2 + 2*a*c^3)*d + 2*(b^3*c + 3*a*b*c^2)*f)*x^13 + 1/10*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e*x^10 + 1/11*(4*(b^3*c +
3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*f)*x^11 + 1/2*(a*b^3 + 3*a^2*b*c)*e*x^8 + 1/9*((b^4 + 12*a*b^2*
c + 6*a^2*c^2)*d + 4*(a*b^3 + 3*a^2*b*c)*f)*x^9 + a^3*b*e*x^4 + 1/3*(3*a^2*b^2 + 2*a^3*c)*e*x^6 + 2/7*(2*(a*b^
3 + 3*a^2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*f)*x^7 + 1/2*a^4*e*x^2 + a^4*d*x + 2/5*(2*a^3*b*f + (3*a^2*b^2 + 2*a^
3*c)*d)*x^5 + 1/3*(4*a^3*b*d + a^4*f)*x^3

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{19} \, c^{4} f x^{19} + \frac {1}{18} \, c^{4} e x^{18} + \frac {1}{17} \, c^{4} d x^{17} + \frac {4}{17} \, b c^{3} f x^{17} + \frac {1}{4} \, b c^{3} e x^{16} + \frac {4}{15} \, b c^{3} d x^{15} + \frac {2}{5} \, b^{2} c^{2} f x^{15} + \frac {4}{15} \, a c^{3} f x^{15} + \frac {3}{7} \, b^{2} c^{2} e x^{14} + \frac {2}{7} \, a c^{3} e x^{14} + \frac {6}{13} \, b^{2} c^{2} d x^{13} + \frac {4}{13} \, a c^{3} d x^{13} + \frac {4}{13} \, b^{3} c f x^{13} + \frac {12}{13} \, a b c^{2} f x^{13} + \frac {1}{3} \, b^{3} c e x^{12} + a b c^{2} e x^{12} + \frac {4}{11} \, b^{3} c d x^{11} + \frac {12}{11} \, a b c^{2} d x^{11} + \frac {1}{11} \, b^{4} f x^{11} + \frac {12}{11} \, a b^{2} c f x^{11} + \frac {6}{11} \, a^{2} c^{2} f x^{11} + \frac {1}{10} \, b^{4} e x^{10} + \frac {6}{5} \, a b^{2} c e x^{10} + \frac {3}{5} \, a^{2} c^{2} e x^{10} + \frac {1}{9} \, b^{4} d x^{9} + \frac {4}{3} \, a b^{2} c d x^{9} + \frac {2}{3} \, a^{2} c^{2} d x^{9} + \frac {4}{9} \, a b^{3} f x^{9} + \frac {4}{3} \, a^{2} b c f x^{9} + \frac {1}{2} \, a b^{3} e x^{8} + \frac {3}{2} \, a^{2} b c e x^{8} + \frac {4}{7} \, a b^{3} d x^{7} + \frac {12}{7} \, a^{2} b c d x^{7} + \frac {6}{7} \, a^{2} b^{2} f x^{7} + \frac {4}{7} \, a^{3} c f x^{7} + a^{2} b^{2} e x^{6} + \frac {2}{3} \, a^{3} c e x^{6} + \frac {6}{5} \, a^{2} b^{2} d x^{5} + \frac {4}{5} \, a^{3} c d x^{5} + \frac {4}{5} \, a^{3} b f x^{5} + a^{3} b e x^{4} + \frac {4}{3} \, a^{3} b d x^{3} + \frac {1}{3} \, a^{4} f x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]

[In]

integrate((c*x^4+b*x^2+a)^3*(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]

1/19*c^4*f*x^19 + 1/18*c^4*e*x^18 + 1/17*c^4*d*x^17 + 4/17*b*c^3*f*x^17 + 1/4*b*c^3*e*x^16 + 4/15*b*c^3*d*x^15
+ 2/5*b^2*c^2*f*x^15 + 4/15*a*c^3*f*x^15 + 3/7*b^2*c^2*e*x^14 + 2/7*a*c^3*e*x^14 + 6/13*b^2*c^2*d*x^13 + 4/13
*a*c^3*d*x^13 + 4/13*b^3*c*f*x^13 + 12/13*a*b*c^2*f*x^13 + 1/3*b^3*c*e*x^12 + a*b*c^2*e*x^12 + 4/11*b^3*c*d*x^
11 + 12/11*a*b*c^2*d*x^11 + 1/11*b^4*f*x^11 + 12/11*a*b^2*c*f*x^11 + 6/11*a^2*c^2*f*x^11 + 1/10*b^4*e*x^10 + 6
/5*a*b^2*c*e*x^10 + 3/5*a^2*c^2*e*x^10 + 1/9*b^4*d*x^9 + 4/3*a*b^2*c*d*x^9 + 2/3*a^2*c^2*d*x^9 + 4/9*a*b^3*f*x
^9 + 4/3*a^2*b*c*f*x^9 + 1/2*a*b^3*e*x^8 + 3/2*a^2*b*c*e*x^8 + 4/7*a*b^3*d*x^7 + 12/7*a^2*b*c*d*x^7 + 6/7*a^2*
b^2*f*x^7 + 4/7*a^3*c*f*x^7 + a^2*b^2*e*x^6 + 2/3*a^3*c*e*x^6 + 6/5*a^2*b^2*d*x^5 + 4/5*a^3*c*d*x^5 + 4/5*a^3*
b*f*x^5 + a^3*b*e*x^4 + 4/3*a^3*b*d*x^3 + 1/3*a^4*f*x^3 + 1/2*a^4*e*x^2 + a^4*d*x

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.96 \[ \int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=x^3\,\left (\frac {f\,a^4}{3}+\frac {4\,b\,d\,a^3}{3}\right )+x^{17}\,\left (\frac {d\,c^4}{17}+\frac {4\,b\,f\,c^3}{17}\right )+x^5\,\left (\frac {4\,f\,a^3\,b}{5}+\frac {4\,c\,d\,a^3}{5}+\frac {6\,d\,a^2\,b^2}{5}\right )+x^{15}\,\left (\frac {2\,f\,b^2\,c^2}{5}+\frac {4\,d\,b\,c^3}{15}+\frac {4\,a\,f\,c^3}{15}\right )+x^9\,\left (\frac {4\,f\,a^2\,b\,c}{3}+\frac {2\,d\,a^2\,c^2}{3}+\frac {4\,f\,a\,b^3}{9}+\frac {4\,d\,a\,b^2\,c}{3}+\frac {d\,b^4}{9}\right )+x^{11}\,\left (\frac {6\,f\,a^2\,c^2}{11}+\frac {12\,f\,a\,b^2\,c}{11}+\frac {12\,d\,a\,b\,c^2}{11}+\frac {f\,b^4}{11}+\frac {4\,d\,b^3\,c}{11}\right )+x^7\,\left (\frac {4\,c\,f\,a^3}{7}+\frac {6\,f\,a^2\,b^2}{7}+\frac {12\,c\,d\,a^2\,b}{7}+\frac {4\,d\,a\,b^3}{7}\right )+x^{13}\,\left (\frac {4\,f\,b^3\,c}{13}+\frac {6\,d\,b^2\,c^2}{13}+\frac {12\,a\,f\,b\,c^2}{13}+\frac {4\,a\,d\,c^3}{13}\right )+\frac {a^4\,e\,x^2}{2}+\frac {c^4\,e\,x^{18}}{18}+\frac {c^4\,f\,x^{19}}{19}+\frac {e\,x^{10}\,\left (6\,a^2\,c^2+12\,a\,b^2\,c+b^4\right )}{10}+a^4\,d\,x+\frac {a^2\,e\,x^6\,\left (3\,b^2+2\,a\,c\right )}{3}+\frac {c^2\,e\,x^{14}\,\left (3\,b^2+2\,a\,c\right )}{7}+a^3\,b\,e\,x^4+\frac {b\,c^3\,e\,x^{16}}{4}+\frac {a\,b\,e\,x^8\,\left (b^2+3\,a\,c\right )}{2}+\frac {b\,c\,e\,x^{12}\,\left (b^2+3\,a\,c\right )}{3} \]

[In]

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

[Out]

x^3*((a^4*f)/3 + (4*a^3*b*d)/3) + x^17*((c^4*d)/17 + (4*b*c^3*f)/17) + x^5*((6*a^2*b^2*d)/5 + (4*a^3*c*d)/5 +
(4*a^3*b*f)/5) + x^15*((2*b^2*c^2*f)/5 + (4*b*c^3*d)/15 + (4*a*c^3*f)/15) + x^9*((b^4*d)/9 + (2*a^2*c^2*d)/3 +
(4*a*b^3*f)/9 + (4*a*b^2*c*d)/3 + (4*a^2*b*c*f)/3) + x^11*((b^4*f)/11 + (6*a^2*c^2*f)/11 + (4*b^3*c*d)/11 + (
12*a*b*c^2*d)/11 + (12*a*b^2*c*f)/11) + x^7*((6*a^2*b^2*f)/7 + (4*a*b^3*d)/7 + (4*a^3*c*f)/7 + (12*a^2*b*c*d)/
7) + x^13*((6*b^2*c^2*d)/13 + (4*a*c^3*d)/13 + (4*b^3*c*f)/13 + (12*a*b*c^2*f)/13) + (a^4*e*x^2)/2 + (c^4*e*x^
18)/18 + (c^4*f*x^19)/19 + (e*x^10*(b^4 + 6*a^2*c^2 + 12*a*b^2*c))/10 + a^4*d*x + (a^2*e*x^6*(2*a*c + 3*b^2))/
3 + (c^2*e*x^14*(2*a*c + 3*b^2))/7 + a^3*b*e*x^4 + (b*c^3*e*x^16)/4 + (a*b*e*x^8*(3*a*c + b^2))/2 + (b*c*e*x^1
2*(3*a*c + b^2))/3