3.515 \(\int \sqrt{x} \sqrt{2-b x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0490268, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{2-b x}-\frac{\sqrt{x} \sqrt{2-b x}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[2 - b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.00149, size = 56, normalized size = 0.86 \[ - \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{2 b} + \frac{\sqrt{x} \sqrt{- b x + 2}}{2 b} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0512328, size = 51, normalized size = 0.78 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{\sqrt{x} \sqrt{2-b x} (b x-1)}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[2 - b*x],x]

[Out]

(Sqrt[x]*Sqrt[2 - b*x]*(-1 + b*x))/(2*b) + ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]/b^(
3/2)

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Maple [A]  time = 0.007, size = 81, normalized size = 1.3 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{1}{2\,b}\sqrt{x}\sqrt{-bx+2}}+{\frac{1}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(-b*x+2)^(1/2),x)

[Out]

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(sqrt(-b*x + 2)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22221, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x - 1\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{2 \, \sqrt{-b} b}, \frac{{\left (b x - 1\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{2 \, b^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.9855, size = 156, normalized size = 2.4 \[ \begin{cases} \frac{i b x^{\frac{5}{2}}}{2 \sqrt{b x - 2}} - \frac{3 i x^{\frac{3}{2}}}{2 \sqrt{b x - 2}} + \frac{i \sqrt{x}}{b \sqrt{b x - 2}} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b x^{\frac{5}{2}}}{2 \sqrt{- b x + 2}} + \frac{3 x^{\frac{3}{2}}}{2 \sqrt{- b x + 2}} - \frac{\sqrt{x}}{b \sqrt{- b x + 2}} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError