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

Optimal. Leaf size=89 \[ -\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}+\frac{2 x^{5/2}}{b \sqrt{2-b x}} \]

[Out]

(2*x^(5/2))/(b*Sqrt[2 - b*x]) + (15*Sqrt[x]*Sqrt[2 - b*x])/(2*b^3) + (5*x^(3/2)*
Sqrt[2 - b*x])/(2*b^2) - (15*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi [A]  time = 0.070138, antiderivative size = 89, 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 \[ -\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}+\frac{2 x^{5/2}}{b \sqrt{2-b x}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(2 - b*x)^(3/2),x]

[Out]

(2*x^(5/2))/(b*Sqrt[2 - b*x]) + (15*Sqrt[x]*Sqrt[2 - b*x])/(2*b^3) + (5*x^(3/2)*
Sqrt[2 - b*x])/(2*b^2) - (15*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi in Sympy [A]  time = 10.5232, size = 82, normalized size = 0.92 \[ \frac{2 x^{\frac{5}{2}}}{b \sqrt{- b x + 2}} + \frac{5 x^{\frac{3}{2}} \sqrt{- b x + 2}}{2 b^{2}} + \frac{15 \sqrt{x} \sqrt{- b x + 2}}{2 b^{3}} - \frac{15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(5/2)/(b*sqrt(-b*x + 2)) + 5*x**(3/2)*sqrt(-b*x + 2)/(2*b**2) + 15*sqrt(x)*
sqrt(-b*x + 2)/(2*b**3) - 15*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(7/2)

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Mathematica [A]  time = 0.10218, size = 60, normalized size = 0.67 \[ -\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{\sqrt{x} \left (b^2 x^2+5 b x-30\right )}{2 b^3 \sqrt{2-b x}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(-30 + 5*b*x + b^2*x^2))/(2*b^3*Sqrt[2 - b*x]) - (15*ArcSin[(Sqrt[b]*S
qrt[x])/Sqrt[2]])/b^(7/2)

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Maple [B]  time = 0.042, size = 138, normalized size = 1.6 \[ -{\frac{ \left ( bx+7 \right ) \left ( bx-2 \right ) }{2\,{b}^{3}}\sqrt{x}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}-{1 \left ({\frac{15}{2}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{7}{2}}}}+8\,{\frac{1}{{b}^{4}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(-b*x+2)^(3/2),x)

[Out]

-1/2*(b*x+7)*x^(1/2)*(b*x-2)/b^3/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/(-b*x+2)^
(1/2)-(15/2/b^(7/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))+8/b^4/(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^(5/2)/(-b*x + 2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226447, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-b x + 2} \sqrt{x} \log \left (\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right ) -{\left (b^{2} x^{3} + 5 \, b x^{2} - 30 \, x\right )} \sqrt{-b}}{2 \, \sqrt{-b x + 2} \sqrt{-b} b^{3} \sqrt{x}}, \frac{30 \, \sqrt{-b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b^{2} x^{3} + 5 \, b x^{2} - 30 \, x\right )} \sqrt{b}}{2 \, \sqrt{-b x + 2} b^{\frac{7}{2}} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(15*sqrt(-b*x + 2)*sqrt(x)*log(sqrt(-b*x + 2)*b*sqrt(x) - (b*x - 1)*sqrt(-b
)) - (b^2*x^3 + 5*b*x^2 - 30*x)*sqrt(-b))/(sqrt(-b*x + 2)*sqrt(-b)*b^3*sqrt(x)),
1/2*(30*sqrt(-b*x + 2)*sqrt(x)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - (b^2*
x^3 + 5*b*x^2 - 30*x)*sqrt(b))/(sqrt(-b*x + 2)*b^(7/2)*sqrt(x))]

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Sympy [A]  time = 82.162, size = 173, normalized size = 1.94 \[ \begin{cases} \frac{i x^{\frac{5}{2}}}{2 b \sqrt{b x - 2}} + \frac{5 i x^{\frac{3}{2}}}{2 b^{2} \sqrt{b x - 2}} - \frac{15 i \sqrt{x}}{b^{3} \sqrt{b x - 2}} + \frac{15 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{x^{\frac{5}{2}}}{2 b \sqrt{- b x + 2}} - \frac{5 x^{\frac{3}{2}}}{2 b^{2} \sqrt{- b x + 2}} + \frac{15 \sqrt{x}}{b^{3} \sqrt{- b x + 2}} - \frac{15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*x**(5/2)/(2*b*sqrt(b*x - 2)) + 5*I*x**(3/2)/(2*b**2*sqrt(b*x - 2))
- 15*I*sqrt(x)/(b**3*sqrt(b*x - 2)) + 15*I*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(
7/2), Abs(b*x)/2 > 1), (-x**(5/2)/(2*b*sqrt(-b*x + 2)) - 5*x**(3/2)/(2*b**2*sqrt
(-b*x + 2)) + 15*sqrt(x)/(b**3*sqrt(-b*x + 2)) - 15*asin(sqrt(2)*sqrt(b)*sqrt(x)
/2)/b**(7/2), True))

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GIAC/XCAS [A]  time = 0.221612, size = 184, normalized size = 2.07 \[ \frac{{\left (\sqrt{{\left (b x - 2\right )} b + 2 \, b} \sqrt{-b x + 2}{\left (\frac{b x - 2}{b^{3}} + \frac{9}{b^{3}}\right )} - \frac{15 \,{\rm ln}\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt{-b} b^{2}} + \frac{64}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt{-b} b}\right )}{\left | b \right |}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt((b*x - 2)*b + 2*b)*sqrt(-b*x + 2)*((b*x - 2)/b^3 + 9/b^3) - 15*ln((sqr
t(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2)/(sqrt(-b)*b^2) + 64/(((sqrt(-
b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*sqrt(-b)*b))*abs(b)/b^2