3.644 \(\int \frac{1}{\sqrt{x} (2-b x)^{5/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{x}}{3 \sqrt{2-b x}}+\frac{\sqrt{x}}{3 (2-b x)^{3/2}} \]

[Out]

Sqrt[x]/(3*(2 - b*x)^(3/2)) + Sqrt[x]/(3*Sqrt[2 - b*x])

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Rubi [A]  time = 0.0233741, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{x}}{3 \sqrt{2-b x}}+\frac{\sqrt{x}}{3 (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(2 - b*x)^(5/2)),x]

[Out]

Sqrt[x]/(3*(2 - b*x)^(3/2)) + Sqrt[x]/(3*Sqrt[2 - b*x])

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Rubi in Sympy [A]  time = 3.71283, size = 29, normalized size = 0.74 \[ \frac{\sqrt{x}}{3 \sqrt{- b x + 2}} + \frac{\sqrt{x}}{3 \left (- b x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/(3*sqrt(-b*x + 2)) + sqrt(x)/(3*(-b*x + 2)**(3/2))

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Mathematica [A]  time = 0.0223988, size = 24, normalized size = 0.62 \[ -\frac{\sqrt{x} (b x-3)}{3 (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*(2 - b*x)^(5/2)),x]

[Out]

-(Sqrt[x]*(-3 + b*x))/(3*(2 - b*x)^(3/2))

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Maple [A]  time = 0.007, size = 19, normalized size = 0.5 \[ -{\frac{bx-3}{3}\sqrt{x} \left ( -bx+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x+2)^(5/2)/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.34482, size = 34, normalized size = 0.87 \[ \frac{{\left (b - \frac{3 \,{\left (b x - 2\right )}}{x}\right )} x^{\frac{3}{2}}}{6 \,{\left (-b x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + 2)^(5/2)*sqrt(x)),x, algorithm="maxima")

[Out]

1/6*(b - 3*(b*x - 2)/x)*x^(3/2)/(-b*x + 2)^(3/2)

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Fricas [A]  time = 0.228981, size = 39, normalized size = 1. \[ \frac{b x^{2} - 3 \, x}{3 \,{\left (b x - 2\right )} \sqrt{-b x + 2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + 2)^(5/2)*sqrt(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 23.8281, size = 165, normalized size = 4.23 \[ \begin{cases} \frac{b x}{3 b^{\frac{3}{2}} x \sqrt{-1 + \frac{2}{b x}} - 6 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}} - \frac{3}{3 b^{\frac{3}{2}} x \sqrt{-1 + \frac{2}{b x}} - 6 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\- \frac{i b^{2} x}{3 b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}} - 6 b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}} + \frac{3 i b}{3 b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}} - 6 b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((b*x/(3*b**(3/2)*x*sqrt(-1 + 2/(b*x)) - 6*sqrt(b)*sqrt(-1 + 2/(b*x)))
- 3/(3*b**(3/2)*x*sqrt(-1 + 2/(b*x)) - 6*sqrt(b)*sqrt(-1 + 2/(b*x))), 2*Abs(1/(b
*x)) > 1), (-I*b**2*x/(3*b**(5/2)*x*sqrt(1 - 2/(b*x)) - 6*b**(3/2)*sqrt(1 - 2/(b
*x))) + 3*I*b/(3*b**(5/2)*x*sqrt(1 - 2/(b*x)) - 6*b**(3/2)*sqrt(1 - 2/(b*x))), T
rue))

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GIAC/XCAS [A]  time = 0.210541, size = 122, normalized size = 3.13 \[ \frac{8 \,{\left (3 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt{-b} b^{2}}{3 \,{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + 2)^(5/2)*sqrt(x)),x, algorithm="giac")

[Out]

8/3*(3*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*sqrt(-b)*b^2
/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)^3*abs(b))