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

Optimal. Leaf size=93 \[ -\frac{15}{4} a^2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{5/2}}{\sqrt{x}}-\frac{5}{2} b \sqrt{x} (a-b x)^{3/2}-\frac{15}{4} a b \sqrt{x} \sqrt{a-b x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0690341, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{15}{4} a^2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{5/2}}{\sqrt{x}}-\frac{5}{2} b \sqrt{x} (a-b x)^{3/2}-\frac{15}{4} a b \sqrt{x} \sqrt{a-b x} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x)^(5/2)/x^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.6614, size = 87, normalized size = 0.94 \[ - \frac{15 a^{2} \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{4} - \frac{15 a b \sqrt{x} \sqrt{a - b x}}{4} - \frac{5 b \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{2} - \frac{2 \left (a - b x\right )^{\frac{5}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0797644, size = 72, normalized size = 0.77 \[ \frac{1}{4} \left (\frac{\sqrt{a-b x} \left (-8 a^2-9 a b x+2 b^2 x^2\right )}{\sqrt{x}}-15 a^2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a - b*x]*(-8*a^2 - 9*a*b*x + 2*b^2*x^2))/Sqrt[x] - 15*a^2*Sqrt[b]*ArcTan[
(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/4

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 88, normalized size = 1. \[ -{\frac{-2\,{b}^{2}{x}^{2}+9\,abx+8\,{a}^{2}}{4}\sqrt{-bx+a}{\frac{1}{\sqrt{x}}}}-{\frac{15\,{a}^{2}}{8}\sqrt{b}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(5/2)/x^(3/2),x)

[Out]

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

_______________________________________________________________________________________

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^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.22458, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{-b} x \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \,{\left (2 \, b^{2} x^{2} - 9 \, a b x - 8 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{8 \, x}, \frac{15 \, a^{2} \sqrt{b} x \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (2 \, b^{2} x^{2} - 9 \, a b x - 8 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{4 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 78.5413, size = 267, normalized size = 2.87 \[ \begin{cases} \frac{2 i a^{\frac{5}{2}}}{\sqrt{x} \sqrt{-1 + \frac{b x}{a}}} + \frac{i a^{\frac{3}{2}} b \sqrt{x}}{4 \sqrt{-1 + \frac{b x}{a}}} - \frac{11 i \sqrt{a} b^{2} x^{\frac{3}{2}}}{4 \sqrt{-1 + \frac{b x}{a}}} + \frac{15 i a^{2} \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4} + \frac{i b^{3} x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{2 a^{\frac{5}{2}}}{\sqrt{x} \sqrt{1 - \frac{b x}{a}}} - \frac{a^{\frac{3}{2}} b \sqrt{x}}{4 \sqrt{1 - \frac{b x}{a}}} + \frac{11 \sqrt{a} b^{2} x^{\frac{3}{2}}}{4 \sqrt{1 - \frac{b x}{a}}} - \frac{15 a^{2} \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4} - \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^(5/2)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError