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]

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 1.14 \[ -\frac {a^3}{2 x^2}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^3,x]

[Out]

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

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fricas [A]  time = 1.14, size = 35, normalized size = 0.97 \[ -\frac {10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \]

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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giac [A]  time = 0.15, size = 35, normalized size = 0.97 \[ -\frac {10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \]

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

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maple [A]  time = 0.00, size = 36, normalized size = 1.00 \[ -\frac {a^{3}}{2 x^{2}}-\frac {a^{2} b}{x^{3}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {b^{3}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.04, size = 35, normalized size = 0.97 \[ -\frac {10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \]

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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mupad [B]  time = 0.03, size = 34, normalized size = 0.94 \[ -\frac {\frac {a^3\,x^3}{2}+a^2\,b\,x^2+\frac {3\,a\,b^2\,x}{4}+\frac {b^3}{5}}{x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^3/x^3,x)

[Out]

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

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sympy [A]  time = 0.24, size = 37, normalized size = 1.03 \[ \frac {- 10 a^{3} x^{3} - 20 a^{2} b x^{2} - 15 a b^{2} x - 4 b^{3}}{20 x^{5}} \]

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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