3.333 \(\int \frac{(-a+b x)^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{15}{4} b^2 \sqrt{b x-a}-\frac{15}{4} \sqrt{a} b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-\frac{(b x-a)^{5/2}}{2 x^2}-\frac{5 b (b x-a)^{3/2}}{4 x} \]

[Out]

(15*b^2*Sqrt[-a + b*x])/4 - (5*b*(-a + b*x)^(3/2))/(4*x) - (-a + b*x)^(5/2)/(2*x
^2) - (15*Sqrt[a]*b^2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/4

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Rubi [A]  time = 0.0726893, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{15}{4} b^2 \sqrt{b x-a}-\frac{15}{4} \sqrt{a} b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-\frac{(b x-a)^{5/2}}{2 x^2}-\frac{5 b (b x-a)^{3/2}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(-a + b*x)^(5/2)/x^3,x]

[Out]

(15*b^2*Sqrt[-a + b*x])/4 - (5*b*(-a + b*x)^(3/2))/(4*x) - (-a + b*x)^(5/2)/(2*x
^2) - (15*Sqrt[a]*b^2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/4

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Rubi in Sympy [A]  time = 10.0026, size = 70, normalized size = 0.81 \[ - \frac{15 \sqrt{a} b^{2} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{4} + \frac{15 b^{2} \sqrt{- a + b x}}{4} - \frac{5 b \left (- a + b x\right )^{\frac{3}{2}}}{4 x} - \frac{\left (- a + b x\right )^{\frac{5}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x-a)**(5/2)/x**3,x)

[Out]

-15*sqrt(a)*b**2*atan(sqrt(-a + b*x)/sqrt(a))/4 + 15*b**2*sqrt(-a + b*x)/4 - 5*b
*(-a + b*x)**(3/2)/(4*x) - (-a + b*x)**(5/2)/(2*x**2)

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Mathematica [A]  time = 0.0728761, size = 68, normalized size = 0.79 \[ \left (-\frac{a^2}{2 x^2}+\frac{9 a b}{4 x}+2 b^2\right ) \sqrt{b x-a}-\frac{15}{4} \sqrt{a} b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-a + b*x)^(5/2)/x^3,x]

[Out]

(2*b^2 - a^2/(2*x^2) + (9*a*b)/(4*x))*Sqrt[-a + b*x] - (15*Sqrt[a]*b^2*ArcTan[Sq
rt[-a + b*x]/Sqrt[a]])/4

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Maple [A]  time = 0.015, size = 70, normalized size = 0.8 \[ 2\,{b}^{2}\sqrt{bx-a}+{\frac{9\,a}{4\,{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{a}^{2}}{4\,{x}^{2}}\sqrt{bx-a}}-{\frac{15\,{b}^{2}}{4}\arctan \left ({1\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(5/2)/x^3,x)

[Out]

2*b^2*(b*x-a)^(1/2)+9/4*a/x^2*(b*x-a)^(3/2)+7/4/x^2*(b*x-a)^(1/2)*a^2-15/4*b^2*a
rctan((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)^(5/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218083, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 9 \, a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{8 \, x^{2}}, -\frac{15 \, \sqrt{a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) -{\left (8 \, b^{2} x^{2} + 9 \, a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{4 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(5/2)/x^3,x, algorithm="fricas")

[Out]

[1/8*(15*sqrt(-a)*b^2*x^2*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2*(8*b
^2*x^2 + 9*a*b*x - 2*a^2)*sqrt(b*x - a))/x^2, -1/4*(15*sqrt(a)*b^2*x^2*arctan(sq
rt(b*x - a)/sqrt(a)) - (8*b^2*x^2 + 9*a*b*x - 2*a^2)*sqrt(b*x - a))/x^2]

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Sympy [A]  time = 6.73167, size = 267, normalized size = 3.1 \[ \begin{cases} - \frac{15 i \sqrt{a} b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4} - \frac{i a^{3}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{11 i a^{2} \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{i a b^{\frac{3}{2}}}{4 \sqrt{x} \sqrt{\frac{a}{b x} - 1}} - \frac{2 i b^{\frac{5}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} - 1}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\\frac{15 \sqrt{a} b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4} + \frac{a^{3}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{11 a^{2} \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{a b^{\frac{3}{2}}}{4 \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} + \frac{2 b^{\frac{5}{2}} \sqrt{x}}{\sqrt{- \frac{a}{b x} + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x-a)**(5/2)/x**3,x)

[Out]

Piecewise((-15*I*sqrt(a)*b**2*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/4 - I*a**3/(2*sqr
t(b)*x**(5/2)*sqrt(a/(b*x) - 1)) + 11*I*a**2*sqrt(b)/(4*x**(3/2)*sqrt(a/(b*x) -
1)) - I*a*b**(3/2)/(4*sqrt(x)*sqrt(a/(b*x) - 1)) - 2*I*b**(5/2)*sqrt(x)/sqrt(a/(
b*x) - 1), Abs(a/(b*x)) > 1), (15*sqrt(a)*b**2*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/4
+ a**3/(2*sqrt(b)*x**(5/2)*sqrt(-a/(b*x) + 1)) - 11*a**2*sqrt(b)/(4*x**(3/2)*sq
rt(-a/(b*x) + 1)) + a*b**(3/2)/(4*sqrt(x)*sqrt(-a/(b*x) + 1)) + 2*b**(5/2)*sqrt(
x)/sqrt(-a/(b*x) + 1), True))

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GIAC/XCAS [A]  time = 0.20907, size = 112, normalized size = 1.3 \[ -\frac{15 \, \sqrt{a} b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) - 8 \, \sqrt{b x - a} b^{3} - \frac{9 \,{\left (b x - a\right )}^{\frac{3}{2}} a b^{3} + 7 \, \sqrt{b x - a} a^{2} b^{3}}{b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(5/2)/x^3,x, algorithm="giac")

[Out]

-1/4*(15*sqrt(a)*b^3*arctan(sqrt(b*x - a)/sqrt(a)) - 8*sqrt(b*x - a)*b^3 - (9*(b
*x - a)^(3/2)*a*b^3 + 7*sqrt(b*x - a)*a^2*b^3)/(b^2*x^2))/b