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

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

[Out]

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

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Rubi [A]  time = 0.0125911, antiderivative size = 33, normalized size of antiderivative = 1., 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 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]

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 \, 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*}

Mathematica [A]  time = 0.0034689, size = 33, normalized size = 1. \[ 3 a^2 b \log (x)+a^3 x-\frac{3 a b^2}{x}-\frac{b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]  time = 0.005, size = 32, normalized size = 1. \begin{align*} -{\frac{{b}^{3}}{2\,{x}^{2}}}-3\,{\frac{{b}^{2}a}{x}}+{a}^{3}x+3\,{a}^{2}b\ln \left ( x \right ) \end{align*}

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 = 0.999899, size = 42, normalized size = 1.27 \begin{align*} a^{3} x + 3 \, a^{2} b \log \left (x\right ) - \frac{3 \, a b^{2}}{x} - \frac{b^{3}}{2 \, x^{2}} \end{align*}

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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Fricas [A]  time = 1.41766, size = 81, normalized size = 2.45 \begin{align*} \frac{2 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} \log \left (x\right ) - 6 \, a b^{2} x - b^{3}}{2 \, x^{2}} \end{align*}

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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Sympy [A]  time = 0.311338, size = 31, normalized size = 0.94 \begin{align*} a^{3} x + 3 a^{2} b \log{\left (x \right )} - \frac{6 a b^{2} x + b^{3}}{2 x^{2}} \end{align*}

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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Giac [A]  time = 1.20756, size = 42, normalized size = 1.27 \begin{align*} a^{3} x + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac{6 \, a b^{2} x + b^{3}}{2 \, x^{2}} \end{align*}

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