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

Optimal. Leaf size=80 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b} \]

[Out]

(-3*a*Sqrt[x]*Sqrt[a - b*x])/(4*b^2) - (x^(3/2)*Sqrt[a - b*x])/(2*b) + (3*a^2*Ar
cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b^(5/2))

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Rubi [A]  time = 0.0551251, antiderivative size = 80, 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 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/Sqrt[a - b*x],x]

[Out]

(-3*a*Sqrt[x]*Sqrt[a - b*x])/(4*b^2) - (x^(3/2)*Sqrt[a - b*x])/(2*b) + (3*a^2*Ar
cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b^(5/2))

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Rubi in Sympy [A]  time = 7.99522, size = 71, normalized size = 0.89 \[ - \frac{3 a^{2} \operatorname{atan}{\left (\frac{\sqrt{a - b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} - \frac{3 a \sqrt{x} \sqrt{a - b x}}{4 b^{2}} - \frac{x^{\frac{3}{2}} \sqrt{a - b x}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(-b*x+a)**(1/2),x)

[Out]

-3*a**2*atan(sqrt(a - b*x)/(sqrt(b)*sqrt(x)))/(4*b**(5/2)) - 3*a*sqrt(x)*sqrt(a
- b*x)/(4*b**2) - x**(3/2)*sqrt(a - b*x)/(2*b)

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Mathematica [A]  time = 0.0677266, size = 65, normalized size = 0.81 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{5/2}}-\frac{\sqrt{x} \sqrt{a-b x} (3 a+2 b x)}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*Sqrt[a - b*x]*(3*a + 2*b*x))/(4*b^2) + (3*a^2*ArcTan[(Sqrt[b]*Sqrt[x])
/Sqrt[a - b*x]])/(4*b^(5/2))

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Maple [A]  time = 0.008, size = 89, normalized size = 1.1 \[ -{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{3\,a}{4\,{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{2}}{8}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(-b*x+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255492, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (2 \, b x + 3 \, a\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{8 \, \sqrt{-b} b^{2}}, -\frac{3 \, a^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (2 \, b x + 3 \, a\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{4 \, b^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*a^2*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) - 2*(2*b*x +
3*a)*sqrt(-b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2), -1/4*(3*a^2*arctan(sqrt(-
b*x + a)/(sqrt(b)*sqrt(x))) + (2*b*x + 3*a)*sqrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(5
/2)]

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Sympy [A]  time = 17.2169, size = 214, normalized size = 2.68 \[ \begin{cases} \frac{3 i a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} - \frac{i x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{3 a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{\sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 - \frac{b x}{a}}} + \frac{3 a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(-b*x+a)**(1/2),x)

[Out]

Piecewise((3*I*a**(3/2)*sqrt(x)/(4*b**2*sqrt(-1 + b*x/a)) - I*sqrt(a)*x**(3/2)/(
4*b*sqrt(-1 + b*x/a)) - 3*I*a**2*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(5/2)) - I
*x**(5/2)/(2*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-3*a**(3/2)*sqrt(x)/(4
*b**2*sqrt(1 - b*x/a)) + sqrt(a)*x**(3/2)/(4*b*sqrt(1 - b*x/a)) + 3*a**2*asin(sq
rt(b)*sqrt(x)/sqrt(a))/(4*b**(5/2)) + x**(5/2)/(2*sqrt(a)*sqrt(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^(3/2)/sqrt(-b*x + a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError