3.1776 \(\int \frac{x^{5/2}}{\sqrt{a+\frac{b}{x}}} \, dx\)

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

[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 = 0.114401, 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 \[ -\frac{32 b^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}{35 a^4}+\frac{16 b^2 x^{3/2} \sqrt{a+\frac{b}{x}}}{35 a^3}-\frac{12 b x^{5/2} \sqrt{a+\frac{b}{x}}}{35 a^2}+\frac{2 x^{7/2} \sqrt{a+\frac{b}{x}}}{7 a} \]

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)

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Rubi in Sympy [A]  time = 9.84751, size = 87, normalized size = 0.87 \[ \frac{2 x^{\frac{7}{2}} \sqrt{a + \frac{b}{x}}}{7 a} - \frac{12 b x^{\frac{5}{2}} \sqrt{a + \frac{b}{x}}}{35 a^{2}} + \frac{16 b^{2} x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}}{35 a^{3}} - \frac{32 b^{3} \sqrt{x} \sqrt{a + \frac{b}{x}}}{35 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(a+b/x)**(1/2),x)

[Out]

2*x**(7/2)*sqrt(a + b/x)/(7*a) - 12*b*x**(5/2)*sqrt(a + b/x)/(35*a**2) + 16*b**2
*x**(3/2)*sqrt(a + b/x)/(35*a**3) - 32*b**3*sqrt(x)*sqrt(a + b/x)/(35*a**4)

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Mathematica [A]  time = 0.0445848, size = 53, normalized size = 0.53 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (5 a^3 x^3-6 a^2 b x^2+8 a b^2 x-16 b^3\right )}{35 a^4} \]

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 = 0.007, size = 55, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 5\,{a}^{3}{x}^{3}-6\,{a}^{2}b{x}^{2}+8\,a{b}^{2}x-16\,{b}^{3} \right ) }{35\,{a}^{4}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

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 = 1.44442, size = 93, normalized size = 0.93 \[ \frac{2 \,{\left (5 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 21 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}} + 35 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2} x^{\frac{3}{2}} - 35 \, \sqrt{a + \frac{b}{x}} b^{3} \sqrt{x}\right )}}{35 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(a + b/x),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 = 0.240728, size = 66, normalized size = 0.66 \[ \frac{2 \,{\left (5 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 8 \, a b^{2} x - 16 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{35 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(a + b/x),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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232322, size = 78, normalized size = 0.78 \[ \frac{32 \, b^{\frac{7}{2}}}{35 \, a^{4}} + \frac{2 \,{\left (5 \,{\left (a x + b\right )}^{\frac{7}{2}} - 21 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} - 35 \, \sqrt{a x + b} b^{3}\right )}}{35 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(a + b/x),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