3.1778 \(\int \frac{x^{5/2}}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=100 \[ \frac{16 b^2 x^{3/2} \sqrt{a+\frac{b}{x}}}{35 a^3}-\frac{32 b^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}{35 a^4}-\frac{12 b x^{5/2} \sqrt{a+\frac{b}{x}}}{35 a^2}+\frac{2 x^{7/2} \sqrt{a+\frac{b}{x}}}{7 a} \]

[Out]

(-32*b^3*Sqrt[a + b/x]*Sqrt[x])/(35*a^4) + (16*b^2*Sqrt[a + b/x]*x^(3/2))/(35*a^3) - (12*b*Sqrt[a + b/x]*x^(5/
2))/(35*a^2) + (2*Sqrt[a + b/x]*x^(7/2))/(7*a)

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Rubi [A]  time = 0.0312053, 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} \[ \frac{16 b^2 x^{3/2} \sqrt{a+\frac{b}{x}}}{35 a^3}-\frac{32 b^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}{35 a^4}-\frac{12 b x^{5/2} \sqrt{a+\frac{b}{x}}}{35 a^2}+\frac{2 x^{7/2} \sqrt{a+\frac{b}{x}}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/Sqrt[a + b/x],x]

[Out]

(-32*b^3*Sqrt[a + b/x]*Sqrt[x])/(35*a^4) + (16*b^2*Sqrt[a + b/x]*x^(3/2))/(35*a^3) - (12*b*Sqrt[a + b/x]*x^(5/
2))/(35*a^2) + (2*Sqrt[a + b/x]*x^(7/2))/(7*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^{5/2}}{\sqrt{a+\frac{b}{x}}} \, dx &=\frac{2 \sqrt{a+\frac{b}{x}} x^{7/2}}{7 a}-\frac{(6 b) \int \frac{x^{3/2}}{\sqrt{a+\frac{b}{x}}} \, dx}{7 a}\\ &=-\frac{12 b \sqrt{a+\frac{b}{x}} x^{5/2}}{35 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{7/2}}{7 a}+\frac{\left (24 b^2\right ) \int \frac{\sqrt{x}}{\sqrt{a+\frac{b}{x}}} \, dx}{35 a^2}\\ &=\frac{16 b^2 \sqrt{a+\frac{b}{x}} x^{3/2}}{35 a^3}-\frac{12 b \sqrt{a+\frac{b}{x}} x^{5/2}}{35 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{7/2}}{7 a}-\frac{\left (16 b^3\right ) \int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx}{35 a^3}\\ &=-\frac{32 b^3 \sqrt{a+\frac{b}{x}} \sqrt{x}}{35 a^4}+\frac{16 b^2 \sqrt{a+\frac{b}{x}} x^{3/2}}{35 a^3}-\frac{12 b \sqrt{a+\frac{b}{x}} x^{5/2}}{35 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{7/2}}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0216306, size = 53, normalized size = 0.53 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (-6 a^2 b x^2+5 a^3 x^3+8 a b^2 x-16 b^3\right )}{35 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/Sqrt[a + b/x],x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-16*b^3 + 8*a*b^2*x - 6*a^2*b*x^2 + 5*a^3*x^3))/(35*a^4)

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Maple [A]  time = 0.004, size = 55, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 5\,{a}^{3}{x}^{3}-6\,{a}^{2}b{x}^{2}+8\,xa{b}^{2}-16\,{b}^{3} \right ) }{35\,{a}^{4}}{\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^(5/2)/(a+b/x)^(1/2),x)

[Out]

2/35*(a*x+b)*(5*a^3*x^3-6*a^2*b*x^2+8*a*b^2*x-16*b^3)/a^4/x^(1/2)/((a*x+b)/x)^(1/2)

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Maxima [A]  time = 0.9635, size = 93, normalized size = 0.93 \begin{align*} \frac{2 \,{\left (5 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 21 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}} + 35 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2} x^{\frac{3}{2}} - 35 \, \sqrt{a + \frac{b}{x}} b^{3} \sqrt{x}\right )}}{35 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(a+b/x)^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*(a + b/x)^(7/2)*x^(7/2) - 21*(a + b/x)^(5/2)*b*x^(5/2) + 35*(a + b/x)^(3/2)*b^2*x^(3/2) - 35*sqrt(a +
b/x)*b^3*sqrt(x))/a^4

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Fricas [A]  time = 1.43166, size = 112, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (5 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 8 \, a b^{2} x - 16 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{35 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(a+b/x)^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*a^3*x^3 - 6*a^2*b*x^2 + 8*a*b^2*x - 16*b^3)*sqrt(x)*sqrt((a*x + b)/x)/a^4

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Sympy [B]  time = 68.4863, size = 452, normalized size = 4.52 \begin{align*} \frac{10 a^{6} b^{\frac{19}{2}} x^{6} \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} + \frac{18 a^{5} b^{\frac{21}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} + \frac{10 a^{4} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} - \frac{10 a^{3} b^{\frac{25}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} - \frac{60 a^{2} b^{\frac{27}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} - \frac{80 a b^{\frac{29}{2}} x \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} - \frac{32 b^{\frac{31}{2}} \sqrt{\frac{a x}{b} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{2} + 105 a^{5} b^{11} x + 35 a^{4} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(a+b/x)**(1/2),x)

[Out]

10*a**6*b**(19/2)*x**6*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**2 + 105*a**5*b**11*x + 35*a**4*b
**12) + 18*a**5*b**(21/2)*x**5*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**2 + 105*a**5*b**11*x + 3
5*a**4*b**12) + 10*a**4*b**(23/2)*x**4*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**2 + 105*a**5*b**
11*x + 35*a**4*b**12) - 10*a**3*b**(25/2)*x**3*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**2 + 105*
a**5*b**11*x + 35*a**4*b**12) - 60*a**2*b**(27/2)*x**2*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**
2 + 105*a**5*b**11*x + 35*a**4*b**12) - 80*a*b**(29/2)*x*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x
**2 + 105*a**5*b**11*x + 35*a**4*b**12) - 32*b**(31/2)*sqrt(a*x/b + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**
2 + 105*a**5*b**11*x + 35*a**4*b**12)

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Giac [A]  time = 1.16846, size = 78, normalized size = 0.78 \begin{align*} \frac{32 \, b^{\frac{7}{2}}}{35 \, a^{4}} + \frac{2 \,{\left (5 \,{\left (a x + b\right )}^{\frac{7}{2}} - 21 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} - 35 \, \sqrt{a x + b} b^{3}\right )}}{35 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(a+b/x)^(1/2),x, algorithm="giac")

[Out]

32/35*b^(7/2)/a^4 + 2/35*(5*(a*x + b)^(7/2) - 21*(a*x + b)^(5/2)*b + 35*(a*x + b)^(3/2)*b^2 - 35*sqrt(a*x + b)
*b^3)/a^4