3.1579 \(\int \frac{(a+\frac{b}{x})^3}{x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{a (a x+b)^4}{20 b^2 x^4}-\frac{(a x+b)^4}{5 b x^5} \]

[Out]

-(b + a*x)^4/(5*b*x^5) + (a*(b + a*x)^4)/(20*b^2*x^4)

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Rubi [A]  time = 0.0071368, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ \frac{a (a x+b)^4}{20 b^2 x^4}-\frac{(a x+b)^4}{5 b x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3/x^3,x]

[Out]

-(b + a*x)^4/(5*b*x^5) + (a*(b + a*x)^4)/(20*b^2*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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^3}{x^3} \, dx &=\int \frac{(b+a x)^3}{x^6} \, dx\\ &=-\frac{(b+a x)^4}{5 b x^5}-\frac{a \int \frac{(b+a x)^3}{x^5} \, dx}{5 b}\\ &=-\frac{(b+a x)^4}{5 b x^5}+\frac{a (b+a x)^4}{20 b^2 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0057052, size = 41, normalized size = 1.14 \[ -\frac{a^2 b}{x^3}-\frac{a^3}{2 x^2}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 36, normalized size = 1. \begin{align*} -{\frac{{a}^{2}b}{{x}^{3}}}-{\frac{{b}^{3}}{5\,{x}^{5}}}-{\frac{3\,{b}^{2}a}{4\,{x}^{4}}}-{\frac{{a}^{3}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981638, size = 47, normalized size = 1.31 \begin{align*} -\frac{10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(10*a^3*x^3 + 20*a^2*b*x^2 + 15*a*b^2*x + 4*b^3)/x^5

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Fricas [A]  time = 1.40144, size = 81, normalized size = 2.25 \begin{align*} -\frac{10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(10*a^3*x^3 + 20*a^2*b*x^2 + 15*a*b^2*x + 4*b^3)/x^5

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Sympy [A]  time = 0.365585, size = 37, normalized size = 1.03 \begin{align*} - \frac{10 a^{3} x^{3} + 20 a^{2} b x^{2} + 15 a b^{2} x + 4 b^{3}}{20 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(10*a**3*x**3 + 20*a**2*b*x**2 + 15*a*b**2*x + 4*b**3)/(20*x**5)

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Giac [A]  time = 1.12576, size = 47, normalized size = 1.31 \begin{align*} -\frac{10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(10*a^3*x^3 + 20*a^2*b*x^2 + 15*a*b^2*x + 4*b^3)/x^5