3.2452 \(\int x^2 (a+b x^n) \, dx\)

Optimal. Leaf size=21 \[ \frac{a x^3}{3}+\frac{b x^{n+3}}{n+3} \]

[Out]

(a*x^3)/3 + (b*x^(3 + n))/(3 + n)

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Rubi [A]  time = 0.0081223, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^3}{3}+\frac{b x^{n+3}}{n+3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^n),x]

[Out]

(a*x^3)/3 + (b*x^(3 + n))/(3 + n)

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

Mathematica [A]  time = 0.0132246, size = 21, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^{n+3}}{n+3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^n),x]

[Out]

(a*x^3)/3 + (b*x^(3 + n))/(3 + n)

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Maple [A]  time = 0.006, size = 23, normalized size = 1.1 \begin{align*}{\frac{b{x}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{3+n}}+{\frac{a{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b/(3+n)*x^3*exp(n*ln(x))+1/3*a*x^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54139, size = 61, normalized size = 2.9 \begin{align*} \frac{3 \, b x^{3} x^{n} +{\left (a n + 3 \, a\right )} x^{3}}{3 \,{\left (n + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b*x^3*x^n + (a*n + 3*a)*x^3)/(n + 3)

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Sympy [A]  time = 0.329752, size = 51, normalized size = 2.43 \begin{align*} \begin{cases} \frac{a n x^{3}}{3 n + 9} + \frac{3 a x^{3}}{3 n + 9} + \frac{3 b x^{3} x^{n}}{3 n + 9} & \text{for}\: n \neq -3 \\\frac{a x^{3}}{3} + b \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*n*x**3/(3*n + 9) + 3*a*x**3/(3*n + 9) + 3*b*x**3*x**n/(3*n + 9), Ne(n, -3)), (a*x**3/3 + b*log(x)
, True))

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Giac [A]  time = 1.17919, size = 39, normalized size = 1.86 \begin{align*} \frac{3 \, b x^{3} x^{n} + a n x^{3} + 3 \, a x^{3}}{3 \,{\left (n + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b*x^3*x^n + a*n*x^3 + 3*a*x^3)/(n + 3)