3.1550 \(\int (a+\frac{b}{x}) x^2 \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^3}{3}+\frac{b x^2}{2} \]

[Out]

(b*x^2)/2 + (a*x^3)/3

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Rubi [A]  time = 0.006411, antiderivative size = 17, 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} \[ \frac{a x^3}{3}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*x^2,x]

[Out]

(b*x^2)/2 + (a*x^3)/3

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

Mathematica [A]  time = 0.0009252, size = 17, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*x^2,x]

[Out]

(b*x^2)/2 + (a*x^3)/3

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*}{\frac{b{x}^{2}}{2}}+{\frac{a{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2+1/3*a*x^3

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Maxima [A]  time = 0.949515, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{2} \, b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/2*b*x^2

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Fricas [A]  time = 1.45693, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{2} \, b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/2*b*x^2

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Sympy [A]  time = 0.053767, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**2/2

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Giac [A]  time = 1.19907, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{2} \, b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/2*b*x^2