3.637 \(\int \frac{\sqrt{x}}{(2-b x)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0347025, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(2 - b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.84384, size = 41, normalized size = 0.91 \[ \frac{2 \sqrt{x}}{b \sqrt{- b x + 2}} - \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(-b*x+2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0628981, size = 45, normalized size = 1. \[ \frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(2 - b*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.046, size = 67, normalized size = 1.5 \[ -2\,{\frac{1}{\sqrt{-b}\sqrt{\pi }b} \left ( 1/2\,{\frac{\sqrt{\pi }\sqrt{x}\sqrt{2} \left ( -b \right ) ^{3/2}}{b\sqrt{-1/2\,bx+1}}}-{\frac{\sqrt{\pi } \left ( -b \right ) ^{3/2}\arcsin \left ( 1/2\,\sqrt{b}\sqrt{x}\sqrt{2} \right ) }{{b}^{3/2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(-b*x+2)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(-b*x + 2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220175, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-b x + 2} \sqrt{x} \log \left (\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right ) + 2 \, \sqrt{-b} x}{\sqrt{-b x + 2} \sqrt{-b} b \sqrt{x}}, \frac{2 \,{\left (\sqrt{-b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) + \sqrt{b} x\right )}}{\sqrt{-b x + 2} b^{\frac{3}{2}} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(-b*x + 2)^(3/2),x, algorithm="fricas")

[Out]

[(sqrt(-b*x + 2)*sqrt(x)*log(sqrt(-b*x + 2)*b*sqrt(x) - (b*x - 1)*sqrt(-b)) + 2*
sqrt(-b)*x)/(sqrt(-b*x + 2)*sqrt(-b)*b*sqrt(x)), 2*(sqrt(-b*x + 2)*sqrt(x)*arcta
n(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) + sqrt(b)*x)/(sqrt(-b*x + 2)*b^(3/2)*sqrt(x)
)]

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Sympy [A]  time = 6.26537, size = 92, normalized size = 2.04 \[ \begin{cases} - \frac{2 i \sqrt{x}}{b \sqrt{b x - 2}} + \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{2 \sqrt{x}}{b \sqrt{- b x + 2}} - \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(-b*x+2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218181, size = 124, normalized size = 2.76 \[ -\frac{{\left (\frac{{\rm ln}\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt{-b}} + \frac{8 \, \sqrt{-b}}{{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b}\right )}{\left | b \right |}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(-b*x + 2)^(3/2),x, algorithm="giac")

[Out]

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