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

Optimal. Leaf size=82 \[ \frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.058617, antiderivative size = 82, 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 \[ \frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - b*x)^(5/2)/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.90206, size = 73, normalized size = 0.89 \[ \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{5}{2}}}{3} + \frac{5 \sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{6} + \frac{5 \sqrt{x} \sqrt{- b x + 2}}{2} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0586113, size = 58, normalized size = 0.71 \[ \frac{1}{6} \sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2-13 b x+33\right )+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[2 - b*x]*(33 - 13*b*x + 2*b^2*x^2))/6 + (5*ArcSin[(Sqrt[b]*Sqrt[x]
)/Sqrt[2]])/Sqrt[b]

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Maple [A]  time = 0.009, size = 91, normalized size = 1.1 \[{\frac{1}{3} \left ( -bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5}{6} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{2}\sqrt{x}\sqrt{-bx+2}}+{\frac{5}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(5/2)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227477, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} - 13 \, b x + 33\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 15 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{6 \, \sqrt{-b}}, \frac{{\left (2 \, b^{2} x^{2} - 13 \, b x + 33\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} - 30 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, \sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*((2*b^2*x^2 - 13*b*x + 33)*sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 15*log(-sqrt(-
b*x + 2)*b*sqrt(x) - (b*x - 1)*sqrt(-b)))/sqrt(-b), 1/6*((2*b^2*x^2 - 13*b*x + 3
3)*sqrt(-b*x + 2)*sqrt(b)*sqrt(x) - 30*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))))
/sqrt(b)]

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Sympy [A]  time = 59.5573, size = 209, normalized size = 2.55 \[ \begin{cases} \frac{i b^{3} x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} - \frac{17 i b^{2} x^{\frac{5}{2}}}{6 \sqrt{b x - 2}} + \frac{59 i b x^{\frac{3}{2}}}{6 \sqrt{b x - 2}} - \frac{11 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b^{3} x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} + \frac{17 b^{2} x^{\frac{5}{2}}}{6 \sqrt{- b x + 2}} - \frac{59 b x^{\frac{3}{2}}}{6 \sqrt{- b x + 2}} + \frac{11 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*b**3*x**(7/2)/(3*sqrt(b*x - 2)) - 17*I*b**2*x**(5/2)/(6*sqrt(b*x -
2)) + 59*I*b*x**(3/2)/(6*sqrt(b*x - 2)) - 11*I*sqrt(x)/sqrt(b*x - 2) - 5*I*acosh
(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), Abs(b*x)/2 > 1), (-b**3*x**(7/2)/(3*sqrt(-b
*x + 2)) + 17*b**2*x**(5/2)/(6*sqrt(-b*x + 2)) - 59*b*x**(3/2)/(6*sqrt(-b*x + 2)
) + 11*sqrt(x)/sqrt(-b*x + 2) + 5*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)^(5/2)/sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError