3.518 \(\int \frac{\sqrt{2-b x}}{x^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{(2-b x)^{3/2}}{3 x^{3/2}} \]

[Out]

-(2 - b*x)^(3/2)/(3*x^(3/2))

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Rubi [A]  time = 0.0115024, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{(2-b x)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 - b*x]/x^(5/2),x]

[Out]

-(2 - b*x)^(3/2)/(3*x^(3/2))

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Rubi in Sympy [A]  time = 2.29331, size = 15, normalized size = 0.79 \[ - \frac{\left (- b x + 2\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-b*x + 2)**(3/2)/(3*x**(3/2))

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Mathematica [A]  time = 0.0160436, size = 19, normalized size = 1. \[ -\frac{(2-b x)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - b*x)^(3/2)/(3*x^(3/2))

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \[ -{\frac{1}{3} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(1/2)/x^(5/2),x)

[Out]

-1/3*(-b*x+2)^(3/2)/x^(3/2)

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Maxima [A]  time = 1.34258, size = 18, normalized size = 0.95 \[ -\frac{{\left (-b x + 2\right )}^{\frac{3}{2}}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 0.209256, size = 24, normalized size = 1.26 \[ \frac{{\left (b x - 2\right )} \sqrt{-b x + 2}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x - 2)*sqrt(-b*x + 2)/x^(3/2)

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Sympy [A]  time = 13.6694, size = 83, normalized size = 4.37 \[ \begin{cases} \frac{b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{3} - \frac{2 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{3 x} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\\frac{i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{3} - \frac{2 i \sqrt{b} \sqrt{1 - \frac{2}{b x}}}{3 x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((b**(3/2)*sqrt(-1 + 2/(b*x))/3 - 2*sqrt(b)*sqrt(-1 + 2/(b*x))/(3*x), 2
*Abs(1/(b*x)) > 1), (I*b**(3/2)*sqrt(1 - 2/(b*x))/3 - 2*I*sqrt(b)*sqrt(1 - 2/(b*
x))/(3*x), True))

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GIAC/XCAS [A]  time = 0.23504, size = 47, normalized size = 2.47 \[ \frac{{\left (b x - 2\right )} \sqrt{-b x + 2} b^{4}}{3 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{3}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x - 2)*sqrt(-b*x + 2)*b^4/(((b*x - 2)*b + 2*b)^(3/2)*abs(b))