3.204 \(\int x (a x^2+b x^3) \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^4}{4}+\frac{b x^5}{5} \]

[Out]

(a*x^4)/4 + (b*x^5)/5

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Rubi [A]  time = 0.0079168, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{a x^4}{4}+\frac{b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x*(a*x^2 + b*x^3),x]

[Out]

(a*x^4)/4 + (b*x^5)/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0010626, size = 17, normalized size = 1. \[ \frac{a x^4}{4}+\frac{b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a*x^2 + b*x^3),x]

[Out]

(a*x^4)/4 + (b*x^5)/5

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{4}}{4}}+{\frac{b{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3+a*x^2),x)

[Out]

1/4*a*x^4+1/5*b*x^5

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Maxima [A]  time = 0.988774, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{5} \, b x^{5} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a*x^2),x, algorithm="maxima")

[Out]

1/5*b*x^5 + 1/4*a*x^4

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Fricas [A]  time = 1.02401, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{5} x^{5} b + \frac{1}{4} x^{4} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a*x^2),x, algorithm="fricas")

[Out]

1/5*x^5*b + 1/4*x^4*a

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Sympy [A]  time = 0.142072, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**3+a*x**2),x)

[Out]

a*x**4/4 + b*x**5/5

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Giac [A]  time = 1.10421, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{5} \, b x^{5} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a*x^2),x, algorithm="giac")

[Out]

1/5*b*x^5 + 1/4*a*x^4