3.514 \(\int x^{3/2} \sqrt{2-b x} \, dx\)

Optimal. Leaf size=87 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{\sqrt{x} \sqrt{2-b x}}{2 b^2}+\frac{1}{3} x^{5/2} \sqrt{2-b x}-\frac{x^{3/2} \sqrt{2-b x}}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.0679356, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{\sqrt{x} \sqrt{2-b x}}{2 b^2}+\frac{1}{3} x^{5/2} \sqrt{2-b x}-\frac{x^{3/2} \sqrt{2-b x}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*Sqrt[2 - b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.36174, size = 76, normalized size = 0.87 \[ - \frac{x^{\frac{3}{2}} \left (- b x + 2\right )^{\frac{3}{2}}}{3 b} - \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{2 b^{2}} + \frac{\sqrt{x} \sqrt{- b x + 2}}{2 b^{2}} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0610147, size = 60, normalized size = 0.69 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{\sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2-b x-3\right )}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 100, normalized size = 1.2 \[ -{\frac{1}{3\,b}{x}^{{\frac{3}{2}}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2\,{b}^{2}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{1}{2\,{b}^{2}}\sqrt{x}\sqrt{-bx+2}}+{\frac{1}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(-b*x+2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219933, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} - b x - 3\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 3 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{6 \, \sqrt{-b} b^{2}}, \frac{{\left (2 \, b^{2} x^{2} - b x - 3\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} - 6 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, b^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 23.156, size = 196, normalized size = 2.25 \[ \begin{cases} \frac{i b x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} - \frac{5 i x^{\frac{5}{2}}}{6 \sqrt{b x - 2}} - \frac{i x^{\frac{3}{2}}}{6 b \sqrt{b x - 2}} + \frac{i \sqrt{x}}{b^{2} \sqrt{b x - 2}} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} + \frac{5 x^{\frac{5}{2}}}{6 \sqrt{- b x + 2}} + \frac{x^{\frac{3}{2}}}{6 b \sqrt{- b x + 2}} - \frac{\sqrt{x}}{b^{2} \sqrt{- b x + 2}} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*b*x**(7/2)/(3*sqrt(b*x - 2)) - 5*I*x**(5/2)/(6*sqrt(b*x - 2)) - I*x
**(3/2)/(6*b*sqrt(b*x - 2)) + I*sqrt(x)/(b**2*sqrt(b*x - 2)) - I*acosh(sqrt(2)*s
qrt(b)*sqrt(x)/2)/b**(5/2), Abs(b*x)/2 > 1), (-b*x**(7/2)/(3*sqrt(-b*x + 2)) + 5
*x**(5/2)/(6*sqrt(-b*x + 2)) + x**(3/2)/(6*b*sqrt(-b*x + 2)) - sqrt(x)/(b**2*sqr
t(-b*x + 2)) + asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(5/2), 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(sqrt(-b*x + 2)*x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError