3.1751 \(\int \sqrt{a+\frac{b}{x}} x^{5/2} \, dx\)

Optimal. Leaf size=74 \[ \frac{16 b^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{8 b x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{35 a^2}+\frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{7 a} \]

[Out]

(16*b^2*(a + b/x)^(3/2)*x^(3/2))/(105*a^3) - (8*b*(a + b/x)^(3/2)*x^(5/2))/(35*a
^2) + (2*(a + b/x)^(3/2)*x^(7/2))/(7*a)

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Rubi [A]  time = 0.0823326, 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 \[ \frac{16 b^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{8 b x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{35 a^2}+\frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{7 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*x^(5/2),x]

[Out]

(16*b^2*(a + b/x)^(3/2)*x^(3/2))/(105*a^3) - (8*b*(a + b/x)^(3/2)*x^(5/2))/(35*a
^2) + (2*(a + b/x)^(3/2)*x^(7/2))/(7*a)

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Rubi in Sympy [A]  time = 6.71071, size = 63, normalized size = 0.85 \[ \frac{2 x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{7 a} - \frac{8 b x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{35 a^{2}} + \frac{16 b^{2} x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{105 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(7/2)*(a + b/x)**(3/2)/(7*a) - 8*b*x**(5/2)*(a + b/x)**(3/2)/(35*a**2) + 16
*b**2*x**(3/2)*(a + b/x)**(3/2)/(105*a**3)

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Mathematica [A]  time = 0.038405, size = 53, normalized size = 0.72 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (15 a^3 x^3+3 a^2 b x^2-4 a b^2 x+8 b^3\right )}{105 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]*x^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(8*b^3 - 4*a*b^2*x + 3*a^2*b*x^2 + 15*a^3*x^3))/(105*a^
3)

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 15\,{a}^{2}{x}^{2}-12\,abx+8\,{b}^{2} \right ) }{105\,{a}^{3}}\sqrt{x}\sqrt{{\frac{ax+b}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)*x^(5/2),x)

[Out]

2/105*(a*x+b)*(15*a^2*x^2-12*a*b*x+8*b^2)*x^(1/2)*((a*x+b)/x)^(1/2)/a^3

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Maxima [A]  time = 1.42952, size = 70, normalized size = 0.95 \[ \frac{2 \,{\left (15 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 42 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}} + 35 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2} x^{\frac{3}{2}}\right )}}{105 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)*x^(5/2),x, algorithm="maxima")

[Out]

2/105*(15*(a + b/x)^(7/2)*x^(7/2) - 42*(a + b/x)^(5/2)*b*x^(5/2) + 35*(a + b/x)^
(3/2)*b^2*x^(3/2))/a^3

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Fricas [A]  time = 0.247368, size = 66, normalized size = 0.89 \[ \frac{2 \,{\left (15 \, a^{3} x^{3} + 3 \, a^{2} b x^{2} - 4 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{105 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)*x^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*a^3*x^3 + 3*a^2*b*x^2 - 4*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x)/a
^3

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235461, size = 68, normalized size = 0.92 \[ -\frac{2}{105} \,{\left (\frac{8 \, b^{\frac{7}{2}}}{a^{3}} - \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}}{a^{3}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)*x^(5/2),x, algorithm="giac")

[Out]

-2/105*(8*b^(7/2)/a^3 - (15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 35*(a*x + b
)^(3/2)*b^2)/a^3)*sign(x)