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

Optimal. Leaf size=74 \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8 \]

[Out]

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

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Rubi [A]  time = 0.0565914, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1628} \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0120258, size = 74, normalized size = 1. \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 61, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{2}}{2}}+{\frac{aB{x}^{3}}{3}}+{\frac{ \left ( Ab+aC \right ){x}^{4}}{4}}+{\frac{bB{x}^{5}}{5}}+{\frac{ \left ( Ac+bC \right ){x}^{6}}{6}}+{\frac{Bc{x}^{7}}{7}}+{\frac{cC{x}^{8}}{8}} \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),x)

[Out]

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

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Maxima [A]  time = 0.961008, size = 81, normalized size = 1.09 \begin{align*} \frac{1}{8} \, C c x^{8} + \frac{1}{7} \, B c x^{7} + \frac{1}{5} \, B b x^{5} + \frac{1}{6} \,{\left (C b + A c\right )} x^{6} + \frac{1}{3} \, B a x^{3} + \frac{1}{4} \,{\left (C a + A b\right )} x^{4} + \frac{1}{2} \, A a 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.12357, size = 169, normalized size = 2.28 \begin{align*} \frac{1}{8} x^{8} c C + \frac{1}{7} x^{7} c B + \frac{1}{6} x^{6} b C + \frac{1}{6} x^{6} c A + \frac{1}{5} x^{5} b B + \frac{1}{4} x^{4} a C + \frac{1}{4} x^{4} b A + \frac{1}{3} x^{3} a B + \frac{1}{2} x^{2} a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.07363, size = 68, normalized size = 0.92 \begin{align*} \frac{A a x^{2}}{2} + \frac{B a x^{3}}{3} + \frac{B b x^{5}}{5} + \frac{B c x^{7}}{7} + \frac{C c x^{8}}{8} + x^{6} \left (\frac{A c}{6} + \frac{C b}{6}\right ) + x^{4} \left (\frac{A b}{4} + \frac{C a}{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),x)

[Out]

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

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Giac [A]  time = 1.08453, size = 86, normalized size = 1.16 \begin{align*} \frac{1}{8} \, C c x^{8} + \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, C b x^{6} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, B b x^{5} + \frac{1}{4} \, C a x^{4} + \frac{1}{4} \, A b x^{4} + \frac{1}{3} \, B a x^{3} + \frac{1}{2} \, A a 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),x, algorithm="giac")

[Out]

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