3.640 \(\int \frac{1}{x^{5/2} (2-b x)^{3/2}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2 \sqrt{2-b x}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{2-b x}}-\frac{2 b \sqrt{2-b x}}{3 \sqrt{x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0384904, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \sqrt{2-b x}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{2-b x}}-\frac{2 b \sqrt{2-b x}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(2 - b*x)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.35548, size = 49, normalized size = 0.88 \[ - \frac{2 b \sqrt{- b x + 2}}{3 \sqrt{x}} - \frac{2 \sqrt{- b x + 2}}{3 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{3}{2}} \sqrt{- b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(-b*x+2)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0300019, size = 33, normalized size = 0.59 \[ \frac{2 b^2 x^2-2 b x-1}{3 x^{3/2} \sqrt{2-b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(2 - b*x)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 28, normalized size = 0.5 \[{\frac{2\,{b}^{2}{x}^{2}-2\,bx-1}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(-b*x+2)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34152, size = 59, normalized size = 1.05 \[ \frac{b^{2} \sqrt{x}}{4 \, \sqrt{-b x + 2}} - \frac{\sqrt{-b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (-b x + 2\right )}^{\frac{3}{2}}}{12 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + 2)^(3/2)*x^(5/2)),x, algorithm="maxima")

[Out]

1/4*b^2*sqrt(x)/sqrt(-b*x + 2) - 1/2*sqrt(-b*x + 2)*b/sqrt(x) - 1/12*(-b*x + 2)^
(3/2)/x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.208235, size = 36, normalized size = 0.64 \[ \frac{2 \, b^{2} x^{2} - 2 \, b x - 1}{3 \, \sqrt{-b x + 2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + 2)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 102.609, size = 355, normalized size = 6.34 \[ \begin{cases} - \frac{2 b^{\frac{15}{2}} x^{3} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} + \frac{6 b^{\frac{13}{2}} x^{2} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{3 b^{\frac{11}{2}} x \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{2 b^{\frac{9}{2}} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\- \frac{2 i b^{\frac{15}{2}} x^{3} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} + \frac{6 i b^{\frac{13}{2}} x^{2} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{3 i b^{\frac{11}{2}} x \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{2 i b^{\frac{9}{2}} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(-b*x+2)**(3/2),x)

[Out]

Piecewise((-2*b**(15/2)*x**3*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12
*b**4*x) + 6*b**(13/2)*x**2*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*
b**4*x) - 3*b**(11/2)*x*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4
*x) - 2*b**(9/2)*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x), 2*
Abs(1/(b*x)) > 1), (-2*I*b**(15/2)*x**3*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5
*x**2 + 12*b**4*x) + 6*I*b**(13/2)*x**2*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5
*x**2 + 12*b**4*x) - 3*I*b**(11/2)*x*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5*x*
*2 + 12*b**4*x) - 2*I*b**(9/2)*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 1
2*b**4*x), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212938, size = 130, normalized size = 2.32 \[ -\frac{\sqrt{-b} b^{3}}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}{\left | b \right |}} - \frac{{\left (5 \,{\left (b x - 2\right )} b^{2}{\left | b \right |} + 12 \, b^{2}{\left | b \right |}\right )} \sqrt{-b x + 2}}{12 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + 2)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

-sqrt(-b)*b^3/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*abs
(b)) - 1/12*(5*(b*x - 2)*b^2*abs(b) + 12*b^2*abs(b))*sqrt(-b*x + 2)/((b*x - 2)*b
+ 2*b)^(3/2)