3.253 \(\int x (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} A c x^4+\frac{1}{5} B c x^5 \]

[Out]

(a*A*x^2)/2 + (a*B*x^3)/3 + (A*c*x^4)/4 + (B*c*x^5)/5

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Rubi [A]  time = 0.0230988, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {766} \[ \frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} A c x^4+\frac{1}{5} B c x^5 \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(a + c*x^2),x]

[Out]

(a*A*x^2)/2 + (a*B*x^3)/3 + (A*c*x^4)/4 + (B*c*x^5)/5

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0018294, size = 37, normalized size = 1. \[ \frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} A c x^4+\frac{1}{5} B c x^5 \]

Antiderivative was successfully verified.

[In]

Integrate[x*(A + B*x)*(a + c*x^2),x]

[Out]

(a*A*x^2)/2 + (a*B*x^3)/3 + (A*c*x^4)/4 + (B*c*x^5)/5

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Maple [A]  time = 0., size = 30, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{2}}{2}}+{\frac{aB{x}^{3}}{3}}+{\frac{Ac{x}^{4}}{4}}+{\frac{Bc{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*A*x^2+1/3*a*B*x^3+1/4*A*c*x^4+1/5*B*c*x^5

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Maxima [A]  time = 1.06966, size = 39, normalized size = 1.05 \begin{align*} \frac{1}{5} \, B c x^{5} + \frac{1}{4} \, A c 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*(B*x+A)*(c*x^2+a),x, algorithm="maxima")

[Out]

1/5*B*c*x^5 + 1/4*A*c*x^4 + 1/3*B*a*x^3 + 1/2*A*a*x^2

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Fricas [A]  time = 1.36387, size = 74, normalized size = 2. \begin{align*} \frac{1}{5} x^{5} c B + \frac{1}{4} x^{4} c 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*(B*x+A)*(c*x^2+a),x, algorithm="fricas")

[Out]

1/5*x^5*c*B + 1/4*x^4*c*A + 1/3*x^3*a*B + 1/2*x^2*a*A

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Sympy [A]  time = 0.072569, size = 32, normalized size = 0.86 \begin{align*} \frac{A a x^{2}}{2} + \frac{A c x^{4}}{4} + \frac{B a x^{3}}{3} + \frac{B c x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x**2/2 + A*c*x**4/4 + B*a*x**3/3 + B*c*x**5/5

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Giac [A]  time = 1.17337, size = 39, normalized size = 1.05 \begin{align*} \frac{1}{5} \, B c x^{5} + \frac{1}{4} \, A c 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*(B*x+A)*(c*x^2+a),x, algorithm="giac")

[Out]

1/5*B*c*x^5 + 1/4*A*c*x^4 + 1/3*B*a*x^3 + 1/2*A*a*x^2