3.2 \(\int x (a x^2+b x^3+c x^4) \, dx\)

Optimal. Leaf size=25 \[ \frac{a x^4}{4}+\frac{b x^5}{5}+\frac{c x^6}{6} \]

[Out]

(a*x^4)/4 + (b*x^5)/5 + (c*x^6)/6

________________________________________________________________________________________

Rubi [A]  time = 0.0073601, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {14} \[ \frac{a x^4}{4}+\frac{b x^5}{5}+\frac{c x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[x*(a*x^2 + b*x^3 + c*x^4),x]

[Out]

(a*x^4)/4 + (b*x^5)/5 + (c*x^6)/6

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+c x^4\right ) \, dx &=\int \left (a x^3+b x^4+c x^5\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{b x^5}{5}+\frac{c x^6}{6}\\ \end{align*}

Mathematica [A]  time = 0.0015328, size = 25, normalized size = 1. \[ \frac{a x^4}{4}+\frac{b x^5}{5}+\frac{c x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a*x^2 + b*x^3 + c*x^4),x]

[Out]

(a*x^4)/4 + (b*x^5)/5 + (c*x^6)/6

________________________________________________________________________________________

Maple [A]  time = 0., size = 20, normalized size = 0.8 \begin{align*}{\frac{a{x}^{4}}{4}}+{\frac{b{x}^{5}}{5}}+{\frac{c{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^4+b*x^3+a*x^2),x)

[Out]

1/4*a*x^4+1/5*b*x^5+1/6*c*x^6

________________________________________________________________________________________

Maxima [A]  time = 1.15779, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{6} \, c x^{6} + \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*(c*x^4+b*x^3+a*x^2),x, algorithm="maxima")

[Out]

1/6*c*x^6 + 1/5*b*x^5 + 1/4*a*x^4

________________________________________________________________________________________

Fricas [A]  time = 1.30137, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{6} x^{6} c + \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*(c*x^4+b*x^3+a*x^2),x, algorithm="fricas")

[Out]

1/6*x^6*c + 1/5*x^5*b + 1/4*x^4*a

________________________________________________________________________________________

Sympy [A]  time = 0.05838, size = 19, normalized size = 0.76 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{5}}{5} + \frac{c x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**4+b*x**3+a*x**2),x)

[Out]

a*x**4/4 + b*x**5/5 + c*x**6/6

________________________________________________________________________________________

Giac [A]  time = 1.10149, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{6} \, c x^{6} + \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*(c*x^4+b*x^3+a*x^2),x, algorithm="giac")

[Out]

1/6*c*x^6 + 1/5*b*x^5 + 1/4*a*x^4