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

Optimal. Leaf size=105 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{7/2}}-\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b} \]

[Out]

(-5*a^2*Sqrt[x]*Sqrt[a - b*x])/(8*b^3) - (5*a*x^(3/2)*Sqrt[a - b*x])/(12*b^2) -
(x^(5/2)*Sqrt[a - b*x])/(3*b) + (5*a^3*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/
(8*b^(7/2))

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Rubi [A]  time = 0.0850979, antiderivative size = 105, 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 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{7/2}}-\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/Sqrt[a - b*x],x]

[Out]

(-5*a^2*Sqrt[x]*Sqrt[a - b*x])/(8*b^3) - (5*a*x^(3/2)*Sqrt[a - b*x])/(12*b^2) -
(x^(5/2)*Sqrt[a - b*x])/(3*b) + (5*a^3*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/
(8*b^(7/2))

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Rubi in Sympy [A]  time = 11.222, size = 94, normalized size = 0.9 \[ \frac{5 a^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{8 b^{\frac{7}{2}}} - \frac{5 a^{2} \sqrt{x} \sqrt{a - b x}}{8 b^{3}} - \frac{5 a x^{\frac{3}{2}} \sqrt{a - b x}}{12 b^{2}} - \frac{x^{\frac{5}{2}} \sqrt{a - b x}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**3*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(8*b**(7/2)) - 5*a**2*sqrt(x)*sqrt(a
- b*x)/(8*b**3) - 5*a*x**(3/2)*sqrt(a - b*x)/(12*b**2) - x**(5/2)*sqrt(a - b*x)/
(3*b)

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Mathematica [A]  time = 0.0880712, size = 76, normalized size = 0.72 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{7/2}}-\frac{\sqrt{x} \sqrt{a-b x} \left (15 a^2+10 a b x+8 b^2 x^2\right )}{24 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*Sqrt[a - b*x]*(15*a^2 + 10*a*b*x + 8*b^2*x^2))/(24*b^3) + (5*a^3*ArcTa
n[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(8*b^(7/2))

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Maple [A]  time = 0.007, size = 108, normalized size = 1. \[ -{\frac{1}{3\,b}{x}^{{\frac{5}{2}}}\sqrt{-bx+a}}-{\frac{5\,a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{5\,{a}^{2}}{8\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{\frac{5\,{a}^{3}}{16}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(-b*x+a)^(1/2),x)

[Out]

-1/3*x^(5/2)*(-b*x+a)^(1/2)/b-5/12*a*x^(3/2)*(-b*x+a)^(1/2)/b^2-5/8*a^2*x^(1/2)*
(-b*x+a)^(1/2)/b^3+5/16*a^3/b^(7/2)*(x*(-b*x+a))^(1/2)/x^(1/2)/(-b*x+a)^(1/2)*ar
ctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*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 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243389, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (8 \, b^{2} x^{2} + 10 \, a b x + 15 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{48 \, \sqrt{-b} b^{3}}, -\frac{15 \, a^{3} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (8 \, b^{2} x^{2} + 10 \, a b x + 15 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{24 \, b^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*a^3*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) - 2*(8*b^2
*x^2 + 10*a*b*x + 15*a^2)*sqrt(-b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3), -1/24
*(15*a^3*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + (8*b^2*x^2 + 10*a*b*x + 15*a
^2)*sqrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(7/2)]

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Sympy [A]  time = 69.2128, size = 270, normalized size = 2.57 \[ \begin{cases} \frac{5 i a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i \sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i a^{3} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} - \frac{i x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{5 a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 - \frac{b x}{a}}} + \frac{5 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{\sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 - \frac{b x}{a}}} + \frac{5 a^{3} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((5*I*a**(5/2)*sqrt(x)/(8*b**3*sqrt(-1 + b*x/a)) - 5*I*a**(3/2)*x**(3/2
)/(24*b**2*sqrt(-1 + b*x/a)) - I*sqrt(a)*x**(5/2)/(12*b*sqrt(-1 + b*x/a)) - 5*I*
a**3*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(7/2)) - I*x**(7/2)/(3*sqrt(a)*sqrt(-1
+ b*x/a)), Abs(b*x/a) > 1), (-5*a**(5/2)*sqrt(x)/(8*b**3*sqrt(1 - b*x/a)) + 5*a
**(3/2)*x**(3/2)/(24*b**2*sqrt(1 - b*x/a)) + sqrt(a)*x**(5/2)/(12*b*sqrt(1 - b*x
/a)) + 5*a**3*asin(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(7/2)) + x**(7/2)/(3*sqrt(a)*s
qrt(1 - b*x/a)), 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 + a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError