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 = 0.005742, antiderivative size = 23, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0166424, size = 23, normalized size = 1. \[ -\frac{2}{3 a x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

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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Maple [A]  time = 0.001, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,ax+2\,b}{3\,a} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}

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 = 0.973414, size = 23, normalized size = 1. \begin{align*} -\frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a x^{\frac{3}{2}}} \end{align*}

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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Fricas [B]  time = 1.49085, size = 84, normalized size = 3.65 \begin{align*} -\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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.16383, size = 28, normalized size = 1.22 \begin{align*} -\frac{2}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} a} + \frac{2}{3 \, a b^{\frac{3}{2}}} \end{align*}

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))