### 3.1 $$\int x^2 (A+B x+C x^2) (a+b x^2+c x^4) \, dx$$

Optimal. Leaf size=74 $\frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{7} x^7 (A c+b C)+\frac{1}{6} b B x^6+\frac{1}{8} B c x^8+\frac{1}{9} c C x^9$

[Out]

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

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Rubi [A]  time = 0.0815531, antiderivative size = 74, 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} $\frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{7} x^7 (A c+b C)+\frac{1}{6} b B x^6+\frac{1}{8} B c x^8+\frac{1}{9} c C x^9$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0154219, size = 74, normalized size = 1. $\frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{7} x^7 (A c+b C)+\frac{1}{6} b B x^6+\frac{1}{8} B c x^8+\frac{1}{9} c C x^9$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 61, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{3}}{3}}+{\frac{aB{x}^{4}}{4}}+{\frac{ \left ( Ab+aC \right ){x}^{5}}{5}}+{\frac{bB{x}^{6}}{6}}+{\frac{ \left ( Ac+bC \right ){x}^{7}}{7}}+{\frac{Bc{x}^{8}}{8}}+{\frac{cC{x}^{9}}{9}} \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),x)

[Out]

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

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Maxima [A]  time = 0.931994, size = 81, normalized size = 1.09 \begin{align*} \frac{1}{9} \, C c x^{9} + \frac{1}{8} \, B c x^{8} + \frac{1}{6} \, B b x^{6} + \frac{1}{7} \,{\left (C b + A c\right )} x^{7} + \frac{1}{4} \, B a x^{4} + \frac{1}{5} \,{\left (C a + A b\right )} x^{5} + \frac{1}{3} \, A a 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.0522, size = 169, normalized size = 2.28 \begin{align*} \frac{1}{9} x^{9} c C + \frac{1}{8} x^{8} c B + \frac{1}{7} x^{7} b C + \frac{1}{7} x^{7} c A + \frac{1}{6} x^{6} b B + \frac{1}{5} x^{5} a C + \frac{1}{5} x^{5} b A + \frac{1}{4} x^{4} a B + \frac{1}{3} x^{3} a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.086636, size = 68, normalized size = 0.92 \begin{align*} \frac{A a x^{3}}{3} + \frac{B a x^{4}}{4} + \frac{B b x^{6}}{6} + \frac{B c x^{8}}{8} + \frac{C c x^{9}}{9} + x^{7} \left (\frac{A c}{7} + \frac{C b}{7}\right ) + x^{5} \left (\frac{A b}{5} + \frac{C a}{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),x)

[Out]

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

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Giac [A]  time = 1.09347, size = 86, normalized size = 1.16 \begin{align*} \frac{1}{9} \, C c x^{9} + \frac{1}{8} \, B c x^{8} + \frac{1}{7} \, C b x^{7} + \frac{1}{7} \, A c x^{7} + \frac{1}{6} \, B b x^{6} + \frac{1}{5} \, C a x^{5} + \frac{1}{5} \, A b x^{5} + \frac{1}{4} \, B a x^{4} + \frac{1}{3} \, A a 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),x, algorithm="giac")

[Out]

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