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

Optimal. Leaf size=68 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(b x-a)^{3/2}}{2 x^2}-\frac{3 b \sqrt{b x-a}}{4 x} \]

[Out]

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

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Rubi [A]  time = 0.0531137, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(b x-a)^{3/2}}{2 x^2}-\frac{3 b \sqrt{b x-a}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(-a + b*x)^(3/2)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.74488, size = 54, normalized size = 0.79 \[ - \frac{3 b \sqrt{- a + b x}}{4 x} - \frac{\left (- a + b x\right )^{\frac{3}{2}}}{2 x^{2}} + \frac{3 b^{2} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0612377, size = 56, normalized size = 0.82 \[ \frac{1}{4} \left (\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{(2 a-5 b x) \sqrt{b x-a}}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 53, normalized size = 0.8 \[ -{\frac{5}{4\,{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a}{4\,{x}^{2}}\sqrt{bx-a}}+{\frac{3\,{b}^{2}}{4}\arctan \left ({1\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(3/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257003, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x^{2} \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b x - a} a}{x}\right ) - 2 \,{\left (5 \, b x - 2 \, a\right )} \sqrt{b x - a} \sqrt{-a}}{8 \, \sqrt{-a} x^{2}}, -\frac{3 \, b^{2} x^{2} \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) +{\left (5 \, b x - 2 \, a\right )} \sqrt{b x - a} \sqrt{a}}{4 \, \sqrt{a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.11389, size = 190, normalized size = 2.79 \[ \begin{cases} \frac{i a^{2}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{7 i a \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{5 i b^{\frac{3}{2}}}{4 \sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\\frac{a \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{2 x^{\frac{3}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{4 \sqrt{x}} - \frac{3 b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*a**2/(2*sqrt(b)*x**(5/2)*sqrt(a/(b*x) - 1)) - 7*I*a*sqrt(b)/(4*x**(
3/2)*sqrt(a/(b*x) - 1)) + 5*I*b**(3/2)/(4*sqrt(x)*sqrt(a/(b*x) - 1)) + 3*I*b**2*
acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*sqrt(a)), Abs(a/(b*x)) > 1), (a*sqrt(b)*sqrt
(-a/(b*x) + 1)/(2*x**(3/2)) - 5*b**(3/2)*sqrt(-a/(b*x) + 1)/(4*sqrt(x)) - 3*b**2
*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.205394, size = 89, normalized size = 1.31 \[ \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{5 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{3} + 3 \, \sqrt{b x - a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*b^3*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a) - (5*(b*x - a)^(3/2)*b^3 + 3*sq
rt(b*x - a)*a*b^3)/(b^2*x^2))/b