3.44 \(\int x^2 (a+b x) \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^3}{3}+\frac{b x^4}{4} \]

[Out]

(a*x^3)/3 + (b*x^4)/4

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Rubi [A]  time = 0.0067273, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a x^3}{3}+\frac{b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x),x]

[Out]

(a*x^3)/3 + (b*x^4)/4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0015769, size = 17, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x),x]

[Out]

(a*x^3)/3 + (b*x^4)/4

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Maple [A]  time = 0.002, size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{b{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a),x)

[Out]

1/3*a*x^3+1/4*b*x^4

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Maxima [A]  time = 1.01881, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a),x, algorithm="maxima")

[Out]

1/4*b*x^4 + 1/3*a*x^3

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Fricas [A]  time = 1.34153, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{4} x^{4} b + \frac{1}{3} x^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a),x, algorithm="fricas")

[Out]

1/4*x^4*b + 1/3*x^3*a

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Sympy [A]  time = 0.088192, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a),x)

[Out]

a*x**3/3 + b*x**4/4

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Giac [A]  time = 1.1926, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a),x, algorithm="giac")

[Out]

1/4*b*x^4 + 1/3*a*x^3