3.1800 \(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^{5/2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2}{3 a x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ -\frac {2}{3 a x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^(5/2)),x]

[Out]

-2/(3*a*(a + b/x)^(3/2)*x^(3/2))

Rule 264

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{5/2}} \, dx &=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 \[ -\frac {2}{3 a x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^(5/2)),x]

[Out]

-2/(3*a*(a + b/x)^(3/2)*x^(3/2))

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fricas [B]  time = 0.98, size = 38, normalized size = 1.65 \[ -\frac {2 \, \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{3} x^{2} + 2 \, a^{2} b x + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(5/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 21, normalized size = 0.91 \[ -\frac {2}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a} + \frac {2}{3 \, a b^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(5/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 25, normalized size = 1.09 \[ -\frac {2 \left (a x +b \right )}{3 \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} a \,x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^(5/2),x)

[Out]

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

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maxima [A]  time = 0.90, size = 17, normalized size = 0.74 \[ -\frac {2}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(5/2),x, algorithm="maxima")

[Out]

-2/3/((a + b/x)^(3/2)*a*x^(3/2))

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mupad [B]  time = 1.49, size = 38, normalized size = 1.65 \[ -\frac {2\,\sqrt {x}\,\sqrt {a+\frac {b}{x}}}{3\,\left (a^3\,x^2+2\,a^2\,b\,x+a\,b^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b/x)^(5/2)),x)

[Out]

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

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sympy [B]  time = 20.02, size = 42, normalized size = 1.83 \[ - \frac {2}{3 a^{2} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**(5/2),x)

[Out]

-2/(3*a**2*sqrt(b)*x*sqrt(a*x/b + 1) + 3*a*b**(3/2)*sqrt(a*x/b + 1))

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