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

Optimal. Leaf size=159 \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{9} x^9 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{7} x^7 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} B x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b B x^6+\frac{1}{11} c x^{11} (A c+2 b C)+\frac{1}{5} b B c x^{10}+\frac{1}{12} B c^2 x^{12}+\frac{1}{13} c^2 C x^{13} \]

[Out]

(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*b*B*x^6)/3 + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^7)/7
+ (B*(b^2 + 2*a*c)*x^8)/8 + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^9)/9 + (b*B*c*x^10)/5 + (c*(A*c + 2*b*C)*x^11)/11 +
(B*c^2*x^12)/12 + (c^2*C*x^13)/13

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Rubi [A]  time = 0.21447, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1628} \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{9} x^9 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{7} x^7 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} B x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b B x^6+\frac{1}{11} c x^{11} (A c+2 b C)+\frac{1}{5} b B c x^{10}+\frac{1}{12} B c^2 x^{12}+\frac{1}{13} c^2 C x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*b*B*x^6)/3 + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^7)/7
+ (B*(b^2 + 2*a*c)*x^8)/8 + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^9)/9 + (b*B*c*x^10)/5 + (c*(A*c + 2*b*C)*x^11)/11 +
(B*c^2*x^12)/12 + (c^2*C*x^13)/13

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^2 \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^2+a^2 B x^3+a (2 A b+a C) x^4+2 a b B x^5+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^6+B \left (b^2+2 a c\right ) x^7+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^8+2 b B c x^9+c (A c+2 b C) x^{10}+B c^2 x^{11}+c^2 C x^{12}\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{5} a (2 A b+a C) x^5+\frac{1}{3} a b B x^6+\frac{1}{7} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^7+\frac{1}{8} B \left (b^2+2 a c\right ) x^8+\frac{1}{9} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^9+\frac{1}{5} b B c x^{10}+\frac{1}{11} c (A c+2 b C) x^{11}+\frac{1}{12} B c^2 x^{12}+\frac{1}{13} c^2 C x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0472205, size = 159, normalized size = 1. \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{9} x^9 \left (2 a c C+2 A b c+b^2 C\right )+\frac{1}{7} x^7 \left (2 a A c+2 a b C+A b^2\right )+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} B x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b B x^6+\frac{1}{11} c x^{11} (A c+2 b C)+\frac{1}{5} b B c x^{10}+\frac{1}{12} B c^2 x^{12}+\frac{1}{13} c^2 C x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*b*B*x^6)/3 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^7)/7
+ (B*(b^2 + 2*a*c)*x^8)/8 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^9)/9 + (b*B*c*x^10)/5 + (c*(A*c + 2*b*C)*x^11)/11 +
(B*c^2*x^12)/12 + (c^2*C*x^13)/13

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Maple [A]  time = 0.002, size = 142, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}C{x}^{13}}{13}}+{\frac{B{c}^{2}{x}^{12}}{12}}+{\frac{ \left ( A{c}^{2}+2\,Cbc \right ){x}^{11}}{11}}+{\frac{bBc{x}^{10}}{5}}+{\frac{ \left ( 2\,Abc+ \left ( 2\,ac+{b}^{2} \right ) C \right ){x}^{9}}{9}}+{\frac{B \left ( 2\,ac+{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( A \left ( 2\,ac+{b}^{2} \right ) +2\,abC \right ){x}^{7}}{7}}+{\frac{abB{x}^{6}}{3}}+{\frac{ \left ( 2\,Aab+C{a}^{2} \right ){x}^{5}}{5}}+{\frac{{a}^{2}B{x}^{4}}{4}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^2*C*x^13+1/12*B*c^2*x^12+1/11*(A*c^2+2*C*b*c)*x^11+1/5*b*B*c*x^10+1/9*(2*A*b*c+(2*a*c+b^2)*C)*x^9+1/8*B
*(2*a*c+b^2)*x^8+1/7*(A*(2*a*c+b^2)+2*a*b*C)*x^7+1/3*a*b*B*x^6+1/5*(2*A*a*b+C*a^2)*x^5+1/4*a^2*B*x^4+1/3*a^2*A
*x^3

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Maxima [A]  time = 0.942833, size = 193, normalized size = 1.21 \begin{align*} \frac{1}{13} \, C c^{2} x^{13} + \frac{1}{12} \, B c^{2} x^{12} + \frac{1}{5} \, B b c x^{10} + \frac{1}{11} \,{\left (2 \, C b c + A c^{2}\right )} x^{11} + \frac{1}{9} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{9} + \frac{1}{3} \, B a b x^{6} + \frac{1}{8} \,{\left (B b^{2} + 2 \, B a c\right )} x^{8} + \frac{1}{7} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{7} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (C a^{2} + 2 \, A a b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*C*c^2*x^13 + 1/12*B*c^2*x^12 + 1/5*B*b*c*x^10 + 1/11*(2*C*b*c + A*c^2)*x^11 + 1/9*(C*b^2 + 2*(C*a + A*b)*
c)*x^9 + 1/3*B*a*b*x^6 + 1/8*(B*b^2 + 2*B*a*c)*x^8 + 1/7*(2*C*a*b + A*b^2 + 2*A*a*c)*x^7 + 1/4*B*a^2*x^4 + 1/3
*A*a^2*x^3 + 1/5*(C*a^2 + 2*A*a*b)*x^5

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Fricas [A]  time = 1.11294, size = 400, normalized size = 2.52 \begin{align*} \frac{1}{13} x^{13} c^{2} C + \frac{1}{12} x^{12} c^{2} B + \frac{2}{11} x^{11} c b C + \frac{1}{11} x^{11} c^{2} A + \frac{1}{5} x^{10} c b B + \frac{1}{9} x^{9} b^{2} C + \frac{2}{9} x^{9} c a C + \frac{2}{9} x^{9} c b A + \frac{1}{8} x^{8} b^{2} B + \frac{1}{4} x^{8} c a B + \frac{2}{7} x^{7} b a C + \frac{1}{7} x^{7} b^{2} A + \frac{2}{7} x^{7} c a A + \frac{1}{3} x^{6} b a B + \frac{1}{5} x^{5} a^{2} C + \frac{2}{5} x^{5} b a A + \frac{1}{4} x^{4} a^{2} B + \frac{1}{3} x^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*c^2*C + 1/12*x^12*c^2*B + 2/11*x^11*c*b*C + 1/11*x^11*c^2*A + 1/5*x^10*c*b*B + 1/9*x^9*b^2*C + 2/9*x
^9*c*a*C + 2/9*x^9*c*b*A + 1/8*x^8*b^2*B + 1/4*x^8*c*a*B + 2/7*x^7*b*a*C + 1/7*x^7*b^2*A + 2/7*x^7*c*a*A + 1/3
*x^6*b*a*B + 1/5*x^5*a^2*C + 2/5*x^5*b*a*A + 1/4*x^4*a^2*B + 1/3*x^3*a^2*A

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Sympy [A]  time = 0.09443, size = 168, normalized size = 1.06 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B a^{2} x^{4}}{4} + \frac{B a b x^{6}}{3} + \frac{B b c x^{10}}{5} + \frac{B c^{2} x^{12}}{12} + \frac{C c^{2} x^{13}}{13} + x^{11} \left (\frac{A c^{2}}{11} + \frac{2 C b c}{11}\right ) + x^{9} \left (\frac{2 A b c}{9} + \frac{2 C a c}{9} + \frac{C b^{2}}{9}\right ) + x^{8} \left (\frac{B a c}{4} + \frac{B b^{2}}{8}\right ) + x^{7} \left (\frac{2 A a c}{7} + \frac{A b^{2}}{7} + \frac{2 C a b}{7}\right ) + x^{5} \left (\frac{2 A a b}{5} + \frac{C a^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**3/3 + B*a**2*x**4/4 + B*a*b*x**6/3 + B*b*c*x**10/5 + B*c**2*x**12/12 + C*c**2*x**13/13 + x**11*(A*c*
*2/11 + 2*C*b*c/11) + x**9*(2*A*b*c/9 + 2*C*a*c/9 + C*b**2/9) + x**8*(B*a*c/4 + B*b**2/8) + x**7*(2*A*a*c/7 +
A*b**2/7 + 2*C*a*b/7) + x**5*(2*A*a*b/5 + C*a**2/5)

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Giac [A]  time = 1.11528, size = 208, normalized size = 1.31 \begin{align*} \frac{1}{13} \, C c^{2} x^{13} + \frac{1}{12} \, B c^{2} x^{12} + \frac{2}{11} \, C b c x^{11} + \frac{1}{11} \, A c^{2} x^{11} + \frac{1}{5} \, B b c x^{10} + \frac{1}{9} \, C b^{2} x^{9} + \frac{2}{9} \, C a c x^{9} + \frac{2}{9} \, A b c x^{9} + \frac{1}{8} \, B b^{2} x^{8} + \frac{1}{4} \, B a c x^{8} + \frac{2}{7} \, C a b x^{7} + \frac{1}{7} \, A b^{2} x^{7} + \frac{2}{7} \, A a c x^{7} + \frac{1}{3} \, B a b x^{6} + \frac{1}{5} \, C a^{2} x^{5} + \frac{2}{5} \, A a b x^{5} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*C*c^2*x^13 + 1/12*B*c^2*x^12 + 2/11*C*b*c*x^11 + 1/11*A*c^2*x^11 + 1/5*B*b*c*x^10 + 1/9*C*b^2*x^9 + 2/9*C
*a*c*x^9 + 2/9*A*b*c*x^9 + 1/8*B*b^2*x^8 + 1/4*B*a*c*x^8 + 2/7*C*a*b*x^7 + 1/7*A*b^2*x^7 + 2/7*A*a*c*x^7 + 1/3
*B*a*b*x^6 + 1/5*C*a^2*x^5 + 2/5*A*a*b*x^5 + 1/4*B*a^2*x^4 + 1/3*A*a^2*x^3