3.544 \(\int \frac{(2-b x)^{3/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=62 \[ 2 b^{3/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 (2-b x)^{3/2}}{3 x^{3/2}}+\frac{2 b \sqrt{2-b x}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0473607, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ 2 b^{3/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 (2-b x)^{3/2}}{3 x^{3/2}}+\frac{2 b \sqrt{2-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - b*x)^(3/2)/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.45197, size = 58, normalized size = 0.94 \[ 2 b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{2 b \sqrt{- b x + 2}}{\sqrt{x}} - \frac{2 \left (- b x + 2\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x+2)**(3/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0486589, size = 50, normalized size = 0.81 \[ 2 b^{3/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+\frac{4 \sqrt{2-b x} (2 b x-1)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - b*x)^(3/2)/x^(5/2),x]

[Out]

(4*Sqrt[2 - b*x]*(-1 + 2*b*x))/(3*x^(3/2)) + 2*b^(3/2)*ArcSin[(Sqrt[b]*Sqrt[x])/
Sqrt[2]]

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Maple [B]  time = 0.028, size = 98, normalized size = 1.6 \[ -{\frac{8\,{b}^{2}{x}^{2}-20\,bx+8}{3}\sqrt{ \left ( -bx+2 \right ) x}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}+{1{b}^{{\frac{3}{2}}}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(3/2)/x^(5/2),x)

[Out]

-4/3*(2*b^2*x^2-5*b*x+2)/x^(3/2)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/(-b*x+2)^
(1/2)+b^(3/2)*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)

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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((-b*x + 2)^(3/2)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220058, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{-b} b x^{2} \log \left (-b x - \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) + 4 \,{\left (2 \, b x - 1\right )} \sqrt{-b x + 2} \sqrt{x}}{3 \, x^{2}}, -\frac{2 \,{\left (3 \, b^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) - 2 \,{\left (2 \, b x - 1\right )} \sqrt{-b x + 2} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x + 2)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 27.2035, size = 184, normalized size = 2.97 \[ \begin{cases} \frac{8 b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{3} + i b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} - 2 i b^{\frac{3}{2}} \log{\left (\frac{1}{\sqrt{b} \sqrt{x}} \right )} + 2 b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{4 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{3 x} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\\frac{8 i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{3} + i b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} - 2 i b^{\frac{3}{2}} \log{\left (\sqrt{1 - \frac{2}{b x}} + 1 \right )} - \frac{4 i \sqrt{b} \sqrt{1 - \frac{2}{b x}}}{3 x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x+2)**(3/2)/x**(5/2),x)

[Out]

Piecewise((8*b**(3/2)*sqrt(-1 + 2/(b*x))/3 + I*b**(3/2)*log(1/(b*x)) - 2*I*b**(3
/2)*log(1/(sqrt(b)*sqrt(x))) + 2*b**(3/2)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) - 4*sq
rt(b)*sqrt(-1 + 2/(b*x))/(3*x), 2*Abs(1/(b*x)) > 1), (8*I*b**(3/2)*sqrt(1 - 2/(b
*x))/3 + I*b**(3/2)*log(1/(b*x)) - 2*I*b**(3/2)*log(sqrt(1 - 2/(b*x)) + 1) - 4*I
*sqrt(b)*sqrt(1 - 2/(b*x))/(3*x), True))

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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^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError