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

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

[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

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Rubi [A]  time = 0.63, antiderivative size = 416, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{11} x^{11} \left (6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+4 b^3 c d+b^4 f\right )+\frac {1}{9} x^9 \left (12 a^2 b c f+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 f+b^4 d\right )+\frac {1}{10} e x^{10} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac {2}{7} a x^7 \left (2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right )+\frac {2}{5} a^2 x^5 \left (2 a b f+2 a c d+3 b^2 d\right )+\frac {1}{3} a^2 e x^6 \left (2 a c+3 b^2\right )+\frac {1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+a^4 d x+\frac {1}{2} a^4 e x^2+\frac {2}{15} c^2 x^{15} \left (2 a c f+3 b^2 f+2 b c d\right )+\frac {2}{13} c x^{13} \left (6 a b c f+2 a c^2 d+3 b^2 c d+2 b^3 f\right )+\frac {1}{7} c^2 e x^{14} \left (2 a c+3 b^2\right )+\frac {1}{3} b c e x^{12} \left (3 a c+b^2\right )+\frac {1}{2} a b e x^8 \left (3 a c+b^2\right )+\frac {1}{17} c^3 x^{17} (4 b f+c d)+\frac {1}{4} b c^3 e x^{16}+\frac {1}{18} c^4 e x^{18}+\frac {1}{19} c^4 f x^{19} \]

Antiderivative was successfully verified.

[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 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 )^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 &=\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*}

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Mathematica [A]  time = 0.12, size = 416, normalized size = 1.00 \[ a^4 d x+\frac {1}{2} a^4 e x^2+\frac {1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac {2}{5} a^2 x^5 \left (2 a b f+2 a c d+3 b^2 d\right )+\frac {1}{3} a^2 e x^6 \left (2 a c+3 b^2\right )+\frac {1}{10} e x^{10} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac {2}{7} a x^7 \left (2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right )+\frac {1}{11} x^{11} \left (6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right )+\frac {1}{9} x^9 \left (12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right )+\frac {2}{15} c^2 x^{15} \left (2 a c f+3 b^2 f+2 b c d\right )+\frac {1}{7} c^2 e x^{14} \left (2 a c+3 b^2\right )+\frac {1}{3} b c e x^{12} \left (3 a c+b^2\right )+\frac {1}{2} a b e x^8 \left (3 a c+b^2\right )+\frac {2}{13} c x^{13} \left (6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right )+\frac {1}{17} c^3 x^{17} (4 b f+c d)+\frac {1}{4} b c^3 e x^{16}+\frac {1}{18} c^4 e x^{18}+\frac {1}{19} c^4 f x^{19} \]

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.81, size = 463, normalized size = 1.11 \[ \frac {1}{19} x^{19} f c^{4} + \frac {1}{18} x^{18} e c^{4} + \frac {1}{17} x^{17} d c^{4} + \frac {4}{17} x^{17} f c^{3} b + \frac {1}{4} x^{16} e c^{3} b + \frac {4}{15} x^{15} d c^{3} b + \frac {2}{5} x^{15} f c^{2} b^{2} + \frac {4}{15} x^{15} f c^{3} a + \frac {3}{7} x^{14} e c^{2} b^{2} + \frac {2}{7} x^{14} e c^{3} a + \frac {6}{13} x^{13} d c^{2} b^{2} + \frac {4}{13} x^{13} f c b^{3} + \frac {4}{13} x^{13} d c^{3} a + \frac {12}{13} x^{13} f c^{2} b a + \frac {1}{3} x^{12} e c b^{3} + x^{12} e c^{2} b a + \frac {4}{11} x^{11} d c b^{3} + \frac {1}{11} x^{11} f b^{4} + \frac {12}{11} x^{11} d c^{2} b a + \frac {12}{11} x^{11} f c b^{2} a + \frac {6}{11} x^{11} f c^{2} a^{2} + \frac {1}{10} x^{10} e b^{4} + \frac {6}{5} x^{10} e c b^{2} a + \frac {3}{5} x^{10} e c^{2} a^{2} + \frac {1}{9} x^{9} d b^{4} + \frac {4}{3} x^{9} d c b^{2} a + \frac {4}{9} x^{9} f b^{3} a + \frac {2}{3} x^{9} d c^{2} a^{2} + \frac {4}{3} x^{9} f c b a^{2} + \frac {1}{2} x^{8} e b^{3} a + \frac {3}{2} x^{8} e c b a^{2} + \frac {4}{7} x^{7} d b^{3} a + \frac {12}{7} x^{7} d c b a^{2} + \frac {6}{7} x^{7} f b^{2} a^{2} + \frac {4}{7} x^{7} f c a^{3} + x^{6} e b^{2} a^{2} + \frac {2}{3} x^{6} e c a^{3} + \frac {6}{5} x^{5} d b^{2} a^{2} + \frac {4}{5} x^{5} d c a^{3} + \frac {4}{5} x^{5} f b a^{3} + x^{4} e b a^{3} + \frac {4}{3} x^{3} d b a^{3} + \frac {1}{3} x^{3} f a^{4} + \frac {1}{2} x^{2} e a^{4} + x d a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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giac [A]  time = 0.43, size = 478, normalized size = 1.15 \[ \frac {1}{19} \, c^{4} f x^{19} + \frac {1}{18} \, c^{4} x^{18} e + \frac {1}{17} \, c^{4} d x^{17} + \frac {4}{17} \, b c^{3} f x^{17} + \frac {1}{4} \, b c^{3} x^{16} e + \frac {4}{15} \, b c^{3} d x^{15} + \frac {2}{5} \, b^{2} c^{2} f x^{15} + \frac {4}{15} \, a c^{3} f x^{15} + \frac {3}{7} \, b^{2} c^{2} x^{14} e + \frac {2}{7} \, a c^{3} x^{14} e + \frac {6}{13} \, b^{2} c^{2} d x^{13} + \frac {4}{13} \, a c^{3} d x^{13} + \frac {4}{13} \, b^{3} c f x^{13} + \frac {12}{13} \, a b c^{2} f x^{13} + \frac {1}{3} \, b^{3} c x^{12} e + a b c^{2} x^{12} e + \frac {4}{11} \, b^{3} c d x^{11} + \frac {12}{11} \, a b c^{2} d x^{11} + \frac {1}{11} \, b^{4} f x^{11} + \frac {12}{11} \, a b^{2} c f x^{11} + \frac {6}{11} \, a^{2} c^{2} f x^{11} + \frac {1}{10} \, b^{4} x^{10} e + \frac {6}{5} \, a b^{2} c x^{10} e + \frac {3}{5} \, a^{2} c^{2} x^{10} e + \frac {1}{9} \, b^{4} d x^{9} + \frac {4}{3} \, a b^{2} c d x^{9} + \frac {2}{3} \, a^{2} c^{2} d x^{9} + \frac {4}{9} \, a b^{3} f x^{9} + \frac {4}{3} \, a^{2} b c f x^{9} + \frac {1}{2} \, a b^{3} x^{8} e + \frac {3}{2} \, a^{2} b c x^{8} e + \frac {4}{7} \, a b^{3} d x^{7} + \frac {12}{7} \, a^{2} b c d x^{7} + \frac {6}{7} \, a^{2} b^{2} f x^{7} + \frac {4}{7} \, a^{3} c f x^{7} + a^{2} b^{2} x^{6} e + \frac {2}{3} \, a^{3} c x^{6} e + \frac {6}{5} \, a^{2} b^{2} d x^{5} + \frac {4}{5} \, a^{3} c d x^{5} + \frac {4}{5} \, a^{3} b f x^{5} + a^{3} b x^{4} e + \frac {4}{3} \, a^{3} b d x^{3} + \frac {1}{3} \, a^{4} f x^{3} + \frac {1}{2} \, a^{4} x^{2} e + a^{4} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

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maple [B]  time = 0.00, size = 829, normalized size = 1.99 \[ \frac {c^{4} f \,x^{19}}{19}+\frac {c^{4} e \,x^{18}}{18}+\frac {b \,c^{3} e \,x^{16}}{4}+\frac {\left (3 b \,c^{3} f +\left (b f +c d \right ) c^{3}\right ) x^{17}}{17}+\frac {\left (3 \left (b f +c d \right ) b \,c^{2}+\left (a f +b d \right ) c^{3}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c f \right ) x^{15}}{15}+\frac {\left (a \,c^{3} e +3 b^{2} c^{2} e +\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c e \right ) x^{14}}{14}+\frac {\left (a \,c^{3} d +3 \left (a f +b d \right ) b \,c^{2}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c f +\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) \left (b f +c d \right )\right ) x^{13}}{13}+\frac {\left (3 a b \,c^{2} e +\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b e +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c e \right ) x^{12}}{12}+\frac {\left (3 a b \,c^{2} d +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c f +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) \left (b f +c d \right )+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) \left (a f +b d \right )\right ) x^{11}}{11}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) a e +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c e \right ) x^{10}}{10}+a^{3} b e \,x^{4}+\frac {\left (3 a^{2} b c f +\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) a d +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) \left (b f +c d \right )+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) \left (a f +b d \right )\right ) x^{9}}{9}+\frac {\left (3 a^{2} b c e +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) a e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b e \right ) x^{8}}{8}+\frac {a^{4} e \,x^{2}}{2}+\frac {\left (a^{3} c f +3 \left (b f +c d \right ) a^{2} b +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) a d +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) \left (a f +b d \right )\right ) x^{7}}{7}+a^{4} d x +\frac {\left (a^{3} c e +3 a^{2} b^{2} e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) a e \right ) x^{6}}{6}+\frac {\left (\left (b f +c d \right ) a^{3}+3 \left (a f +b d \right ) a^{2} b +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) a d \right ) x^{5}}{5}+\frac {\left (3 a^{3} b d +\left (a f +b d \right ) a^{3}\right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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]

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

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maxima [A]  time = 0.52, size = 418, normalized size = 1.00 \[ \frac {1}{19} \, c^{4} f x^{19} + \frac {1}{18} \, c^{4} e x^{18} + \frac {1}{4} \, b c^{3} e x^{16} + \frac {1}{17} \, {\left (c^{4} d + 4 \, b c^{3} f\right )} x^{17} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e x^{14} + \frac {2}{15} \, {\left (2 \, b c^{3} d + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} f\right )} x^{15} + \frac {1}{3} \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e x^{12} + \frac {2}{13} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} f\right )} x^{13} + \frac {1}{10} \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e x^{10} + \frac {1}{11} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} f\right )} x^{11} + \frac {1}{2} \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e x^{8} + \frac {1}{9} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} f\right )} x^{9} + a^{3} b e x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e x^{6} + \frac {2}{7} \, {\left (2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} f\right )} x^{7} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x + \frac {2}{5} \, {\left (2 \, a^{3} b f + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} b d + a^{4} f\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.38, size = 398, normalized size = 0.96 \[ x^3\,\left (\frac {f\,a^4}{3}+\frac {4\,b\,d\,a^3}{3}\right )+x^{17}\,\left (\frac {d\,c^4}{17}+\frac {4\,b\,f\,c^3}{17}\right )+x^5\,\left (\frac {4\,f\,a^3\,b}{5}+\frac {4\,c\,d\,a^3}{5}+\frac {6\,d\,a^2\,b^2}{5}\right )+x^{15}\,\left (\frac {2\,f\,b^2\,c^2}{5}+\frac {4\,d\,b\,c^3}{15}+\frac {4\,a\,f\,c^3}{15}\right )+x^9\,\left (\frac {4\,f\,a^2\,b\,c}{3}+\frac {2\,d\,a^2\,c^2}{3}+\frac {4\,f\,a\,b^3}{9}+\frac {4\,d\,a\,b^2\,c}{3}+\frac {d\,b^4}{9}\right )+x^{11}\,\left (\frac {6\,f\,a^2\,c^2}{11}+\frac {12\,f\,a\,b^2\,c}{11}+\frac {12\,d\,a\,b\,c^2}{11}+\frac {f\,b^4}{11}+\frac {4\,d\,b^3\,c}{11}\right )+x^7\,\left (\frac {4\,c\,f\,a^3}{7}+\frac {6\,f\,a^2\,b^2}{7}+\frac {12\,c\,d\,a^2\,b}{7}+\frac {4\,d\,a\,b^3}{7}\right )+x^{13}\,\left (\frac {4\,f\,b^3\,c}{13}+\frac {6\,d\,b^2\,c^2}{13}+\frac {12\,a\,f\,b\,c^2}{13}+\frac {4\,a\,d\,c^3}{13}\right )+\frac {a^4\,e\,x^2}{2}+\frac {c^4\,e\,x^{18}}{18}+\frac {c^4\,f\,x^{19}}{19}+\frac {e\,x^{10}\,\left (6\,a^2\,c^2+12\,a\,b^2\,c+b^4\right )}{10}+a^4\,d\,x+\frac {a^2\,e\,x^6\,\left (3\,b^2+2\,a\,c\right )}{3}+\frac {c^2\,e\,x^{14}\,\left (3\,b^2+2\,a\,c\right )}{7}+a^3\,b\,e\,x^4+\frac {b\,c^3\,e\,x^{16}}{4}+\frac {a\,b\,e\,x^8\,\left (b^2+3\,a\,c\right )}{2}+\frac {b\,c\,e\,x^{12}\,\left (b^2+3\,a\,c\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.16, size = 503, normalized size = 1.21 \[ a^{4} d x + \frac {a^{4} e x^{2}}{2} + a^{3} b e x^{4} + \frac {b c^{3} e x^{16}}{4} + \frac {c^{4} e x^{18}}{18} + \frac {c^{4} f x^{19}}{19} + x^{17} \left (\frac {4 b c^{3} f}{17} + \frac {c^{4} d}{17}\right ) + x^{15} \left (\frac {4 a c^{3} f}{15} + \frac {2 b^{2} c^{2} f}{5} + \frac {4 b c^{3} d}{15}\right ) + x^{14} \left (\frac {2 a c^{3} e}{7} + \frac {3 b^{2} c^{2} e}{7}\right ) + x^{13} \left (\frac {12 a b c^{2} f}{13} + \frac {4 a c^{3} d}{13} + \frac {4 b^{3} c f}{13} + \frac {6 b^{2} c^{2} d}{13}\right ) + x^{12} \left (a b c^{2} e + \frac {b^{3} c e}{3}\right ) + x^{11} \left (\frac {6 a^{2} c^{2} f}{11} + \frac {12 a b^{2} c f}{11} + \frac {12 a b c^{2} d}{11} + \frac {b^{4} f}{11} + \frac {4 b^{3} c d}{11}\right ) + x^{10} \left (\frac {3 a^{2} c^{2} e}{5} + \frac {6 a b^{2} c e}{5} + \frac {b^{4} e}{10}\right ) + x^{9} \left (\frac {4 a^{2} b c f}{3} + \frac {2 a^{2} c^{2} d}{3} + \frac {4 a b^{3} f}{9} + \frac {4 a b^{2} c d}{3} + \frac {b^{4} d}{9}\right ) + x^{8} \left (\frac {3 a^{2} b c e}{2} + \frac {a b^{3} e}{2}\right ) + x^{7} \left (\frac {4 a^{3} c f}{7} + \frac {6 a^{2} b^{2} f}{7} + \frac {12 a^{2} b c d}{7} + \frac {4 a b^{3} d}{7}\right ) + x^{6} \left (\frac {2 a^{3} c e}{3} + a^{2} b^{2} e\right ) + x^{5} \left (\frac {4 a^{3} b f}{5} + \frac {4 a^{3} c d}{5} + \frac {6 a^{2} b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{4} f}{3} + \frac {4 a^{3} b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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