3.1763 \(\int (a+\frac{b}{x})^{3/2} x^{3/2} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{5 a} \]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

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Rubi [A]  time = 0.0056947, 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} \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*x^(3/2),x]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

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

Mathematica [A]  time = 0.0134587, size = 23, normalized size = 1. \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{5 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*x^(3/2),x]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956766, size = 23, normalized size = 1. \begin{align*} \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}}}{5 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a + b/x)^(5/2)*x^(5/2)/a

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Fricas [B]  time = 1.45245, size = 80, normalized size = 3.48 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{5 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 46.8789, size = 63, normalized size = 2.74 \begin{align*} \frac{2 a \sqrt{b} x^{2} \sqrt{\frac{a x}{b} + 1}}{5} + \frac{4 b^{\frac{3}{2}} x \sqrt{\frac{a x}{b} + 1}}{5} + \frac{2 b^{\frac{5}{2}} \sqrt{\frac{a x}{b} + 1}}{5 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17474, size = 16, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (a x + b\right )}^{\frac{5}{2}}}{5 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a*x + b)^(5/2)/a