### 3.12 $$\int x (A+B x+C x^2) (a+b x^2+c x^4)^2 \, dx$$

Optimal. Leaf size=159 $\frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{8} x^8 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{6} x^6 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{4} a x^4 (a C+2 A b)+\frac{1}{7} B x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b B x^5+\frac{1}{10} c x^{10} (A c+2 b C)+\frac{2}{9} b B c x^9+\frac{1}{11} B c^2 x^{11}+\frac{1}{12} c^2 C x^{12}$

[Out]

(a^2*A*x^2)/2 + (a^2*B*x^3)/3 + (a*(2*A*b + a*C)*x^4)/4 + (2*a*b*B*x^5)/5 + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^6)/
6 + (B*(b^2 + 2*a*c)*x^7)/7 + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^8)/8 + (2*b*B*c*x^9)/9 + (c*(A*c + 2*b*C)*x^10)/1
0 + (B*c^2*x^11)/11 + (c^2*C*x^12)/12

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Rubi [A]  time = 0.143461, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.038, Rules used = {1628} $\frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{8} x^8 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{6} x^6 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{4} a x^4 (a C+2 A b)+\frac{1}{7} B x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b B x^5+\frac{1}{10} c x^{10} (A c+2 b C)+\frac{2}{9} b B c x^9+\frac{1}{11} B c^2 x^{11}+\frac{1}{12} c^2 C x^{12}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]

[Out]

(a^2*A*x^2)/2 + (a^2*B*x^3)/3 + (a*(2*A*b + a*C)*x^4)/4 + (2*a*b*B*x^5)/5 + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^6)/
6 + (B*(b^2 + 2*a*c)*x^7)/7 + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^8)/8 + (2*b*B*c*x^9)/9 + (c*(A*c + 2*b*C)*x^10)/1
0 + (B*c^2*x^11)/11 + (c^2*C*x^12)/12

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int x \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 A x+a^2 B x^2+a (2 A b+a C) x^3+2 a b B x^4+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^5+B \left (b^2+2 a c\right ) x^6+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^7+2 b B c x^8+c (A c+2 b C) x^9+B c^2 x^{10}+c^2 C x^{11}\right ) \, dx\\ &=\frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{4} a (2 A b+a C) x^4+\frac{2}{5} a b B x^5+\frac{1}{6} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^6+\frac{1}{7} B \left (b^2+2 a c\right ) x^7+\frac{1}{8} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^8+\frac{2}{9} b B c x^9+\frac{1}{10} c (A c+2 b C) x^{10}+\frac{1}{11} B c^2 x^{11}+\frac{1}{12} c^2 C x^{12}\\ \end{align*}

Mathematica [A]  time = 0.035737, size = 159, normalized size = 1. $\frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{8} x^8 \left (2 a c C+2 A b c+b^2 C\right )+\frac{1}{6} x^6 \left (2 a A c+2 a b C+A b^2\right )+\frac{1}{4} a x^4 (a C+2 A b)+\frac{1}{7} B x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b B x^5+\frac{1}{10} c x^{10} (A c+2 b C)+\frac{2}{9} b B c x^9+\frac{1}{11} B c^2 x^{11}+\frac{1}{12} c^2 C x^{12}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]

[Out]

(a^2*A*x^2)/2 + (a^2*B*x^3)/3 + (a*(2*A*b + a*C)*x^4)/4 + (2*a*b*B*x^5)/5 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^6)/
6 + (B*(b^2 + 2*a*c)*x^7)/7 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^8)/8 + (2*b*B*c*x^9)/9 + (c*(A*c + 2*b*C)*x^10)/1
0 + (B*c^2*x^11)/11 + (c^2*C*x^12)/12

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Maple [A]  time = 0.001, size = 142, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}C{x}^{12}}{12}}+{\frac{B{c}^{2}{x}^{11}}{11}}+{\frac{ \left ( A{c}^{2}+2\,Cbc \right ){x}^{10}}{10}}+{\frac{2\,bBc{x}^{9}}{9}}+{\frac{ \left ( 2\,Abc+ \left ( 2\,ac+{b}^{2} \right ) C \right ){x}^{8}}{8}}+{\frac{B \left ( 2\,ac+{b}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( A \left ( 2\,ac+{b}^{2} \right ) +2\,abC \right ){x}^{6}}{6}}+{\frac{2\,abB{x}^{5}}{5}}+{\frac{ \left ( 2\,Aab+C{a}^{2} \right ){x}^{4}}{4}}+{\frac{{a}^{2}B{x}^{3}}{3}}+{\frac{{a}^{2}A{x}^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^2,x)

[Out]

1/12*c^2*C*x^12+1/11*B*c^2*x^11+1/10*(A*c^2+2*C*b*c)*x^10+2/9*b*B*c*x^9+1/8*(2*A*b*c+(2*a*c+b^2)*C)*x^8+1/7*B*
(2*a*c+b^2)*x^7+1/6*(A*(2*a*c+b^2)+2*a*b*C)*x^6+2/5*a*b*B*x^5+1/4*(2*A*a*b+C*a^2)*x^4+1/3*a^2*B*x^3+1/2*a^2*A*
x^2

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Maxima [A]  time = 0.949302, size = 193, normalized size = 1.21 \begin{align*} \frac{1}{12} \, C c^{2} x^{12} + \frac{1}{11} \, B c^{2} x^{11} + \frac{2}{9} \, B b c x^{9} + \frac{1}{10} \,{\left (2 \, C b c + A c^{2}\right )} x^{10} + \frac{1}{8} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{8} + \frac{2}{5} \, B a b x^{5} + \frac{1}{7} \,{\left (B b^{2} + 2 \, B a c\right )} x^{7} + \frac{1}{6} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{6} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{2} \, A a^{2} x^{2} + \frac{1}{4} \,{\left (C a^{2} + 2 \, A a b\right )} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*C*c^2*x^12 + 1/11*B*c^2*x^11 + 2/9*B*b*c*x^9 + 1/10*(2*C*b*c + A*c^2)*x^10 + 1/8*(C*b^2 + 2*(C*a + A*b)*c
)*x^8 + 2/5*B*a*b*x^5 + 1/7*(B*b^2 + 2*B*a*c)*x^7 + 1/6*(2*C*a*b + A*b^2 + 2*A*a*c)*x^6 + 1/3*B*a^2*x^3 + 1/2*
A*a^2*x^2 + 1/4*(C*a^2 + 2*A*a*b)*x^4

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Fricas [A]  time = 1.12957, size = 397, normalized size = 2.5 \begin{align*} \frac{1}{12} x^{12} c^{2} C + \frac{1}{11} x^{11} c^{2} B + \frac{1}{5} x^{10} c b C + \frac{1}{10} x^{10} c^{2} A + \frac{2}{9} x^{9} c b B + \frac{1}{8} x^{8} b^{2} C + \frac{1}{4} x^{8} c a C + \frac{1}{4} x^{8} c b A + \frac{1}{7} x^{7} b^{2} B + \frac{2}{7} x^{7} c a B + \frac{1}{3} x^{6} b a C + \frac{1}{6} x^{6} b^{2} A + \frac{1}{3} x^{6} c a A + \frac{2}{5} x^{5} b a B + \frac{1}{4} x^{4} a^{2} C + \frac{1}{2} x^{4} b a A + \frac{1}{3} x^{3} a^{2} B + \frac{1}{2} x^{2} a^{2} A \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*c^2*C + 1/11*x^11*c^2*B + 1/5*x^10*c*b*C + 1/10*x^10*c^2*A + 2/9*x^9*c*b*B + 1/8*x^8*b^2*C + 1/4*x^8
*c*a*C + 1/4*x^8*c*b*A + 1/7*x^7*b^2*B + 2/7*x^7*c*a*B + 1/3*x^6*b*a*C + 1/6*x^6*b^2*A + 1/3*x^6*c*a*A + 2/5*x
^5*b*a*B + 1/4*x^4*a^2*C + 1/2*x^4*b*a*A + 1/3*x^3*a^2*B + 1/2*x^2*a^2*A

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Sympy [A]  time = 0.095328, size = 163, normalized size = 1.03 \begin{align*} \frac{A a^{2} x^{2}}{2} + \frac{B a^{2} x^{3}}{3} + \frac{2 B a b x^{5}}{5} + \frac{2 B b c x^{9}}{9} + \frac{B c^{2} x^{11}}{11} + \frac{C c^{2} x^{12}}{12} + x^{10} \left (\frac{A c^{2}}{10} + \frac{C b c}{5}\right ) + x^{8} \left (\frac{A b c}{4} + \frac{C a c}{4} + \frac{C b^{2}}{8}\right ) + x^{7} \left (\frac{2 B a c}{7} + \frac{B b^{2}}{7}\right ) + x^{6} \left (\frac{A a c}{3} + \frac{A b^{2}}{6} + \frac{C a b}{3}\right ) + x^{4} \left (\frac{A a b}{2} + \frac{C a^{2}}{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(C*x**2+B*x+A)*(c*x**4+b*x**2+a)**2,x)

[Out]

A*a**2*x**2/2 + B*a**2*x**3/3 + 2*B*a*b*x**5/5 + 2*B*b*c*x**9/9 + B*c**2*x**11/11 + C*c**2*x**12/12 + x**10*(A
*c**2/10 + C*b*c/5) + x**8*(A*b*c/4 + C*a*c/4 + C*b**2/8) + x**7*(2*B*a*c/7 + B*b**2/7) + x**6*(A*a*c/3 + A*b*
*2/6 + C*a*b/3) + x**4*(A*a*b/2 + C*a**2/4)

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Giac [A]  time = 1.10008, size = 208, normalized size = 1.31 \begin{align*} \frac{1}{12} \, C c^{2} x^{12} + \frac{1}{11} \, B c^{2} x^{11} + \frac{1}{5} \, C b c x^{10} + \frac{1}{10} \, A c^{2} x^{10} + \frac{2}{9} \, B b c x^{9} + \frac{1}{8} \, C b^{2} x^{8} + \frac{1}{4} \, C a c x^{8} + \frac{1}{4} \, A b c x^{8} + \frac{1}{7} \, B b^{2} x^{7} + \frac{2}{7} \, B a c x^{7} + \frac{1}{3} \, C a b x^{6} + \frac{1}{6} \, A b^{2} x^{6} + \frac{1}{3} \, A a c x^{6} + \frac{2}{5} \, B a b x^{5} + \frac{1}{4} \, C a^{2} x^{4} + \frac{1}{2} \, A a b x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{2} \, A a^{2} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*C*c^2*x^12 + 1/11*B*c^2*x^11 + 1/5*C*b*c*x^10 + 1/10*A*c^2*x^10 + 2/9*B*b*c*x^9 + 1/8*C*b^2*x^8 + 1/4*C*a
*c*x^8 + 1/4*A*b*c*x^8 + 1/7*B*b^2*x^7 + 2/7*B*a*c*x^7 + 1/3*C*a*b*x^6 + 1/6*A*b^2*x^6 + 1/3*A*a*c*x^6 + 2/5*B
*a*b*x^5 + 1/4*C*a^2*x^4 + 1/2*A*a*b*x^4 + 1/3*B*a^2*x^3 + 1/2*A*a^2*x^2