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

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

[Out]

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

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Rubi [A]  time = 0.0573771, antiderivative size = 84, 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 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (2-b x)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{2-b x}-\frac{\sqrt{x} \sqrt{2-b x}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(2 - b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.15491, size = 73, normalized size = 0.87 \[ - \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{5}{2}}}{3 b} + \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{6 b} + \frac{\sqrt{x} \sqrt{- b x + 2}}{2 b} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*(-b*x + 2)**(5/2)/(3*b) + sqrt(x)*(-b*x + 2)**(3/2)/(6*b) + sqrt(x)*sqr
t(-b*x + 2)/(2*b) + asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2)

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Mathematica [A]  time = 0.0727008, size = 60, normalized size = 0.71 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{\sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2-7 b x+3\right )}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*Sqrt[2 - b*x]*(3 - 7*b*x + 2*b^2*x^2))/(6*b) + ArcSin[(Sqrt[b]*Sqrt[x]
)/Sqrt[2]]/b^(3/2)

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Maple [A]  time = 0.007, size = 94, normalized size = 1.1 \[{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{1}{2\,b}\sqrt{x}\sqrt{-bx+2}}+{\frac{1}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(3/2)*x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239696, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, b^{2} x^{2} - 7 \, b x + 3\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} - 3 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{6 \, \sqrt{-b} b}, -\frac{{\left (2 \, b^{2} x^{2} - 7 \, b x + 3\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} + 6 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, b^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*((2*b^2*x^2 - 7*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), -1/6*((2*b^2*x^2 - 7*b*x +
3)*sqrt(-b*x + 2)*sqrt(b)*sqrt(x) + 6*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))))
/b^(3/2)]

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Sympy [A]  time = 21.9239, size = 199, normalized size = 2.37 \[ \begin{cases} - \frac{i b^{2} x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} + \frac{11 i b x^{\frac{5}{2}}}{6 \sqrt{b x - 2}} - \frac{17 i x^{\frac{3}{2}}}{6 \sqrt{b x - 2}} + \frac{i \sqrt{x}}{b \sqrt{b x - 2}} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} - \frac{11 b x^{\frac{5}{2}}}{6 \sqrt{- b x + 2}} + \frac{17 x^{\frac{3}{2}}}{6 \sqrt{- b x + 2}} - \frac{\sqrt{x}}{b \sqrt{- b x + 2}} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*b**2*x**(7/2)/(3*sqrt(b*x - 2)) + 11*I*b*x**(5/2)/(6*sqrt(b*x - 2)
) - 17*I*x**(3/2)/(6*sqrt(b*x - 2)) + I*sqrt(x)/(b*sqrt(b*x - 2)) - I*acosh(sqrt
(2)*sqrt(b)*sqrt(x)/2)/b**(3/2), Abs(b*x)/2 > 1), (b**2*x**(7/2)/(3*sqrt(-b*x +
2)) - 11*b*x**(5/2)/(6*sqrt(-b*x + 2)) + 17*x**(3/2)/(6*sqrt(-b*x + 2)) - sqrt(x
)/(b*sqrt(-b*x + 2)) + asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2), 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((-b*x + 2)^(3/2)*sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError