3.1787 \(\int \frac{x^{3/2}}{(a+\frac{b}{x})^{3/2}} \, dx\)

Optimal. Leaf size=100 \[ \frac{32 b^3}{5 a^4 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{16 b^2 \sqrt{x}}{5 a^3 \sqrt{a+\frac{b}{x}}}-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}} \]

[Out]

(32*b^3)/(5*a^4*Sqrt[a + b/x]*Sqrt[x]) + (16*b^2*Sqrt[x])/(5*a^3*Sqrt[a + b/x]) - (4*b*x^(3/2))/(5*a^2*Sqrt[a
+ b/x]) + (2*x^(5/2))/(5*a*Sqrt[a + b/x])

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Rubi [A]  time = 0.0333058, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{32 b^3}{5 a^4 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{16 b^2 \sqrt{x}}{5 a^3 \sqrt{a+\frac{b}{x}}}-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a + b/x)^(3/2),x]

[Out]

(32*b^3)/(5*a^4*Sqrt[a + b/x]*Sqrt[x]) + (16*b^2*Sqrt[x])/(5*a^3*Sqrt[a + b/x]) - (4*b*x^(3/2))/(5*a^2*Sqrt[a
+ b/x]) + (2*x^(5/2))/(5*a*Sqrt[a + b/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{x^{3/2}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx &=\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}}-\frac{(6 b) \int \frac{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx}{5 a}\\ &=-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}}+\frac{\left (8 b^2\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}} \, dx}{5 a^2}\\ &=\frac{16 b^2 \sqrt{x}}{5 a^3 \sqrt{a+\frac{b}{x}}}-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}}-\frac{\left (16 b^3\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{3/2}} \, dx}{5 a^3}\\ &=\frac{32 b^3}{5 a^4 \sqrt{a+\frac{b}{x}} \sqrt{x}}+\frac{16 b^2 \sqrt{x}}{5 a^3 \sqrt{a+\frac{b}{x}}}-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0136722, size = 52, normalized size = 0.52 \[ \frac{2 \left (-2 a^2 b x^2+a^3 x^3+8 a b^2 x+16 b^3\right )}{5 a^4 \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(a + b/x)^(3/2),x]

[Out]

(2*(16*b^3 + 8*a*b^2*x - 2*a^2*b*x^2 + a^3*x^3))/(5*a^4*Sqrt[a + b/x]*Sqrt[x])

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Maple [A]  time = 0.004, size = 54, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{3}{x}^{3}-2\,{a}^{2}b{x}^{2}+8\,xa{b}^{2}+16\,{b}^{3} \right ) }{5\,{a}^{4}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a*x+b)*(a^3*x^3-2*a^2*b*x^2+8*a*b^2*x+16*b^3)/a^4/x^(3/2)/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 0.967283, size = 97, normalized size = 0.97 \begin{align*} \frac{2 \, b^{3}}{\sqrt{a + \frac{b}{x}} a^{4} \sqrt{x}} + \frac{2 \,{\left ({\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}} + 15 \, \sqrt{a + \frac{b}{x}} b^{2} \sqrt{x}\right )}}{5 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43338, size = 124, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} - 2 \, a^{2} b x^{2} + 8 \, a b^{2} x + 16 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{5 \,{\left (a^{5} x + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a^3*x^3 - 2*a^2*b*x^2 + 8*a*b^2*x + 16*b^3)*sqrt(x)*sqrt((a*x + b)/x)/(a^5*x + a^4*b)

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Sympy [B]  time = 14.6493, size = 320, normalized size = 3.2 \begin{align*} \frac{2 a^{5} b^{\frac{19}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{10 a^{3} b^{\frac{23}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{60 a^{2} b^{\frac{25}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{80 a b^{\frac{27}{2}} x \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{32 b^{\frac{29}{2}} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**5*b**(19/2)*x**5*sqrt(a*x/b + 1)/(5*a**7*b**9*x**3 + 15*a**6*b**10*x**2 + 15*a**5*b**11*x + 5*a**4*b**12)
+ 10*a**3*b**(23/2)*x**3*sqrt(a*x/b + 1)/(5*a**7*b**9*x**3 + 15*a**6*b**10*x**2 + 15*a**5*b**11*x + 5*a**4*b*
*12) + 60*a**2*b**(25/2)*x**2*sqrt(a*x/b + 1)/(5*a**7*b**9*x**3 + 15*a**6*b**10*x**2 + 15*a**5*b**11*x + 5*a**
4*b**12) + 80*a*b**(27/2)*x*sqrt(a*x/b + 1)/(5*a**7*b**9*x**3 + 15*a**6*b**10*x**2 + 15*a**5*b**11*x + 5*a**4*
b**12) + 32*b**(29/2)*sqrt(a*x/b + 1)/(5*a**7*b**9*x**3 + 15*a**6*b**10*x**2 + 15*a**5*b**11*x + 5*a**4*b**12)

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Giac [A]  time = 1.18531, size = 76, normalized size = 0.76 \begin{align*} -\frac{32 \, b^{\frac{5}{2}}}{5 \, a^{4}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x + b} b^{2} + \frac{5 \, b^{3}}{\sqrt{a x + b}}\right )}}{5 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/5*b^(5/2)/a^4 + 2/5*((a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b + 15*sqrt(a*x + b)*b^2 + 5*b^3/sqrt(a*x + b))/a
^4