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

Optimal. Leaf size=75 \[ -\frac{2 \sqrt{2-b x}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{2-b x}}+\frac{1}{3 x^{3/2} (2-b x)^{3/2}}-\frac{2 b \sqrt{2-b x}}{3 \sqrt{x}} \]

[Out]

1/(3*x^(3/2)*(2 - b*x)^(3/2)) + 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])

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Rubi [A]  time = 0.0478922, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \sqrt{2-b x}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{2-b x}}+\frac{1}{3 x^{3/2} (2-b x)^{3/2}}-\frac{2 b \sqrt{2-b x}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

1/(3*x^(3/2)*(2 - b*x)^(3/2)) + 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])

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Rubi in Sympy [A]  time = 6.17764, size = 66, normalized size = 0.88 \[ - \frac{2 b \sqrt{- b x + 2}}{3 \sqrt{x}} - \frac{2 \sqrt{- b x + 2}}{3 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{3}{2}} \sqrt{- b x + 2}} + \frac{1}{3 x^{\frac{3}{2}} \left (- b x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.0331432, size = 41, normalized size = 0.55 \[ -\frac{2 b^3 x^3-6 b^2 x^2+3 b x+1}{3 x^{3/2} (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 + 3*b*x - 6*b^2*x^2 + 2*b^3*x^3)/(3*x^(3/2)*(2 - b*x)^(3/2))

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Maple [A]  time = 0.006, size = 36, normalized size = 0.5 \[ -{\frac{2\,{b}^{3}{x}^{3}-6\,{b}^{2}{x}^{2}+3\,bx+1}{3}{x}^{-{\frac{3}{2}}} \left ( -bx+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(-b*x+2)^(5/2),x)

[Out]

-1/3*(2*b^3*x^3-6*b^2*x^2+3*b*x+1)/x^(3/2)/(-b*x+2)^(3/2)

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Maxima [A]  time = 1.3435, size = 78, normalized size = 1.04 \[ -\frac{3 \, \sqrt{-b x + 2} b}{8 \, \sqrt{x}} + \frac{{\left (b^{3} - \frac{9 \,{\left (b x - 2\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{24 \,{\left (-b x + 2\right )}^{\frac{3}{2}}} - \frac{{\left (-b x + 2\right )}^{\frac{3}{2}}}{24 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*sqrt(-b*x + 2)*b/sqrt(x) + 1/24*(b^3 - 9*(b*x - 2)*b^2/x)*x^(3/2)/(-b*x + 2
)^(3/2) - 1/24*(-b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 0.24165, size = 62, normalized size = 0.83 \[ \frac{2 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 3 \, b x + 1}{3 \,{\left (b x^{2} - 2 \, x\right )} \sqrt{-b x + 2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*b^3*x^3 - 6*b^2*x^2 + 3*b*x + 1)/((b*x^2 - 2*x)*sqrt(-b*x + 2)*sqrt(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228596, size = 247, normalized size = 3.29 \[ -\frac{{\left (4 \,{\left (b x - 2\right )} b^{2}{\left | b \right |} + 9 \, b^{2}{\left | b \right |}\right )} \sqrt{-b x + 2}}{12 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt{-b} b^{3} - 18 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} \sqrt{-b} b^{4} + 16 \, \sqrt{-b} b^{5}}{3 \,{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(4*(b*x - 2)*b^2*abs(b) + 9*b^2*abs(b))*sqrt(-b*x + 2)/((b*x - 2)*b + 2*b)
^(3/2) - 1/3*(3*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^4*sqrt(-b)*b
^3 - 18*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2*sqrt(-b)*b^4 + 16*
sqrt(-b)*b^5)/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)^3*a
bs(b))