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

Optimal. Leaf size=89 \[ \frac{10 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{2-b x}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{2-b x}}+\frac{2 x^{5/2}}{3 b (2-b x)^{3/2}} \]

[Out]

(2*x^(5/2))/(3*b*(2 - b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[2 - b*x]) - (5*Sqrt
[x]*Sqrt[2 - b*x])/b^3 + (10*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi [A]  time = 0.0688159, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{10 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{2-b x}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{2-b x}}+\frac{2 x^{5/2}}{3 b (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(2 - b*x)^(5/2),x]

[Out]

(2*x^(5/2))/(3*b*(2 - b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[2 - b*x]) - (5*Sqrt
[x]*Sqrt[2 - b*x])/b^3 + (10*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi in Sympy [A]  time = 11.5827, size = 82, normalized size = 0.92 \[ \frac{2 x^{\frac{5}{2}}}{3 b \left (- b x + 2\right )^{\frac{3}{2}}} - \frac{10 x^{\frac{3}{2}}}{3 b^{2} \sqrt{- b x + 2}} - \frac{5 \sqrt{x} \sqrt{- b x + 2}}{b^{3}} + \frac{10 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(5/2)/(3*b*(-b*x + 2)**(3/2)) - 10*x**(3/2)/(3*b**2*sqrt(-b*x + 2)) - 5*sqr
t(x)*sqrt(-b*x + 2)/b**3 + 10*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(7/2)

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Mathematica [A]  time = 0.141551, size = 61, normalized size = 0.69 \[ \frac{10 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{\sqrt{x} \left (3 b^2 x^2-40 b x+60\right )}{3 b^3 (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(60 - 40*b*x + 3*b^2*x^2))/(3*b^3*(2 - b*x)^(3/2)) + (10*ArcSin[(Sqrt[
b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Maple [B]  time = 0.05, size = 168, normalized size = 1.9 \[{\frac{bx-2}{{b}^{3}}\sqrt{x}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}+{1 \left ( 5\,{\frac{1}{{b}^{7/2}}\arctan \left ({\frac{\sqrt{b}}{\sqrt{-b{x}^{2}+2\,x}} \left ( x-{b}^{-1} \right ) } \right ) }+{\frac{8}{3\,{b}^{5}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-2}}+{\frac{28}{3\,{b}^{4}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(-b*x+2)^(5/2),x)

[Out]

1/b^3*x^(1/2)*(b*x-2)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/(-b*x+2)^(1/2)+(5/b^
(7/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))+8/3/b^5/(x-2/b)^2*(-b*(x-2/b)^2
-2*x+4/b)^(1/2)+28/3/b^4/(x-2/b)*(-b*(x-2/b)^2-2*x+4/b)^(1/2))*((-b*x+2)*x)^(1/2
)/x^(1/2)/(-b*x+2)^(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)/(-b*x + 2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226206, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b x - 2\right )} \sqrt{-b x + 2} \sqrt{x} \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right ) +{\left (3 \, b^{2} x^{3} - 40 \, b x^{2} + 60 \, x\right )} \sqrt{-b}}{3 \,{\left (b^{4} x - 2 \, b^{3}\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x}}, -\frac{30 \,{\left (b x - 2\right )} \sqrt{-b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{3} - 40 \, b x^{2} + 60 \, x\right )} \sqrt{b}}{3 \,{\left (b^{4} x - 2 \, b^{3}\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(15*(b*x - 2)*sqrt(-b*x + 2)*sqrt(x)*log(-sqrt(-b*x + 2)*b*sqrt(x) - (b*x -
1)*sqrt(-b)) + (3*b^2*x^3 - 40*b*x^2 + 60*x)*sqrt(-b))/((b^4*x - 2*b^3)*sqrt(-b
*x + 2)*sqrt(-b)*sqrt(x)), -1/3*(30*(b*x - 2)*sqrt(-b*x + 2)*sqrt(x)*arctan(sqrt
(-b*x + 2)/(sqrt(b)*sqrt(x))) - (3*b^2*x^3 - 40*b*x^2 + 60*x)*sqrt(b))/((b^4*x -
2*b^3)*sqrt(-b*x + 2)*sqrt(b)*sqrt(x))]

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Sympy [A]  time = 81.5704, size = 753, normalized size = 8.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*I*b**(23/2)*x**15/(3*b**(27/2)*x**(27/2)*sqrt(b*x - 2) - 6*b**(25/
2)*x**(25/2)*sqrt(b*x - 2)) + 40*I*b**(21/2)*x**14/(3*b**(27/2)*x**(27/2)*sqrt(b
*x - 2) - 6*b**(25/2)*x**(25/2)*sqrt(b*x - 2)) - 60*I*b**(19/2)*x**13/(3*b**(27/
2)*x**(27/2)*sqrt(b*x - 2) - 6*b**(25/2)*x**(25/2)*sqrt(b*x - 2)) - 30*I*b**10*x
**(27/2)*sqrt(b*x - 2)*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(3*b**(27/2)*x**(27/2)*s
qrt(b*x - 2) - 6*b**(25/2)*x**(25/2)*sqrt(b*x - 2)) + 15*pi*b**10*x**(27/2)*sqrt
(b*x - 2)/(3*b**(27/2)*x**(27/2)*sqrt(b*x - 2) - 6*b**(25/2)*x**(25/2)*sqrt(b*x
- 2)) + 60*I*b**9*x**(25/2)*sqrt(b*x - 2)*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(3*b*
*(27/2)*x**(27/2)*sqrt(b*x - 2) - 6*b**(25/2)*x**(25/2)*sqrt(b*x - 2)) - 30*pi*b
**9*x**(25/2)*sqrt(b*x - 2)/(3*b**(27/2)*x**(27/2)*sqrt(b*x - 2) - 6*b**(25/2)*x
**(25/2)*sqrt(b*x - 2)), Abs(b*x)/2 > 1), (3*b**(23/2)*x**15/(3*b**(27/2)*x**(27
/2)*sqrt(-b*x + 2) - 6*b**(25/2)*x**(25/2)*sqrt(-b*x + 2)) - 40*b**(21/2)*x**14/
(3*b**(27/2)*x**(27/2)*sqrt(-b*x + 2) - 6*b**(25/2)*x**(25/2)*sqrt(-b*x + 2)) +
60*b**(19/2)*x**13/(3*b**(27/2)*x**(27/2)*sqrt(-b*x + 2) - 6*b**(25/2)*x**(25/2)
*sqrt(-b*x + 2)) + 30*b**10*x**(27/2)*sqrt(-b*x + 2)*asin(sqrt(2)*sqrt(b)*sqrt(x
)/2)/(3*b**(27/2)*x**(27/2)*sqrt(-b*x + 2) - 6*b**(25/2)*x**(25/2)*sqrt(-b*x + 2
)) - 60*b**9*x**(25/2)*sqrt(-b*x + 2)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/(3*b**(27/
2)*x**(27/2)*sqrt(-b*x + 2) - 6*b**(25/2)*x**(25/2)*sqrt(-b*x + 2)), True))

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GIAC/XCAS [A]  time = 0.229447, size = 270, normalized size = 3.03 \[ \frac{{\left (\frac{15 \,{\rm ln}\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt{-b} b^{2}} - \frac{3 \, \sqrt{{\left (b x - 2\right )} b + 2 \, b} \sqrt{-b x + 2}}{b^{3}} - \frac{16 \,{\left (9 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} - 24 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} b + 28 \, b^{2}\right )}}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3} \sqrt{-b} b}\right )}{\left | b \right |}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(15*ln((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2)/(sqrt(-b)*b^2)
- 3*sqrt((b*x - 2)*b + 2*b)*sqrt(-b*x + 2)/b^3 - 16*(9*(sqrt(-b*x + 2)*sqrt(-b)
- sqrt((b*x - 2)*b + 2*b))^4 - 24*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b +
2*b))^2*b + 28*b^2)/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2
*b)^3*sqrt(-b)*b))*abs(b)/b^2