3.1715 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]

[Out]

(-2*(a + b/x)^(7/2))/(7*b)

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Rubi [A]  time = 0.0263247, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/x^2,x]

[Out]

(-2*(a + b/x)^(7/2))/(7*b)

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Rubi in Sympy [A]  time = 2.18791, size = 14, normalized size = 0.78 \[ - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a + b/x)**(7/2)/(7*b)

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Mathematica [A]  time = 0.024816, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b/x)^(7/2))/(7*b)

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Maple [A]  time = 0.009, size = 25, normalized size = 1.4 \[ -{\frac{2\,ax+2\,b}{7\,bx} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x^2,x)

[Out]

-2/7/x*(a*x+b)/b*((a*x+b)/x)^(5/2)

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Maxima [A]  time = 1.42586, size = 19, normalized size = 1.06 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*(a + b/x)^(7/2)/b

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Fricas [A]  time = 0.22674, size = 62, normalized size = 3.44 \[ -\frac{2 \,{\left (a^{3} x^{3} + 3 \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{7 \, b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*(a^3*x^3 + 3*a^2*b*x^2 + 3*a*b^2*x + b^3)*sqrt((a*x + b)/x)/(b*x^3)

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Sympy [A]  time = 10.6268, size = 80, normalized size = 4.44 \[ \begin{cases} - \frac{2 a^{3} \sqrt{a + \frac{b}{x}}}{7 b} - \frac{6 a^{2} \sqrt{a + \frac{b}{x}}}{7 x} - \frac{6 a b \sqrt{a + \frac{b}{x}}}{7 x^{2}} - \frac{2 b^{2} \sqrt{a + \frac{b}{x}}}{7 x^{3}} & \text{for}\: b \neq 0 \\- \frac{a^{\frac{5}{2}}}{x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a**3*sqrt(a + b/x)/(7*b) - 6*a**2*sqrt(a + b/x)/(7*x) - 6*a*b*sqrt
(a + b/x)/(7*x**2) - 2*b**2*sqrt(a + b/x)/(7*x**3), Ne(b, 0)), (-a**(5/2)/x, Tru
e))

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GIAC/XCAS [A]  time = 0.264229, size = 279, normalized size = 15.5 \[ \frac{2 \,{\left (7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3}{\rm sign}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b{\rm sign}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{2}{\rm sign}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{4}{\rm sign}\left (x\right ) + 7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{5}{\rm sign}\left (x\right ) + b^{6}{\rm sign}\left (x\right )\right )}}{7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(7*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sign(x) + 21*(sqrt(a)*x - sqrt(a*x^
2 + b*x))^5*a^(5/2)*b*sign(x) + 35*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^2*sig
n(x) + 35*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^3*sign(x) + 21*(sqrt(a)*x
- sqrt(a*x^2 + b*x))^2*a*b^4*sign(x) + 7*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)
*b^5*sign(x) + b^6*sign(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^7