3.1737 \(\int \frac {1}{(a+\frac {b}{x})^{3/2} x^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^3),x]

[Out]

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

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 25, normalized size = 0.74 \[ -\frac {2 (2 a x+b)}{b^2 x \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*x^3),x]

[Out]

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

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fricas [A]  time = 0.62, size = 31, normalized size = 0.91 \[ -\frac {2 \, {\left (2 \, a x + b\right )} \sqrt {\frac {a x + b}{x}}}{a b^{2} x + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 30, normalized size = 0.88 \[ -\frac {2 \, {\left (\frac {a}{\sqrt {\frac {a x + b}{x}}} + \sqrt {\frac {a x + b}{x}}\right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^3,x, algorithm="giac")

[Out]

-2*(a/sqrt((a*x + b)/x) + sqrt((a*x + b)/x))/b^2

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maple [A]  time = 0.00, size = 31, normalized size = 0.91 \[ -\frac {2 \left (a x +b \right ) \left (2 a x +b \right )}{\left (\frac {a x +b}{x}\right )^{\frac {3}{2}} b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2)/x^3,x)

[Out]

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

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maxima [A]  time = 0.98, size = 30, normalized size = 0.88 \[ -\frac {2 \, \sqrt {a + \frac {b}{x}}}{b^{2}} - \frac {2 \, a}{\sqrt {a + \frac {b}{x}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^3,x, algorithm="maxima")

[Out]

-2*sqrt(a + b/x)/b^2 - 2*a/(sqrt(a + b/x)*b^2)

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mupad [B]  time = 1.26, size = 36, normalized size = 1.06 \[ -\frac {x\,\sqrt {a+\frac {b}{x}}\,\left (\frac {2}{b}+\frac {4\,a\,x}{b^2}\right )}{a\,x^2+b\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x)^(3/2)),x)

[Out]

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

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sympy [A]  time = 3.37, size = 42, normalized size = 1.24 \[ \begin {cases} - \frac {4 a}{b^{2} \sqrt {a + \frac {b}{x}}} - \frac {2}{b x \sqrt {a + \frac {b}{x}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {3}{2}} x^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2)/x**3,x)

[Out]

Piecewise((-4*a/(b**2*sqrt(a + b/x)) - 2/(b*x*sqrt(a + b/x)), Ne(b, 0)), (-1/(2*a**(3/2)*x**2), True))

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