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]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.01, size = 39, normalized size = 2.44 \[ -\frac {a^3}{x}-\frac {3 a^2 b}{2 x^2}-\frac {a b^2}{x^3}-\frac {b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^2,x]

[Out]

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

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fricas [B]  time = 1.07, size = 33, normalized size = 2.06 \[ -\frac {4 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a b^{2} x + b^{3}}{4 \, x^{4}} \]

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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giac [A]  time = 0.15, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x}\right )}^{4}}{4 \, b} \]

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

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maple [B]  time = 0.00, size = 36, normalized size = 2.25 \[ -\frac {a^{3}}{x}-\frac {3 a^{2} b}{2 x^{2}}-\frac {a \,b^{2}}{x^{3}}-\frac {b^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.02, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x}\right )}^{4}}{4 \, b} \]

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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mupad [B]  time = 0.03, size = 33, normalized size = 2.06 \[ -\frac {a^3\,x^3+\frac {3\,a^2\,b\,x^2}{2}+a\,b^2\,x+\frac {b^3}{4}}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^3/x^2,x)

[Out]

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

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sympy [B]  time = 0.22, size = 36, normalized size = 2.25 \[ \frac {- 4 a^{3} x^{3} - 6 a^{2} b x^{2} - 4 a b^{2} x - b^{3}}{4 x^{4}} \]

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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