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

Optimal. Leaf size=84 \[ 10 b^{3/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+5 b^2 \sqrt{x} \sqrt{2-b x}-\frac{2 (2-b x)^{5/2}}{3 x^{3/2}}+\frac{10 b (2-b x)^{3/2}}{3 \sqrt{x}} \]

[Out]

5*b^2*Sqrt[x]*Sqrt[2 - b*x] + (10*b*(2 - b*x)^(3/2))/(3*Sqrt[x]) - (2*(2 - b*x)^
(5/2))/(3*x^(3/2)) + 10*b^(3/2)*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi [A]  time = 0.061431, antiderivative size = 84, 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 \[ 10 b^{3/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+5 b^2 \sqrt{x} \sqrt{2-b x}-\frac{2 (2-b x)^{5/2}}{3 x^{3/2}}+\frac{10 b (2-b x)^{3/2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - b*x)^(5/2)/x^(5/2),x]

[Out]

5*b^2*Sqrt[x]*Sqrt[2 - b*x] + (10*b*(2 - b*x)^(3/2))/(3*Sqrt[x]) - (2*(2 - b*x)^
(5/2))/(3*x^(3/2)) + 10*b^(3/2)*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi in Sympy [A]  time = 10.0569, size = 78, normalized size = 0.93 \[ 10 b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + 5 b^{2} \sqrt{x} \sqrt{- b x + 2} + \frac{10 b \left (- b x + 2\right )^{\frac{3}{2}}}{3 \sqrt{x}} - \frac{2 \left (- b x + 2\right )^{\frac{5}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*b**(3/2)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) + 5*b**2*sqrt(x)*sqrt(-b*x + 2) + 10
*b*(-b*x + 2)**(3/2)/(3*sqrt(x)) - 2*(-b*x + 2)**(5/2)/(3*x**(3/2))

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Mathematica [A]  time = 0.0537578, size = 58, normalized size = 0.69 \[ 10 b^{3/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+\frac{\sqrt{2-b x} \left (3 b^2 x^2+28 b x-8\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 - b*x]*(-8 + 28*b*x + 3*b^2*x^2))/(3*x^(3/2)) + 10*b^(3/2)*ArcSin[(Sqrt[
b]*Sqrt[x])/Sqrt[2]]

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Maple [A]  time = 0.03, size = 107, normalized size = 1.3 \[ -{\frac{3\,{b}^{3}{x}^{3}+22\,{b}^{2}{x}^{2}-64\,bx+16}{3}\sqrt{ \left ( -bx+2 \right ) x}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}+5\,{\frac{{b}^{3/2}\sqrt{ \left ( -bx+2 \right ) x}}{\sqrt{x}\sqrt{-bx+2}}\arctan \left ({\frac{\sqrt{b}}{\sqrt{-b{x}^{2}+2\,x}} \left ( x-{b}^{-1} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(5/2)/x^(5/2),x)

[Out]

-1/3*(3*b^3*x^3+22*b^2*x^2-64*b*x+16)/x^(3/2)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1
/2)/(-b*x+2)^(1/2)+5*b^(3/2)*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^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228301, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-b} b x^{2} \log \left (-b x - \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) +{\left (3 \, b^{2} x^{2} + 28 \, b x - 8\right )} \sqrt{-b x + 2} \sqrt{x}}{3 \, x^{2}}, -\frac{30 \, b^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{2} + 28 \, b x - 8\right )} \sqrt{-b x + 2} \sqrt{x}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(15*sqrt(-b)*b*x^2*log(-b*x - sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1) + (3*b^2
*x^2 + 28*b*x - 8)*sqrt(-b*x + 2)*sqrt(x))/x^2, -1/3*(30*b^(3/2)*x^2*arctan(sqrt
(-b*x + 2)/(sqrt(b)*sqrt(x))) - (3*b^2*x^2 + 28*b*x - 8)*sqrt(-b*x + 2)*sqrt(x))
/x^2]

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Sympy [A]  time = 78.0075, size = 223, normalized size = 2.65 \[ \begin{cases} b^{\frac{5}{2}} x \sqrt{-1 + \frac{2}{b x}} + \frac{28 b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{3} + 5 i b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} - 10 i b^{\frac{3}{2}} \log{\left (\frac{1}{\sqrt{b} \sqrt{x}} \right )} + 10 b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{8 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{3 x} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\i b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}} + \frac{28 i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{3} + 5 i b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} - 10 i b^{\frac{3}{2}} \log{\left (\sqrt{1 - \frac{2}{b x}} + 1 \right )} - \frac{8 i \sqrt{b} \sqrt{1 - \frac{2}{b x}}}{3 x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((b**(5/2)*x*sqrt(-1 + 2/(b*x)) + 28*b**(3/2)*sqrt(-1 + 2/(b*x))/3 + 5*
I*b**(3/2)*log(1/(b*x)) - 10*I*b**(3/2)*log(1/(sqrt(b)*sqrt(x))) + 10*b**(3/2)*a
sin(sqrt(2)*sqrt(b)*sqrt(x)/2) - 8*sqrt(b)*sqrt(-1 + 2/(b*x))/(3*x), 2*Abs(1/(b*
x)) > 1), (I*b**(5/2)*x*sqrt(1 - 2/(b*x)) + 28*I*b**(3/2)*sqrt(1 - 2/(b*x))/3 +
5*I*b**(3/2)*log(1/(b*x)) - 10*I*b**(3/2)*log(sqrt(1 - 2/(b*x)) + 1) - 8*I*sqrt(
b)*sqrt(1 - 2/(b*x))/(3*x), 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^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError