3.1799 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{4 b}{3 a^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{2}{a \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(-4*b)/(3*a^2*(a + b/x)^(3/2)*x^(3/2)) - 2/(a*(a + b/x)^(3/2)*Sqrt[x])

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Rubi [A]  time = 0.0132287, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ -\frac{4 b}{3 a^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{2}{a \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^(3/2)),x]

[Out]

(-4*b)/(3*a^2*(a + b/x)^(3/2)*x^(3/2)) - 2/(a*(a + b/x)^(3/2)*Sqrt[x])

Rule 271

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

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^{3/2}} \, dx &=-\frac{2}{a \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}+\frac{(2 b) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{5/2}} \, dx}{a}\\ &=-\frac{4 b}{3 a^2 \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}-\frac{2}{a \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0252999, size = 38, normalized size = 0.83 \[ -\frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (3 a x+2 b)}{3 a^2 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 33, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,ax+2\,b \right ) }{3\,{a}^{2}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.941047, size = 42, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )} x - b\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2} 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^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45885, size = 105, normalized size = 2.28 \begin{align*} -\frac{2 \,{\left (3 \, a x + 2 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 92.1562, size = 94, normalized size = 2.04 \begin{align*} - \frac{6 a x}{3 a^{3} \sqrt{b} x \sqrt{\frac{a x}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{4 b}{3 a^{3} \sqrt{b} x \sqrt{\frac{a x}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**(3/2),x)

[Out]

-6*a*x/(3*a**3*sqrt(b)*x*sqrt(a*x/b + 1) + 3*a**2*b**(3/2)*sqrt(a*x/b + 1)) - 4*b/(3*a**3*sqrt(b)*x*sqrt(a*x/b
+ 1) + 3*a**2*b**(3/2)*sqrt(a*x/b + 1))

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Giac [A]  time = 1.22525, size = 39, normalized size = 0.85 \begin{align*} -\frac{2 \,{\left (3 \, a x + 2 \, b\right )}}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} a^{2}} + \frac{4}{3 \, a^{2} \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(3/2),x, algorithm="giac")

[Out]

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