\(\int (a+b x^2+c x^4)^2 (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6) \, dx\) [61]

Optimal result

Integrand size = 63, antiderivative size = 259 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{3} a^2 (3 b d+a f) x^3+\frac {3}{4} a^2 b e x^4+\frac {3}{5} a \left (b^2 d+a c d+a b f\right ) x^5+\frac {1}{2} a \left (b^2+a c\right ) e x^6+\frac {1}{7} \left (b^3 d+6 a b c d+3 a b^2 f+3 a^2 c f\right ) x^7+\frac {1}{8} b \left (b^2+6 a c\right ) e x^8+\frac {1}{9} \left (3 b^2 c d+3 a c^2 d+b^3 f+6 a b c f\right ) x^9+\frac {3}{10} c \left (b^2+a c\right ) e x^{10}+\frac {3}{11} c \left (b c d+b^2 f+a c f\right ) x^{11}+\frac {1}{4} b c^2 e x^{12}+\frac {1}{13} c^2 (c d+3 b f) x^{13}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 259, 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} \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac {1}{3} a^2 x^3 (a f+3 b d)+\frac {3}{4} a^2 b e x^4+\frac {3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac {3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac {3}{10} c e x^{10} \left (a c+b^2\right )+\frac {1}{8} b e x^8 \left (6 a c+b^2\right )+\frac {1}{2} a e x^6 \left (a c+b^2\right )+\frac {1}{9} x^9 \left (6 a b c f+3 a c^2 d+b^3 f+3 b^2 c d\right )+\frac {1}{13} c^2 x^{13} (3 b f+c d)+\frac {1}{4} b c^2 e x^{12}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15} \]

[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 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^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] (verified)

Time = 0.03 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{3} a^2 (3 b d+a f) x^3+\frac {3}{4} a^2 b e x^4+\frac {3}{5} a \left (b^2 d+a c d+a b f\right ) x^5+\frac {1}{2} a \left (b^2+a c\right ) e x^6+\frac {1}{7} \left (b^3 d+6 a b c d+3 a b^2 f+3 a^2 c f\right ) x^7+\frac {1}{8} b \left (b^2+6 a c\right ) e x^8+\frac {1}{9} \left (3 b^2 c d+3 a c^2 d+b^3 f+6 a b c f\right ) x^9+\frac {3}{10} c \left (b^2+a c\right ) e x^{10}+\frac {3}{11} c \left (b c d+b^2 f+a c f\right ) x^{11}+\frac {1}{4} b c^2 e x^{12}+\frac {1}{13} c^2 (c d+3 b f) x^{13}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15} \]

[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

Maple [A] (verified)

Time = 27.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00

method	result	size
norman	\(\left (\frac {1}{3} a^{3} f +a^{2} b d \right ) x^{3}+\left (\frac {3}{10} a \,c^{2} e +\frac {3}{10} b^{2} c e \right ) x^{10}+\left (\frac {1}{2} a^{2} c e +\frac {1}{2} a \,b^{2} e \right ) x^{6}+\left (\frac {3}{13} b \,c^{2} f +\frac {1}{13} c^{3} d \right ) x^{13}+\left (\frac {3}{4} a b c e +\frac {1}{8} b^{3} e \right ) x^{8}+\left (\frac {3}{11} a \,c^{2} f +\frac {3}{11} b^{2} c f +\frac {3}{11} b \,c^{2} d \right ) x^{11}+\left (\frac {3}{5} a^{2} b f +\frac {3}{5} a^{2} c d +\frac {3}{5} a \,b^{2} d \right ) x^{5}+\left (\frac {3}{7} a^{2} c f +\frac {3}{7} a \,b^{2} f +\frac {6}{7} a b c d +\frac {1}{7} b^{3} d \right ) x^{7}+\left (\frac {2}{3} a b c f +\frac {1}{3} a \,c^{2} d +\frac {1}{9} b^{3} f +\frac {1}{3} b^{2} c d \right ) x^{9}+a^{3} d x +\frac {a^{3} e \,x^{2}}{2}+\frac {c^{3} e \,x^{14}}{14}+\frac {c^{3} f \,x^{15}}{15}+\frac {3 a^{2} b e \,x^{4}}{4}+\frac {b \,c^{2} e \,x^{12}}{4}\)	\(259\)
risch	\(a^{3} d x +\frac {3}{4} a^{2} b e \,x^{4}+\frac {1}{2} a \,b^{2} e \,x^{6}+\frac {1}{8} b^{3} e \,x^{8}+\frac {1}{4} b \,c^{2} e \,x^{12}+\frac {1}{2} a^{3} e \,x^{2}+\frac {1}{14} c^{3} e \,x^{14}+\frac {1}{15} c^{3} f \,x^{15}+\frac {3}{13} x^{13} b \,c^{2} f +\frac {3}{11} x^{11} a \,c^{2} f +\frac {3}{11} x^{11} b^{2} c f +\frac {3}{11} x^{11} b \,c^{2} d +\frac {3}{10} x^{10} a \,c^{2} e +\frac {3}{10} x^{10} b^{2} c e +\frac {1}{3} x^{9} a \,c^{2} d +\frac {1}{3} x^{9} b^{2} c d +\frac {3}{7} x^{7} a^{2} c f +\frac {3}{7} x^{7} a \,b^{2} f +\frac {1}{2} x^{6} a^{2} c e +\frac {3}{5} x^{5} a^{2} b f +\frac {3}{5} x^{5} a^{2} c d +\frac {3}{5} x^{5} a \,b^{2} d +x^{3} a^{2} b d +\frac {1}{13} x^{13} c^{3} d +\frac {1}{9} x^{9} b^{3} f +\frac {1}{7} x^{7} b^{3} d +\frac {1}{3} x^{3} a^{3} f +\frac {2}{3} x^{9} a b c f +\frac {3}{4} x^{8} a b c e +\frac {6}{7} x^{7} a b c d\)	\(286\)
parallelrisch	\(a^{3} d x +\frac {3}{4} a^{2} b e \,x^{4}+\frac {1}{2} a \,b^{2} e \,x^{6}+\frac {1}{8} b^{3} e \,x^{8}+\frac {1}{4} b \,c^{2} e \,x^{12}+\frac {1}{2} a^{3} e \,x^{2}+\frac {1}{14} c^{3} e \,x^{14}+\frac {1}{15} c^{3} f \,x^{15}+\frac {3}{13} x^{13} b \,c^{2} f +\frac {3}{11} x^{11} a \,c^{2} f +\frac {3}{11} x^{11} b^{2} c f +\frac {3}{11} x^{11} b \,c^{2} d +\frac {3}{10} x^{10} a \,c^{2} e +\frac {3}{10} x^{10} b^{2} c e +\frac {1}{3} x^{9} a \,c^{2} d +\frac {1}{3} x^{9} b^{2} c d +\frac {3}{7} x^{7} a^{2} c f +\frac {3}{7} x^{7} a \,b^{2} f +\frac {1}{2} x^{6} a^{2} c e +\frac {3}{5} x^{5} a^{2} b f +\frac {3}{5} x^{5} a^{2} c d +\frac {3}{5} x^{5} a \,b^{2} d +x^{3} a^{2} b d +\frac {1}{13} x^{13} c^{3} d +\frac {1}{9} x^{9} b^{3} f +\frac {1}{7} x^{7} b^{3} d +\frac {1}{3} x^{3} a^{3} f +\frac {2}{3} x^{9} a b c f +\frac {3}{4} x^{8} a b c e +\frac {6}{7} x^{7} a b c d\)	\(286\)
gosper	\(\frac {x \left (24024 c^{3} f \,x^{14}+25740 c^{3} e \,x^{13}+83160 b \,c^{2} f \,x^{12}+27720 c^{3} d \,x^{12}+90090 b \,c^{2} e \,x^{11}+98280 a \,c^{2} f \,x^{10}+98280 b^{2} c f \,x^{10}+98280 b \,c^{2} d \,x^{10}+108108 a \,c^{2} e \,x^{9}+108108 b^{2} c e \,x^{9}+240240 a b c f \,x^{8}+120120 a \,c^{2} d \,x^{8}+40040 b^{3} f \,x^{8}+120120 b^{2} c d \,x^{8}+270270 a b c e \,x^{7}+45045 b^{3} e \,x^{7}+154440 a^{2} c f \,x^{6}+154440 a \,b^{2} f \,x^{6}+308880 a b c d \,x^{6}+51480 b^{3} d \,x^{6}+180180 a^{2} c e \,x^{5}+180180 a \,b^{2} e \,x^{5}+216216 a^{2} b f \,x^{4}+216216 a^{2} c d \,x^{4}+216216 a \,b^{2} d \,x^{4}+270270 a^{2} b e \,x^{3}+120120 a^{3} f \,x^{2}+360360 a^{2} b d \,x^{2}+180180 a^{3} e x +360360 a^{3} d \right )}{360360}\)	\(288\)
default	\(\frac {c^{3} f \,x^{15}}{15}+\frac {c^{3} e \,x^{14}}{14}+\frac {\left (2 b \,c^{2} f +c^{2} \left (b f +c d \right )\right ) x^{13}}{13}+\frac {b \,c^{2} e \,x^{12}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) c f +2 b c \left (b f +c d \right )+c^{2} \left (a f +b d \right )\right ) x^{11}}{11}+\frac {\left (\left (2 a c +b^{2}\right ) e c +2 b^{2} c e +a \,c^{2} e \right ) x^{10}}{10}+\frac {\left (2 a b c f +\left (2 a c +b^{2}\right ) \left (b f +c d \right )+2 b c \left (a f +b d \right )+a \,c^{2} d \right ) x^{9}}{9}+\frac {\left (4 a b c e +\left (2 a c +b^{2}\right ) b e \right ) x^{8}}{8}+\frac {\left (a^{2} c f +2 a b \left (b f +c d \right )+\left (2 a c +b^{2}\right ) \left (a f +b d \right )+2 a b c d \right ) x^{7}}{7}+\frac {\left (a^{2} c e +2 a \,b^{2} e +\left (2 a c +b^{2}\right ) a e \right ) x^{6}}{6}+\frac {\left (a^{2} \left (b f +c d \right )+2 a b \left (a f +b d \right )+\left (2 a c +b^{2}\right ) d a \right ) x^{5}}{5}+\frac {3 a^{2} b e \,x^{4}}{4}+\frac {\left (a^{2} \left (a f +b d \right )+2 a^{2} b d \right ) x^{3}}{3}+\frac {a^{3} e \,x^{2}}{2}+a^{3} d x\)	\(354\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{15} \, c^{3} f x^{15} + \frac {1}{14} \, c^{3} e x^{14} + \frac {1}{4} \, b c^{2} e x^{12} + \frac {1}{13} \, {\left (c^{3} d + 3 \, b c^{2} f\right )} x^{13} + \frac {3}{10} \, {\left (b^{2} c + a c^{2}\right )} e x^{10} + \frac {3}{11} \, {\left (b c^{2} d + {\left (b^{2} c + a c^{2}\right )} f\right )} x^{11} + \frac {1}{8} \, {\left (b^{3} + 6 \, a b c\right )} e x^{8} + \frac {1}{9} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} f\right )} x^{9} + \frac {3}{4} \, a^{2} b e x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} e x^{6} + \frac {1}{7} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} f\right )} x^{7} + \frac {1}{2} \, a^{3} e x^{2} + \frac {3}{5} \, {\left (a^{2} b f + {\left (a b^{2} + a^{2} c\right )} d\right )} x^{5} + a^{3} d x + \frac {1}{3} \, {\left (3 \, a^{2} b d + a^{3} f\right )} x^{3} \]

[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*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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.19 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^{3} d x + \frac {a^{3} e x^{2}}{2} + \frac {3 a^{2} b e x^{4}}{4} + \frac {b c^{2} e x^{12}}{4} + \frac {c^{3} e x^{14}}{14} + \frac {c^{3} f x^{15}}{15} + x^{13} \cdot \left (\frac {3 b c^{2} f}{13} + \frac {c^{3} d}{13}\right ) + x^{11} \cdot \left (\frac {3 a c^{2} f}{11} + \frac {3 b^{2} c f}{11} + \frac {3 b c^{2} d}{11}\right ) + x^{10} \cdot \left (\frac {3 a c^{2} e}{10} + \frac {3 b^{2} c e}{10}\right ) + x^{9} \cdot \left (\frac {2 a b c f}{3} + \frac {a c^{2} d}{3} + \frac {b^{3} f}{9} + \frac {b^{2} c d}{3}\right ) + x^{8} \cdot \left (\frac {3 a b c e}{4} + \frac {b^{3} e}{8}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c f}{7} + \frac {3 a b^{2} f}{7} + \frac {6 a b c d}{7} + \frac {b^{3} d}{7}\right ) + x^{6} \left (\frac {a^{2} c e}{2} + \frac {a b^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b f}{5} + \frac {3 a^{2} c d}{5} + \frac {3 a b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{3} f}{3} + a^{2} b d\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{15} \, c^{3} f x^{15} + \frac {1}{14} \, c^{3} e x^{14} + \frac {1}{4} \, b c^{2} e x^{12} + \frac {1}{13} \, {\left (c^{3} d + 3 \, b c^{2} f\right )} x^{13} + \frac {3}{10} \, {\left (b^{2} c + a c^{2}\right )} e x^{10} + \frac {3}{11} \, {\left (b c^{2} d + {\left (b^{2} c + a c^{2}\right )} f\right )} x^{11} + \frac {1}{8} \, {\left (b^{3} + 6 \, a b c\right )} e x^{8} + \frac {1}{9} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} f\right )} x^{9} + \frac {3}{4} \, a^{2} b e x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} e x^{6} + \frac {1}{7} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} f\right )} x^{7} + \frac {1}{2} \, a^{3} e x^{2} + \frac {3}{5} \, {\left (a^{2} b f + {\left (a b^{2} + a^{2} c\right )} d\right )} x^{5} + a^{3} d x + \frac {1}{3} \, {\left (3 \, a^{2} b d + a^{3} f\right )} x^{3} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.10 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{15} \, c^{3} f x^{15} + \frac {1}{14} \, c^{3} e x^{14} + \frac {1}{13} \, c^{3} d x^{13} + \frac {3}{13} \, b c^{2} f x^{13} + \frac {1}{4} \, b c^{2} e x^{12} + \frac {3}{11} \, b c^{2} d x^{11} + \frac {3}{11} \, b^{2} c f x^{11} + \frac {3}{11} \, a c^{2} f x^{11} + \frac {3}{10} \, b^{2} c e x^{10} + \frac {3}{10} \, a c^{2} e x^{10} + \frac {1}{3} \, b^{2} c d x^{9} + \frac {1}{3} \, a c^{2} d x^{9} + \frac {1}{9} \, b^{3} f x^{9} + \frac {2}{3} \, a b c f x^{9} + \frac {1}{8} \, b^{3} e x^{8} + \frac {3}{4} \, a b c e x^{8} + \frac {1}{7} \, b^{3} d x^{7} + \frac {6}{7} \, a b c d x^{7} + \frac {3}{7} \, a b^{2} f x^{7} + \frac {3}{7} \, a^{2} c f x^{7} + \frac {1}{2} \, a b^{2} e x^{6} + \frac {1}{2} \, a^{2} c e x^{6} + \frac {3}{5} \, a b^{2} d x^{5} + \frac {3}{5} \, a^{2} c d x^{5} + \frac {3}{5} \, a^{2} b f x^{5} + \frac {3}{4} \, a^{2} b e x^{4} + a^{2} b d x^{3} + \frac {1}{3} \, a^{3} f x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]

[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*e*x^14 + 1/13*c^3*d*x^13 + 3/13*b*c^2*f*x^13 + 1/4*b*c^2*e*x^12 + 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*e*x^10 + 3/10*a*c^2*e*x^10 + 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*e*x^8 + 3/4*a*b*c*e*x^8 + 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*e*x^6 + 1/2*a^2*c*e*x^6 + 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*e*x^4 + a^2*b*d*x^3 + 1/3*a^3*f*x^3 + 1/2*a^3*e*x^2 + a^3*d*x

Mupad [B] (verification not implemented)

Time = 8.11 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=x^3\,\left (\frac {f\,a^3}{3}+b\,d\,a^2\right )+x^{13}\,\left (\frac {d\,c^3}{13}+\frac {3\,b\,f\,c^2}{13}\right )+x^5\,\left (\frac {3\,f\,a^2\,b}{5}+\frac {3\,c\,d\,a^2}{5}+\frac {3\,d\,a\,b^2}{5}\right )+x^{11}\,\left (\frac {3\,f\,b^2\,c}{11}+\frac {3\,d\,b\,c^2}{11}+\frac {3\,a\,f\,c^2}{11}\right )+x^7\,\left (\frac {3\,c\,f\,a^2}{7}+\frac {3\,f\,a\,b^2}{7}+\frac {6\,c\,d\,a\,b}{7}+\frac {d\,b^3}{7}\right )+x^9\,\left (\frac {f\,b^3}{9}+\frac {d\,b^2\,c}{3}+\frac {2\,a\,f\,b\,c}{3}+\frac {a\,d\,c^2}{3}\right )+\frac {a^3\,e\,x^2}{2}+\frac {c^3\,e\,x^{14}}{14}+\frac {c^3\,f\,x^{15}}{15}+a^3\,d\,x+\frac {a\,e\,x^6\,\left (b^2+a\,c\right )}{2}+\frac {b\,e\,x^8\,\left (b^2+6\,a\,c\right )}{8}+\frac {3\,c\,e\,x^{10}\,\left (b^2+a\,c\right )}{10}+\frac {3\,a^2\,b\,e\,x^4}{4}+\frac {b\,c^2\,e\,x^{12}}{4} \]

[In]

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