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

Optimal. Leaf size=63 \[ \frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0433811, antiderivative size = 63, 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 \[ \frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - b*x)^(3/2)/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.98751, size = 56, normalized size = 0.89 \[ \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{2} + \frac{3 \sqrt{x} \sqrt{- b x + 2}}{2} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

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

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Mathematica [A]  time = 0.049853, size = 49, normalized size = 0.78 \[ \frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}}-\frac{1}{2} \sqrt{x} \sqrt{2-b x} (b x-5) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*Sqrt[2 - b*x]*(-5 + b*x))/2 + (3*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/Sq
rt[b]

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Maple [A]  time = 0.007, size = 78, normalized size = 1.2 \[{\frac{1}{2} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3}{2}\sqrt{x}\sqrt{-bx+2}}+{\frac{3}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(3/2)/x^(1/2),x)

[Out]

1/2*(-b*x+2)^(3/2)*x^(1/2)+3/2*x^(1/2)*(-b*x+2)^(1/2)+3/2*((-b*x+2)*x)^(1/2)/(-b
*x+2)^(1/2)/x^(1/2)/b^(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 = 0.24767, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (b x - 5\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} - 3 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{2 \, \sqrt{-b}}, -\frac{{\left (b x - 5\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} + 6 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{2 \, \sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.7703, size = 167, normalized size = 2.65 \[ \begin{cases} - \frac{i b^{2} x^{\frac{5}{2}}}{2 \sqrt{b x - 2}} + \frac{7 i b x^{\frac{3}{2}}}{2 \sqrt{b x - 2}} - \frac{5 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{b^{2} x^{\frac{5}{2}}}{2 \sqrt{- b x + 2}} - \frac{7 b x^{\frac{3}{2}}}{2 \sqrt{- b x + 2}} + \frac{5 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

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