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

Optimal. Leaf size=65 \[ -\frac{6 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{2-b x}}{b^2}+\frac{2 x^{3/2}}{b \sqrt{2-b x}} \]

[Out]

(2*x^(3/2))/(b*Sqrt[2 - b*x]) + (3*Sqrt[x]*Sqrt[2 - b*x])/b^2 - (6*ArcSin[(Sqrt[
b]*Sqrt[x])/Sqrt[2]])/b^(5/2)

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Rubi [A]  time = 0.0520545, antiderivative size = 65, 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 \[ -\frac{6 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{2-b x}}{b^2}+\frac{2 x^{3/2}}{b \sqrt{2-b x}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(2 - b*x)^(3/2),x]

[Out]

(2*x^(3/2))/(b*Sqrt[2 - b*x]) + (3*Sqrt[x]*Sqrt[2 - b*x])/b^2 - (6*ArcSin[(Sqrt[
b]*Sqrt[x])/Sqrt[2]])/b^(5/2)

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Rubi in Sympy [A]  time = 8.70608, size = 60, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}}}{b \sqrt{- b x + 2}} + \frac{3 \sqrt{x} \sqrt{- b x + 2}}{b^{2}} - \frac{6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/(b*sqrt(-b*x + 2)) + 3*sqrt(x)*sqrt(-b*x + 2)/b**2 - 6*asin(sqrt(2)*s
qrt(b)*sqrt(x)/2)/b**(5/2)

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Mathematica [A]  time = 0.0757573, size = 50, normalized size = 0.77 \[ \frac{\sqrt{x} (6-b x)}{b^2 \sqrt{2-b x}}-\frac{6 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(6 - b*x))/(b^2*Sqrt[2 - b*x]) - (6*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/
b^(5/2)

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Maple [B]  time = 0.036, size = 133, normalized size = 2.1 \[ -{\frac{bx-2}{{b}^{2}}\sqrt{x}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}-{1 \left ( 3\,{\frac{1}{{b}^{5/2}}\arctan \left ({\frac{\sqrt{b}}{\sqrt{-b{x}^{2}+2\,x}} \left ( x-{b}^{-1} \right ) } \right ) }+4\,{\frac{1}{{b}^{3}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(-b*x+2)^(3/2),x)

[Out]

-1/b^2*x^(1/2)*(b*x-2)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/(-b*x+2)^(1/2)-(3/b
^(5/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))+4/b^3/(x-2/b)*(-b*(x-2/b)^2-2*
x+4/b)^(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(x^(3/2)/(-b*x + 2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22152, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{-b x + 2} \sqrt{x} \log \left (\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right ) -{\left (b x^{2} - 6 \, x\right )} \sqrt{-b}}{\sqrt{-b x + 2} \sqrt{-b} b^{2} \sqrt{x}}, \frac{6 \, \sqrt{-b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b x^{2} - 6 \, x\right )} \sqrt{b}}{\sqrt{-b x + 2} b^{\frac{5}{2}} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(3*sqrt(-b*x + 2)*sqrt(x)*log(sqrt(-b*x + 2)*b*sqrt(x) - (b*x - 1)*sqrt(-b)) -
(b*x^2 - 6*x)*sqrt(-b))/(sqrt(-b*x + 2)*sqrt(-b)*b^2*sqrt(x)), (6*sqrt(-b*x + 2)
*sqrt(x)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - (b*x^2 - 6*x)*sqrt(b))/(sqrt
(-b*x + 2)*b^(5/2)*sqrt(x))]

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Sympy [A]  time = 14.7449, size = 128, normalized size = 1.97 \[ \begin{cases} \frac{i x^{\frac{3}{2}}}{b \sqrt{b x - 2}} - \frac{6 i \sqrt{x}}{b^{2} \sqrt{b x - 2}} + \frac{6 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{x^{\frac{3}{2}}}{b \sqrt{- b x + 2}} + \frac{6 \sqrt{x}}{b^{2} \sqrt{- b x + 2}} - \frac{6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*x**(3/2)/(b*sqrt(b*x - 2)) - 6*I*sqrt(x)/(b**2*sqrt(b*x - 2)) + 6*I
*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(5/2), Abs(b*x)/2 > 1), (-x**(3/2)/(b*sqrt(
-b*x + 2)) + 6*sqrt(x)/(b**2*sqrt(-b*x + 2)) - 6*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)
/b**(5/2), True))

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GIAC/XCAS [A]  time = 0.218386, size = 162, normalized size = 2.49 \[ \frac{{\left (\frac{3 \, \sqrt{-b}{\rm ln}\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{b} + \frac{\sqrt{{\left (b x - 2\right )} b + 2 \, b} \sqrt{-b x + 2}}{b} - \frac{16 \, \sqrt{-b}}{{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b}\right )}{\left | b \right |}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*sqrt(-b)*ln((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2)/b + sqrt((
b*x - 2)*b + 2*b)*sqrt(-b*x + 2)/b - 16*sqrt(-b)/((sqrt(-b*x + 2)*sqrt(-b) - sqr
t((b*x - 2)*b + 2*b))^2 - 2*b))*abs(b)/b^3