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

Optimal. Leaf size=66 \[ -\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-3 b \sqrt{x} \sqrt{a-b x}-3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0498914, antiderivative size = 66, 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 \[ -\frac{2 (a-b x)^{3/2}}{\sqrt{x}}-3 b \sqrt{x} \sqrt{a-b x}-3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x)^(3/2)/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.51781, size = 60, normalized size = 0.91 \[ 3 a \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a - b x}}{\sqrt{b} \sqrt{x}} \right )} - 3 b \sqrt{x} \sqrt{a - b x} - \frac{2 \left (a - b x\right )^{\frac{3}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0568354, size = 55, normalized size = 0.83 \[ -\frac{\sqrt{a-b x} (2 a+b x)}{\sqrt{x}}-3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[a - b*x]*(2*a + b*x))/Sqrt[x]) - 3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sq
rt[a - b*x]]

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Maple [A]  time = 0.026, size = 74, normalized size = 1.1 \[ -{(bx+2\,a)\sqrt{-bx+a}{\frac{1}{\sqrt{x}}}}-{\frac{3\,a}{2}\sqrt{b}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(3/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219793, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{-b} x \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) - 2 \,{\left (b x + 2 \, a\right )} \sqrt{-b x + a} \sqrt{x}}{2 \, x}, \frac{3 \, a \sqrt{b} x \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b x + 2 \, a\right )} \sqrt{-b x + a} \sqrt{x}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.8959, size = 197, normalized size = 2.98 \[ \begin{cases} \frac{2 i a^{\frac{3}{2}}}{\sqrt{x} \sqrt{-1 + \frac{b x}{a}}} - \frac{i \sqrt{a} b \sqrt{x}}{\sqrt{-1 + \frac{b x}{a}}} + 3 i a \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{i b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 - \frac{b x}{a}}} + \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 - \frac{b x}{a}}} - 3 a \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError