### 3.3 $$\int (A+B x+C x^2) (a+b x^2+c x^4) \, dx$$

Optimal. Leaf size=69 $\frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7$

[Out]

a*A*x + (a*B*x^2)/2 + ((A*b + a*C)*x^3)/3 + (b*B*x^4)/4 + ((A*c + b*C)*x^5)/5 + (B*c*x^6)/6 + (c*C*x^7)/7

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Rubi [A]  time = 0.0355215, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.043, Rules used = {1657} $\frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*A*x + (a*B*x^2)/2 + ((A*b + a*C)*x^3)/3 + (b*B*x^4)/4 + ((A*c + b*C)*x^5)/5 + (B*c*x^6)/6 + (c*C*x^7)/7

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.0127978, size = 69, normalized size = 1. $\frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*A*x + (a*B*x^2)/2 + ((A*b + a*C)*x^3)/3 + (b*B*x^4)/4 + ((A*c + b*C)*x^5)/5 + (B*c*x^6)/6 + (c*C*x^7)/7

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Maple [A]  time = 0., size = 58, normalized size = 0.8 \begin{align*} aAx+{\frac{aB{x}^{2}}{2}}+{\frac{ \left ( Ab+aC \right ){x}^{3}}{3}}+{\frac{bB{x}^{4}}{4}}+{\frac{ \left ( Ac+bC \right ){x}^{5}}{5}}+{\frac{Bc{x}^{6}}{6}}+{\frac{cC{x}^{7}}{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.94561, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{7} \, C c x^{7} + \frac{1}{6} \, B c x^{6} + \frac{1}{4} \, B b x^{4} + \frac{1}{5} \,{\left (C b + A c\right )} x^{5} + \frac{1}{2} \, B a x^{2} + \frac{1}{3} \,{\left (C a + A b\right )} x^{3} + A a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.12176, size = 161, normalized size = 2.33 \begin{align*} \frac{1}{7} x^{7} c C + \frac{1}{6} x^{6} c B + \frac{1}{5} x^{5} b C + \frac{1}{5} x^{5} c A + \frac{1}{4} x^{4} b B + \frac{1}{3} x^{3} a C + \frac{1}{3} x^{3} b A + \frac{1}{2} x^{2} a B + x a A \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c*C + 1/6*x^6*c*B + 1/5*x^5*b*C + 1/5*x^5*c*A + 1/4*x^4*b*B + 1/3*x^3*a*C + 1/3*x^3*b*A + 1/2*x^2*a*B
+ x*a*A

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Sympy [A]  time = 0.11015, size = 65, normalized size = 0.94 \begin{align*} A a x + \frac{B a x^{2}}{2} + \frac{B b x^{4}}{4} + \frac{B c x^{6}}{6} + \frac{C c x^{7}}{7} + x^{5} \left (\frac{A c}{5} + \frac{C b}{5}\right ) + x^{3} \left (\frac{A b}{3} + \frac{C a}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x + B*a*x**2/2 + B*b*x**4/4 + B*c*x**6/6 + C*c*x**7/7 + x**5*(A*c/5 + C*b/5) + x**3*(A*b/3 + C*a/3)

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Giac [A]  time = 1.09284, size = 82, normalized size = 1.19 \begin{align*} \frac{1}{7} \, C c x^{7} + \frac{1}{6} \, B c x^{6} + \frac{1}{5} \, C b x^{5} + \frac{1}{5} \, A c x^{5} + \frac{1}{4} \, B b x^{4} + \frac{1}{3} \, C a x^{3} + \frac{1}{3} \, A b x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*C*c*x^7 + 1/6*B*c*x^6 + 1/5*C*b*x^5 + 1/5*A*c*x^5 + 1/4*B*b*x^4 + 1/3*C*a*x^3 + 1/3*A*b*x^3 + 1/2*B*a*x^2
+ A*a*x