3.1581 \(\int \frac{(a+\frac{b}{x})^3}{x^5} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}-\frac{a b^2}{2 x^6}-\frac{b^3}{7 x^7} \]

[Out]

-b^3/(7*x^7) - (a*b^2)/(2*x^6) - (3*a^2*b)/(5*x^5) - a^3/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0149061, antiderivative size = 43, 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} \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}-\frac{a b^2}{2 x^6}-\frac{b^3}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3/x^5,x]

[Out]

-b^3/(7*x^7) - (a*b^2)/(2*x^6) - (3*a^2*b)/(5*x^5) - a^3/(4*x^4)

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^5} \, dx &=\int \frac{(b+a x)^3}{x^8} \, dx\\ &=\int \left (\frac{b^3}{x^8}+\frac{3 a b^2}{x^7}+\frac{3 a^2 b}{x^6}+\frac{a^3}{x^5}\right ) \, dx\\ &=-\frac{b^3}{7 x^7}-\frac{a b^2}{2 x^6}-\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0032526, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}-\frac{a b^2}{2 x^6}-\frac{b^3}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^5,x]

[Out]

-b^3/(7*x^7) - (a*b^2)/(2*x^6) - (3*a^2*b)/(5*x^5) - a^3/(4*x^4)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{7\,{x}^{7}}}-{\frac{{b}^{2}a}{2\,{x}^{6}}}-{\frac{3\,{a}^{2}b}{5\,{x}^{5}}}-{\frac{{a}^{3}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^3/x^5,x)

[Out]

-1/7*b^3/x^7-1/2*a*b^2/x^6-3/5*a^2*b/x^5-1/4*a^3/x^4

________________________________________________________________________________________

Maxima [A]  time = 1.013, size = 47, normalized size = 1.09 \begin{align*} -\frac{35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^5,x, algorithm="maxima")

[Out]

-1/140*(35*a^3*x^3 + 84*a^2*b*x^2 + 70*a*b^2*x + 20*b^3)/x^7

________________________________________________________________________________________

Fricas [A]  time = 1.4386, size = 84, normalized size = 1.95 \begin{align*} -\frac{35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^5,x, algorithm="fricas")

[Out]

-1/140*(35*a^3*x^3 + 84*a^2*b*x^2 + 70*a*b^2*x + 20*b^3)/x^7

________________________________________________________________________________________

Sympy [A]  time = 0.393709, size = 37, normalized size = 0.86 \begin{align*} - \frac{35 a^{3} x^{3} + 84 a^{2} b x^{2} + 70 a b^{2} x + 20 b^{3}}{140 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**3/x**5,x)

[Out]

-(35*a**3*x**3 + 84*a**2*b*x**2 + 70*a*b**2*x + 20*b**3)/(140*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.21555, size = 47, normalized size = 1.09 \begin{align*} -\frac{35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^5,x, algorithm="giac")

[Out]

-1/140*(35*a^3*x^3 + 84*a^2*b*x^2 + 70*a*b^2*x + 20*b^3)/x^7