3.1576 \(\int (a+\frac {b}{x})^3 \, dx\)

Optimal. Leaf size=33 \[ a^3 x+3 a^2 b \log (x)-\frac {3 a b^2}{x}-\frac {b^3}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {193, 43} \[ 3 a^2 b \log (x)+a^3 x-\frac {3 a b^2}{x}-\frac {b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3,x]

[Out]

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

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])

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 33, normalized size = 1.00 \[ a^3 x+3 a^2 b \log (x)-\frac {3 a b^2}{x}-\frac {b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3,x]

[Out]

-1/2*b^3/x^2 - (3*a*b^2)/x + a^3*x + 3*a^2*b*Log[x]

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fricas [A]  time = 0.91, size = 37, normalized size = 1.12 \[ \frac {2 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} \log \relax (x) - 6 \, a b^{2} x - b^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 31, normalized size = 0.94 \[ a^{3} x + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac {6 \, a b^{2} x + b^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 32, normalized size = 0.97 \[ a^{3} x +3 a^{2} b \ln \relax (x )-\frac {3 a \,b^{2}}{x}-\frac {b^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 31, normalized size = 0.94 \[ a^{3} x + 3 \, a^{2} b \log \relax (x) - \frac {3 \, a b^{2}}{x} - \frac {b^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 32, normalized size = 0.97 \[ a^3\,x-\frac {\frac {b^3}{2}+3\,a\,x\,b^2}{x^2}+3\,a^2\,b\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^3,x)

[Out]

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

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sympy [A]  time = 0.18, size = 32, normalized size = 0.97 \[ a^{3} x + 3 a^{2} b \log {\relax (x )} + \frac {- 6 a b^{2} x - b^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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