3.1754 \(\int \sqrt{a+\frac{b}{x}} x^{3/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{5 a}-\frac{4 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{15 a^2} \]

[Out]

(-4*b*(a + b/x)^(3/2)*x^(3/2))/(15*a^2) + (2*(a + b/x)^(3/2)*x^(5/2))/(5*a)

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Rubi [A]  time = 0.0123544, antiderivative size = 48, 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} \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{5 a}-\frac{4 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{15 a^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]*x^(3/2),x]

[Out]

(-4*b*(a + b/x)^(3/2)*x^(3/2))/(15*a^2) + (2*(a + b/x)^(3/2)*x^(5/2))/(5*a)

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 \sqrt{a+\frac{b}{x}} x^{3/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{5/2}}{5 a}-\frac{(2 b) \int \sqrt{a+\frac{b}{x}} \sqrt{x} \, dx}{5 a}\\ &=-\frac{4 b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}{15 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{5/2}}{5 a}\\ \end{align*}

Mathematica [A]  time = 0.0116979, size = 36, normalized size = 0.75 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b) (3 a x-2 b)}{15 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]*x^(3/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)*(-2*b + 3*a*x))/(15*a^2)

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Maple [A]  time = 0.002, size = 33, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,ax-2\,b \right ) }{15\,{a}^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(a*x+b)*(3*a*x-2*b)*x^(1/2)*((a*x+b)/x)^(1/2)/a^2

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Maxima [A]  time = 0.954613, size = 47, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}} - 5 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b x^{\frac{3}{2}}\right )}}{15 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(a + b/x)^(5/2)*x^(5/2) - 5*(a + b/x)^(3/2)*b*x^(3/2))/a^2

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Fricas [A]  time = 1.46204, size = 86, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (3 \, a^{2} x^{2} + a b x - 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{15 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a^2*x^2 + a*b*x - 2*b^2)*sqrt(x)*sqrt((a*x + b)/x)/a^2

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Sympy [A]  time = 18.2156, size = 65, normalized size = 1.35 \begin{align*} \frac{2 \sqrt{b} x^{2} \sqrt{\frac{a x}{b} + 1}}{5} + \frac{2 b^{\frac{3}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a} - \frac{4 b^{\frac{5}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b)*x**2*sqrt(a*x/b + 1)/5 + 2*b**(3/2)*x*sqrt(a*x/b + 1)/(15*a) - 4*b**(5/2)*sqrt(a*x/b + 1)/(15*a**2)

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Giac [A]  time = 1.24995, size = 50, normalized size = 1.04 \begin{align*} \frac{2}{15} \,{\left (\frac{2 \, b^{\frac{5}{2}}}{a^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b}{a^{2}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*b^(5/2)/a^2 + (3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)/a^2)*sgn(x)