3.1779 \(\int \frac{x^{3/2}}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=74 \[ \frac{16 b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{15 a^3}-\frac{8 b x^{3/2} \sqrt{a+\frac{b}{x}}}{15 a^2}+\frac{2 x^{5/2} \sqrt{a+\frac{b}{x}}}{5 a} \]

[Out]

(16*b^2*Sqrt[a + b/x]*Sqrt[x])/(15*a^3) - (8*b*Sqrt[a + b/x]*x^(3/2))/(15*a^2) + (2*Sqrt[a + b/x]*x^(5/2))/(5*
a)

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Rubi [A]  time = 0.0210913, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{16 b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{15 a^3}-\frac{8 b x^{3/2} \sqrt{a+\frac{b}{x}}}{15 a^2}+\frac{2 x^{5/2} \sqrt{a+\frac{b}{x}}}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[a + b/x],x]

[Out]

(16*b^2*Sqrt[a + b/x]*Sqrt[x])/(15*a^3) - (8*b*Sqrt[a + b/x]*x^(3/2))/(15*a^2) + (2*Sqrt[a + b/x]*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 \frac{x^{3/2}}{\sqrt{a+\frac{b}{x}}} \, dx &=\frac{2 \sqrt{a+\frac{b}{x}} x^{5/2}}{5 a}-\frac{(4 b) \int \frac{\sqrt{x}}{\sqrt{a+\frac{b}{x}}} \, dx}{5 a}\\ &=-\frac{8 b \sqrt{a+\frac{b}{x}} x^{3/2}}{15 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{5/2}}{5 a}+\frac{\left (8 b^2\right ) \int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx}{15 a^2}\\ &=\frac{16 b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}}{15 a^3}-\frac{8 b \sqrt{a+\frac{b}{x}} x^{3/2}}{15 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{5/2}}{5 a}\\ \end{align*}

Mathematica [A]  time = 0.0183902, size = 42, normalized size = 0.57 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2-4 a b x+8 b^2\right )}{15 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[a + b/x],x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(8*b^2 - 4*a*b*x + 3*a^2*x^2))/(15*a^3)

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Maple [A]  time = 0.005, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,{a}^{2}{x}^{2}-4\,xab+8\,{b}^{2} \right ) }{15\,{a}^{3}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(a*x+b)*(3*a^2*x^2-4*a*b*x+8*b^2)/a^3/x^(1/2)/((a*x+b)/x)^(1/2)

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Maxima [A]  time = 0.975295, size = 70, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}} - 10 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b x^{\frac{3}{2}} + 15 \, \sqrt{a + \frac{b}{x}} b^{2} \sqrt{x}\right )}}{15 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(a + b/x)^(5/2)*x^(5/2) - 10*(a + b/x)^(3/2)*b*x^(3/2) + 15*sqrt(a + b/x)*b^2*sqrt(x))/a^3

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Fricas [A]  time = 1.45286, size = 89, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (3 \, a^{2} x^{2} - 4 \, a b x + 8 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{15 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a^2*x^2 - 4*a*b*x + 8*b^2)*sqrt(x)*sqrt((a*x + b)/x)/a^3

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Sympy [B]  time = 8.19572, size = 260, normalized size = 3.51 \begin{align*} \frac{6 a^{4} b^{\frac{9}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{4 a^{3} b^{\frac{11}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{6 a^{2} b^{\frac{13}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{24 a b^{\frac{15}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{16 b^{\frac{17}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*a**4*b**(9/2)*x**4*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**6) + 4*a**3*b**(11/2)*x*
*3*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**6) + 6*a**2*b**(13/2)*x**2*sqrt(a*x/b + 1)
/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**6) + 24*a*b**(15/2)*x*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 3
0*a**4*b**5*x + 15*a**3*b**6) + 16*b**(17/2)*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**
6)

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Giac [A]  time = 1.18205, size = 62, normalized size = 0.84 \begin{align*} -\frac{16 \, b^{\frac{5}{2}}}{15 \, a^{3}} + \frac{2 \,{\left (3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 10 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x + b} b^{2}\right )}}{15 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/15*b^(5/2)/a^3 + 2/15*(3*(a*x + b)^(5/2) - 10*(a*x + b)^(3/2)*b + 15*sqrt(a*x + b)*b^2)/a^3