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

Optimal. Leaf size=71 \[ -\frac{3 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}+\frac{2 x^{3/2}}{b \sqrt{a-b x}} \]

[Out]

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

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Rubi [A]  time = 0.055527, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}+\frac{2 x^{3/2}}{b \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a - b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.01517, size = 63, normalized size = 0.89 \[ - \frac{3 a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{b^{\frac{5}{2}}} + \frac{2 x^{\frac{3}{2}}}{b \sqrt{a - b x}} + \frac{3 \sqrt{x} \sqrt{a - b x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0913292, size = 58, normalized size = 0.82 \[ \frac{\sqrt{x} (3 a-b x)}{b^2 \sqrt{a-b x}}-\frac{3 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 114, normalized size = 1.6 \[{\frac{1}{{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{1 \left ( -{\frac{3\,a}{2}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}}-2\,{\frac{a}{{b}^{3}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}- \left ( x-{\frac{a}{b}} \right ) a} \left ( x-{\frac{a}{b}} \right ) ^{-1}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(-b*x+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227751, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{-b x + a} a \sqrt{x} \log \left (2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (b x^{2} - 3 \, a x\right )} \sqrt{-b}}{2 \, \sqrt{-b x + a} \sqrt{-b} b^{2} \sqrt{x}}, \frac{3 \, \sqrt{-b x + a} a \sqrt{x} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b x^{2} - 3 \, a x\right )} \sqrt{b}}{\sqrt{-b x + a} b^{\frac{5}{2}} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.6603, size = 155, normalized size = 2.18 \[ \begin{cases} - \frac{3 i \sqrt{a} \sqrt{x}}{b^{2} \sqrt{-1 + \frac{b x}{a}}} + \frac{3 i a \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{i x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{3 \sqrt{a} \sqrt{x}}{b^{2} \sqrt{1 - \frac{b x}{a}}} - \frac{3 a \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*I*sqrt(a)*sqrt(x)/(b**2*sqrt(-1 + b*x/a)) + 3*I*a*acosh(sqrt(b)*sq
rt(x)/sqrt(a))/b**(5/2) + I*x**(3/2)/(sqrt(a)*b*sqrt(-1 + b*x/a)), Abs(b*x/a) >
1), (3*sqrt(a)*sqrt(x)/(b**2*sqrt(1 - b*x/a)) - 3*a*asin(sqrt(b)*sqrt(x)/sqrt(a)
)/b**(5/2) - x**(3/2)/(sqrt(a)*b*sqrt(1 - b*x/a)), True))

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GIAC/XCAS [A]  time = 0.222663, size = 176, normalized size = 2.48 \[ -\frac{{\left (\frac{8 \, a^{2} \sqrt{-b}}{{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b} + \frac{3 \, a{\rm ln}\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt{-b}} - \frac{2 \, \sqrt{{\left (b x - a\right )} b + a b} \sqrt{-b x + a}}{b}\right )}{\left | b \right |}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(8*a^2*sqrt(-b)/((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*
b) + 3*a*ln((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/sqrt(-b) - 2*
sqrt((b*x - a)*b + a*b)*sqrt(-b*x + a)/b)*abs(b)/b^3