3.1577 \(\int \frac{(a+\frac{b}{x})^3}{x} \, dx\)

Optimal. Leaf size=37 \[ -\frac{3 a^2 b}{x}+a^3 \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{3 x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.014899, antiderivative size = 37, 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} \[ -\frac{3 a^2 b}{x}+a^3 \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3/x,x]

[Out]

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

Mathematica [A]  time = 0.0035974, size = 37, normalized size = 1. \[ -\frac{3 a^2 b}{x}+a^3 \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{3\,{x}^{3}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}b}{x}}+{a}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.963716, size = 46, normalized size = 1.24 \begin{align*} a^{3} \log \left (x\right ) - \frac{18 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*log(x) - 1/6*(18*a^2*b*x^2 + 9*a*b^2*x + 2*b^3)/x^3

________________________________________________________________________________________

Fricas [A]  time = 1.45002, size = 85, normalized size = 2.3 \begin{align*} \frac{6 \, a^{3} x^{3} \log \left (x\right ) - 18 \, a^{2} b x^{2} - 9 \, a b^{2} x - 2 \, b^{3}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*a^3*x^3*log(x) - 18*a^2*b*x^2 - 9*a*b^2*x - 2*b^3)/x^3

________________________________________________________________________________________

Sympy [A]  time = 0.336971, size = 34, normalized size = 0.92 \begin{align*} a^{3} \log{\left (x \right )} - \frac{18 a^{2} b x^{2} + 9 a b^{2} x + 2 b^{3}}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x) - (18*a**2*b*x**2 + 9*a*b**2*x + 2*b**3)/(6*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.58979, size = 47, normalized size = 1.27 \begin{align*} a^{3} \log \left ({\left | x \right |}\right ) - \frac{18 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*log(abs(x)) - 1/6*(18*a^2*b*x^2 + 9*a*b^2*x + 2*b^3)/x^3