3.1574 \(\int (a+\frac{b}{x})^3 x^2 \, dx\)

Optimal. Leaf size=35 \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+3 a b^2 x+b^3 \log (x) \]

[Out]

3*a*b^2*x + (3*a^2*b*x^2)/2 + (a^3*x^3)/3 + b^3*Log[x]

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Rubi [A]  time = 0.011661, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+3 a b^2 x+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3*x^2,x]

[Out]

3*a*b^2*x + (3*a^2*b*x^2)/2 + (a^3*x^3)/3 + b^3*Log[x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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

Mathematica [A]  time = 0.0027684, size = 35, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+3 a b^2 x+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3*x^2,x]

[Out]

3*a*b^2*x + (3*a^2*b*x^2)/2 + (a^3*x^3)/3 + b^3*Log[x]

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Maple [A]  time = 0.002, size = 32, normalized size = 0.9 \begin{align*} 3\,xa{b}^{2}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{{a}^{3}{x}^{3}}{3}}+{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*a*b^2+3/2*a^2*b*x^2+1/3*a^3*x^3+b^3*ln(x)

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Maxima [A]  time = 0.997642, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3 + 3/2*a^2*b*x^2 + 3*a*b^2*x + b^3*log(x)

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Fricas [A]  time = 1.4567, size = 73, normalized size = 2.09 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3 + 3/2*a^2*b*x^2 + 3*a*b^2*x + b^3*log(x)

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Sympy [A]  time = 0.257519, size = 34, normalized size = 0.97 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} x + b^{3} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 3*a**2*b*x**2/2 + 3*a*b**2*x + b**3*log(x)

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Giac [A]  time = 1.17491, size = 43, normalized size = 1.23 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3 + 3/2*a^2*b*x^2 + 3*a*b^2*x + b^3*log(abs(x))