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

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

[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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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])

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^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*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ -\frac {a^3}{4 x^4}-\frac {3 a^2 b}{5 x^5}-\frac {a b^2}{2 x^6}-\frac {b^3}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^5,x]

[Out]

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

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fricas [A]  time = 0.62, size = 35, normalized size = 0.81 \[ -\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \]

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

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giac [A]  time = 0.15, size = 35, normalized size = 0.81 \[ -\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \]

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

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ -\frac {a^{3}}{4 x^{4}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {b^{3}}{7 x^{7}} \]

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

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maxima [A]  time = 1.03, size = 35, normalized size = 0.81 \[ -\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \]

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

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mupad [B]  time = 0.03, size = 35, normalized size = 0.81 \[ -\frac {\frac {a^3\,x^3}{4}+\frac {3\,a^2\,b\,x^2}{5}+\frac {a\,b^2\,x}{2}+\frac {b^3}{7}}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^3/x^5,x)

[Out]

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

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sympy [A]  time = 0.28, size = 37, normalized size = 0.86 \[ \frac {- 35 a^{3} x^{3} - 84 a^{2} b x^{2} - 70 a b^{2} x - 20 b^{3}}{140 x^{7}} \]

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)

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