3.1578 \(\int \frac{(a+\frac{b}{x})^3}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x}\right )^4}{4 b} \]

[Out]

-(a + b/x)^4/(4*b)

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Rubi [A]  time = 0.0034446, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{\left (a+\frac{b}{x}\right )^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3/x^2,x]

[Out]

-(a + b/x)^4/(4*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [B]  time = 0.0033146, size = 39, normalized size = 2.44 \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{x}-\frac{a b^2}{x^3}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^2,x]

[Out]

-b^3/(4*x^4) - (a*b^2)/x^3 - (3*a^2*b)/(2*x^2) - a^3/x

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Maple [B]  time = 0.005, size = 36, normalized size = 2.3 \begin{align*} -{\frac{{b}^{2}a}{{x}^{3}}}-{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}-{\frac{{a}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*b^2/x^3-1/4*b^3/x^4-3/2*a^2*b/x^2-a^3/x

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Maxima [A]  time = 0.963718, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x}\right )}^{4}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a + b/x)^4/b

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Fricas [B]  time = 1.42868, size = 73, normalized size = 4.56 \begin{align*} -\frac{4 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a b^{2} x + b^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a^3*x^3 + 6*a^2*b*x^2 + 4*a*b^2*x + b^3)/x^4

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Sympy [B]  time = 0.344411, size = 36, normalized size = 2.25 \begin{align*} - \frac{4 a^{3} x^{3} + 6 a^{2} b x^{2} + 4 a b^{2} x + b^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*a**3*x**3 + 6*a**2*b*x**2 + 4*a*b**2*x + b**3)/(4*x**4)

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Giac [A]  time = 1.16773, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x}\right )}^{4}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a + b/x)^4/b