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

Optimal. Leaf size=82 \[ -\frac{2 (2-b x)^{5/2}}{\sqrt{x}}-\frac{5}{2} b \sqrt{x} (2-b x)^{3/2}-\frac{15}{2} b \sqrt{x} \sqrt{2-b x}-15 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

[Out]

(-15*b*Sqrt[x]*Sqrt[2 - b*x])/2 - (5*b*Sqrt[x]*(2 - b*x)^(3/2))/2 - (2*(2 - b*x)
^(5/2))/Sqrt[x] - 15*Sqrt[b]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi [A]  time = 0.0595271, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 (2-b x)^{5/2}}{\sqrt{x}}-\frac{5}{2} b \sqrt{x} (2-b x)^{3/2}-\frac{15}{2} b \sqrt{x} \sqrt{2-b x}-15 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 - b*x)^(5/2)/x^(3/2),x]

[Out]

(-15*b*Sqrt[x]*Sqrt[2 - b*x])/2 - (5*b*Sqrt[x]*(2 - b*x)^(3/2))/2 - (2*(2 - b*x)
^(5/2))/Sqrt[x] - 15*Sqrt[b]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi in Sympy [A]  time = 9.78192, size = 78, normalized size = 0.95 \[ - 15 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{5 b \sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{2} - \frac{15 b \sqrt{x} \sqrt{- b x + 2}}{2} - \frac{2 \left (- b x + 2\right )^{\frac{5}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) - 5*b*sqrt(x)*(-b*x + 2)**(3/2)/2 -
15*b*sqrt(x)*sqrt(-b*x + 2)/2 - 2*(-b*x + 2)**(5/2)/sqrt(x)

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Mathematica [A]  time = 0.0501641, size = 57, normalized size = 0.7 \[ \frac{\sqrt{2-b x} \left (b^2 x^2-9 b x-16\right )}{2 \sqrt{x}}-15 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 - b*x]*(-16 - 9*b*x + b^2*x^2))/(2*Sqrt[x]) - 15*Sqrt[b]*ArcSin[(Sqrt[b]
*Sqrt[x])/Sqrt[2]]

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Maple [A]  time = 0.027, size = 106, normalized size = 1.3 \[ -{\frac{{b}^{3}{x}^{3}-11\,{b}^{2}{x}^{2}+2\,bx+32}{2}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}}-{\frac{15}{2}\sqrt{b}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(5/2)/x^(3/2),x)

[Out]

-1/2*(b^3*x^3-11*b^2*x^2+2*b*x+32)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/x^(1/2)
/(-b*x+2)^(1/2)-15/2*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)

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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)^(5/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227175, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-b} x \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) +{\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt{-b x + 2} \sqrt{x}}{2 \, x}, \frac{30 \, \sqrt{b} x \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) +{\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt{-b x + 2} \sqrt{x}}{2 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(15*sqrt(-b)*x*log(-b*x + sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1) + (b^2*x^2 -
9*b*x - 16)*sqrt(-b*x + 2)*sqrt(x))/x, 1/2*(30*sqrt(b)*x*arctan(sqrt(-b*x + 2)/
(sqrt(b)*sqrt(x))) + (b^2*x^2 - 9*b*x - 16)*sqrt(-b*x + 2)*sqrt(x))/x]

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Sympy [A]  time = 76.389, size = 202, normalized size = 2.46 \[ \begin{cases} 15 i \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{i b^{3} x^{\frac{5}{2}}}{2 \sqrt{b x - 2}} - \frac{11 i b^{2} x^{\frac{3}{2}}}{2 \sqrt{b x - 2}} + \frac{i b \sqrt{x}}{\sqrt{b x - 2}} + \frac{16 i}{\sqrt{x} \sqrt{b x - 2}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- 15 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{- b x + 2}} + \frac{11 b^{2} x^{\frac{3}{2}}}{2 \sqrt{- b x + 2}} - \frac{b \sqrt{x}}{\sqrt{- b x + 2}} - \frac{16}{\sqrt{x} \sqrt{- b x + 2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((15*I*sqrt(b)*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2) + I*b**3*x**(5/2)/(2*sq
rt(b*x - 2)) - 11*I*b**2*x**(3/2)/(2*sqrt(b*x - 2)) + I*b*sqrt(x)/sqrt(b*x - 2)
+ 16*I/(sqrt(x)*sqrt(b*x - 2)), Abs(b*x)/2 > 1), (-15*sqrt(b)*asin(sqrt(2)*sqrt(
b)*sqrt(x)/2) - b**3*x**(5/2)/(2*sqrt(-b*x + 2)) + 11*b**2*x**(3/2)/(2*sqrt(-b*x
+ 2)) - b*sqrt(x)/sqrt(-b*x + 2) - 16/(sqrt(x)*sqrt(-b*x + 2)), 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)^(5/2)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError