3.1777 \(\int \frac{x^{3/2}}{\sqrt{a+\frac{b}{x}}} \, dx\)

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 = 0.0822299, 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 \[ \frac{16 b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{15 a^3}-\frac{8 b x^{3/2} \sqrt{a+\frac{b}{x}}}{15 a^2}+\frac{2 x^{5/2} \sqrt{a+\frac{b}{x}}}{5 a} \]

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)

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Rubi in Sympy [A]  time = 6.8025, size = 63, normalized size = 0.85 \[ \frac{2 x^{\frac{5}{2}} \sqrt{a + \frac{b}{x}}}{5 a} - \frac{8 b x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}}{15 a^{2}} + \frac{16 b^{2} \sqrt{x} \sqrt{a + \frac{b}{x}}}{15 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(a+b/x)**(1/2),x)

[Out]

2*x**(5/2)*sqrt(a + b/x)/(5*a) - 8*b*x**(3/2)*sqrt(a + b/x)/(15*a**2) + 16*b**2*
sqrt(x)*sqrt(a + b/x)/(15*a**3)

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Mathematica [A]  time = 0.0412756, 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 = 0.007, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,{a}^{2}{x}^{2}-4\,abx+8\,{b}^{2} \right ) }{15\,{a}^{3}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

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 = 1.43946, size = 70, normalized size = 0.95 \[ \frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}} - 10 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b x^{\frac{3}{2}} + 15 \, \sqrt{a + \frac{b}{x}} b^{2} \sqrt{x}\right )}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(a + b/x),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 = 0.239447, size = 51, normalized size = 0.69 \[ \frac{2 \,{\left (3 \, a^{2} x^{2} - 4 \, a b x + 8 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(a + b/x),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 [A]  time = 43.8497, size = 260, normalized size = 3.51 \[ \frac{6 a^{4} b^{\frac{9}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{4 a^{3} b^{\frac{11}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{6 a^{2} b^{\frac{13}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{24 a b^{\frac{15}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac{16 b^{\frac{17}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} \]

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 + 30*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/XCAS [A]  time = 0.228059, size = 62, normalized size = 0.84 \[ -\frac{16 \, b^{\frac{5}{2}}}{15 \, a^{3}} + \frac{2 \,{\left (3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 10 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x + b} b^{2}\right )}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(a + b/x),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