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

Optimal. Leaf size=91 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{2-b x}}{2 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{6 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{3 b} \]

[Out]

(-5*Sqrt[x]*Sqrt[2 - b*x])/(2*b^3) - (5*x^(3/2)*Sqrt[2 - b*x])/(6*b^2) - (x^(5/2
)*Sqrt[2 - b*x])/(3*b) + (5*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi [A]  time = 0.0702088, antiderivative size = 91, 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 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{2-b x}}{2 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{6 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/Sqrt[2 - b*x],x]

[Out]

(-5*Sqrt[x]*Sqrt[2 - b*x])/(2*b^3) - (5*x^(3/2)*Sqrt[2 - b*x])/(6*b^2) - (x^(5/2
)*Sqrt[2 - b*x])/(3*b) + (5*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi in Sympy [A]  time = 9.61548, size = 82, normalized size = 0.9 \[ - \frac{x^{\frac{5}{2}} \sqrt{- b x + 2}}{3 b} - \frac{5 x^{\frac{3}{2}} \sqrt{- b x + 2}}{6 b^{2}} - \frac{5 \sqrt{x} \sqrt{- b x + 2}}{2 b^{3}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(-b*x+2)**(1/2),x)

[Out]

-x**(5/2)*sqrt(-b*x + 2)/(3*b) - 5*x**(3/2)*sqrt(-b*x + 2)/(6*b**2) - 5*sqrt(x)*
sqrt(-b*x + 2)/(2*b**3) + 5*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(7/2)

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Mathematica [A]  time = 0.071209, size = 61, normalized size = 0.67 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{\sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2+5 b x+15\right )}{6 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*Sqrt[2 - b*x]*(15 + 5*b*x + 2*b^2*x^2))/(6*b^3) + (5*ArcSin[(Sqrt[b]*S
qrt[x])/Sqrt[2]])/b^(7/2)

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Maple [A]  time = 0.007, size = 100, normalized size = 1.1 \[ -{\frac{1}{3\,b}{x}^{{\frac{5}{2}}}\sqrt{-bx+2}}-{\frac{5}{6\,{b}^{2}}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{5}{2\,{b}^{3}}\sqrt{x}\sqrt{-bx+2}}+{\frac{5}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(-b*x+2)^(1/2),x)

[Out]

-1/3*x^(5/2)*(-b*x+2)^(1/2)/b-5/6*x^(3/2)*(-b*x+2)^(1/2)/b^2-5/2*x^(1/2)*(-b*x+2
)^(1/2)/b^3+5/2/b^(7/2)*((-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(x^(5/2)/sqrt(-b*x + 2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244485, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, b^{2} x^{2} + 5 \, b x + 15\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} - 15 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{6 \, \sqrt{-b} b^{3}}, -\frac{{\left (2 \, b^{2} x^{2} + 5 \, b x + 15\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} + 30 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, b^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*((2*b^2*x^2 + 5*b*x + 15)*sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) - 15*log(-sqrt(-
b*x + 2)*b*sqrt(x) - (b*x - 1)*sqrt(-b)))/(sqrt(-b)*b^3), -1/6*((2*b^2*x^2 + 5*b
*x + 15)*sqrt(-b*x + 2)*sqrt(b)*sqrt(x) + 30*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt
(x))))/b^(7/2)]

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Sympy [A]  time = 65.4969, size = 206, normalized size = 2.26 \[ \begin{cases} - \frac{i x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} - \frac{i x^{\frac{5}{2}}}{6 b \sqrt{b x - 2}} - \frac{5 i x^{\frac{3}{2}}}{6 b^{2} \sqrt{b x - 2}} + \frac{5 i \sqrt{x}}{b^{3} \sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} + \frac{x^{\frac{5}{2}}}{6 b \sqrt{- b x + 2}} + \frac{5 x^{\frac{3}{2}}}{6 b^{2} \sqrt{- b x + 2}} - \frac{5 \sqrt{x}}{b^{3} \sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(-b*x+2)**(1/2),x)

[Out]

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

[Out]

Exception raised: NotImplementedError