3.1706 \(\int \frac{(a+\frac{b}{x})^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

[Out]

(-2*(a + b/x)^(5/2))/(5*b)

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Rubi [A]  time = 0.0059618, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)/x^2,x]

[Out]

(-2*(a + b/x)^(5/2))/(5*b)

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0085436, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)/x^2,x]

[Out]

(-2*(a + b/x)^(5/2))/(5*b)

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Maple [A]  time = 0.003, size = 25, normalized size = 1.4 \begin{align*} -{\frac{2\,ax+2\,b}{5\,bx} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.38087, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(a + b/x)^(5/2)/b

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Fricas [B]  time = 1.73605, size = 78, normalized size = 4.33 \begin{align*} -\frac{2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{5 \, b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.998756, size = 65, normalized size = 3.61 \begin{align*} - \frac{2 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}}{5 b} - \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{5 x} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{5 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**(5/2)*sqrt(1 + b/(a*x))/(5*b) - 4*a**(3/2)*sqrt(1 + b/(a*x))/(5*x) - 2*sqrt(a)*b*sqrt(1 + b/(a*x))/(5*x*
*2)

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Giac [B]  time = 1.17028, size = 196, normalized size = 10.89 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{2} \mathrm{sgn}\left (x\right ) + 5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{3} \mathrm{sgn}\left (x\right ) + b^{4} \mathrm{sgn}\left (x\right )\right )}}{5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(5*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*sgn(x) + 10*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b*sgn(x) +
10*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2*sgn(x) + 5*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*sgn(x) + b^4
*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^5