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

Optimal. Leaf size=60 \[ -\frac{2 (2-b x)^{3/2}}{\sqrt{x}}-3 b \sqrt{x} \sqrt{2-b x}-6 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

[Out]

-3*b*Sqrt[x]*Sqrt[2 - b*x] - (2*(2 - b*x)^(3/2))/Sqrt[x] - 6*Sqrt[b]*ArcSin[(Sqr
t[b]*Sqrt[x])/Sqrt[2]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0435846, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 (2-b x)^{3/2}}{\sqrt{x}}-3 b \sqrt{x} \sqrt{2-b x}-6 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 - b*x)^(3/2)/x^(3/2),x]

[Out]

-3*b*Sqrt[x]*Sqrt[2 - b*x] - (2*(2 - b*x)^(3/2))/Sqrt[x] - 6*Sqrt[b]*ArcSin[(Sqr
t[b]*Sqrt[x])/Sqrt[2]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.14736, size = 58, normalized size = 0.97 \[ - 6 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - 3 b \sqrt{x} \sqrt{- b x + 2} - \frac{2 \left (- b x + 2\right )^{\frac{3}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*sqrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) - 3*b*sqrt(x)*sqrt(-b*x + 2) - 2*(-b*
x + 2)**(3/2)/sqrt(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0461966, size = 47, normalized size = 0.78 \[ -\frac{\sqrt{2-b x} (b x+4)}{\sqrt{x}}-6 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2 - b*x]*(4 + b*x))/Sqrt[x]) - 6*Sqrt[b]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2
]]

_______________________________________________________________________________________

Maple [B]  time = 0.029, size = 97, normalized size = 1.6 \[{({b}^{2}{x}^{2}+2\,bx-8)\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}}-3\,{\frac{\sqrt{b}\sqrt{ \left ( -bx+2 \right ) x}}{\sqrt{x}\sqrt{-bx+2}}\arctan \left ({\frac{\sqrt{b}}{\sqrt{-b{x}^{2}+2\,x}} \left ( x-{b}^{-1} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(3/2)/x^(3/2),x)

[Out]

(b^2*x^2+2*b*x-8)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/x^(1/2)/(-b*x+2)^(1/2)-3
*b^(1/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))*((-b*x+2)*x)^(1/2)/x^(1/2)/(
-b*x+2)^(1/2)

_______________________________________________________________________________________

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((-b*x + 2)^(3/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.247444, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{-b} x \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) -{\left (b x + 4\right )} \sqrt{-b x + 2} \sqrt{x}}{x}, \frac{6 \, \sqrt{b} x \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b x + 4\right )} \sqrt{-b x + 2} \sqrt{x}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(3*sqrt(-b)*x*log(-b*x + sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1) - (b*x + 4)*sqrt(
-b*x + 2)*sqrt(x))/x, (6*sqrt(b)*x*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - (b
*x + 4)*sqrt(-b*x + 2)*sqrt(x))/x]

_______________________________________________________________________________________

Sympy [A]  time = 12.9476, size = 160, normalized size = 2.67 \[ \begin{cases} 6 i \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{i b^{2} x^{\frac{3}{2}}}{\sqrt{b x - 2}} - \frac{2 i b \sqrt{x}}{\sqrt{b x - 2}} + \frac{8 i}{\sqrt{x} \sqrt{b x - 2}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- 6 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{- b x + 2}} + \frac{2 b \sqrt{x}}{\sqrt{- b x + 2}} - \frac{8}{\sqrt{x} \sqrt{- b x + 2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((6*I*sqrt(b)*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2) - I*b**2*x**(3/2)/sqrt(b
*x - 2) - 2*I*b*sqrt(x)/sqrt(b*x - 2) + 8*I/(sqrt(x)*sqrt(b*x - 2)), Abs(b*x)/2
> 1), (-6*sqrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) + b**2*x**(3/2)/sqrt(-b*x + 2)
+ 2*b*sqrt(x)/sqrt(-b*x + 2) - 8/(sqrt(x)*sqrt(-b*x + 2)), True))

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError