3.519 \(\int \frac{\sqrt{2-b x}}{x^{7/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{b (2-b x)^{3/2}}{15 x^{3/2}}-\frac{(2-b x)^{3/2}}{5 x^{5/2}} \]

[Out]

-(2 - b*x)^(3/2)/(5*x^(5/2)) - (b*(2 - b*x)^(3/2))/(15*x^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0241741, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{b (2-b x)^{3/2}}{15 x^{3/2}}-\frac{(2-b x)^{3/2}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 - b*x]/x^(7/2),x]

[Out]

-(2 - b*x)^(3/2)/(5*x^(5/2)) - (b*(2 - b*x)^(3/2))/(15*x^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.2404, size = 32, normalized size = 0.8 \[ - \frac{b \left (- b x + 2\right )^{\frac{3}{2}}}{15 x^{\frac{3}{2}}} - \frac{\left (- b x + 2\right )^{\frac{3}{2}}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x+2)**(1/2)/x**(7/2),x)

[Out]

-b*(-b*x + 2)**(3/2)/(15*x**(3/2)) - (-b*x + 2)**(3/2)/(5*x**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0149605, size = 31, normalized size = 0.78 \[ \frac{\sqrt{2-b x} \left (b^2 x^2+b x-6\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 - b*x]/x^(7/2),x]

[Out]

(Sqrt[2 - b*x]*(-6 + b*x + b^2*x^2))/(15*x^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 19, normalized size = 0.5 \[ -{\frac{bx+3}{15} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(1/2)/x^(7/2),x)

[Out]

-1/15*(b*x+3)*(-b*x+2)^(3/2)/x^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.34254, size = 38, normalized size = 0.95 \[ -\frac{{\left (-b x + 2\right )}^{\frac{3}{2}} b}{6 \, x^{\frac{3}{2}}} - \frac{{\left (-b x + 2\right )}^{\frac{5}{2}}}{10 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x + 2)/x^(7/2),x, algorithm="maxima")

[Out]

-1/6*(-b*x + 2)^(3/2)*b/x^(3/2) - 1/10*(-b*x + 2)^(5/2)/x^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.212782, size = 34, normalized size = 0.85 \[ \frac{{\left (b^{2} x^{2} + b x - 6\right )} \sqrt{-b x + 2}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x + 2)/x^(7/2),x, algorithm="fricas")

[Out]

1/15*(b^2*x^2 + b*x - 6)*sqrt(-b*x + 2)/x^(5/2)

_______________________________________________________________________________________

Sympy [A]  time = 138.413, size = 196, normalized size = 4.9 \[ \begin{cases} \frac{b^{\frac{5}{2}} \sqrt{-1 + \frac{2}{b x}}}{15} + \frac{b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{15 x} - \frac{2 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{5 x^{2}} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\\frac{i b^{\frac{9}{2}} x^{2} \sqrt{1 - \frac{2}{b x}}}{15 b^{2} x^{2} - 30 b x} - \frac{i b^{\frac{7}{2}} x \sqrt{1 - \frac{2}{b x}}}{15 b^{2} x^{2} - 30 b x} - \frac{8 i b^{\frac{5}{2}} \sqrt{1 - \frac{2}{b x}}}{15 b^{2} x^{2} - 30 b x} + \frac{12 i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{x \left (15 b^{2} x^{2} - 30 b x\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x+2)**(1/2)/x**(7/2),x)

[Out]

Piecewise((b**(5/2)*sqrt(-1 + 2/(b*x))/15 + b**(3/2)*sqrt(-1 + 2/(b*x))/(15*x) -
2*sqrt(b)*sqrt(-1 + 2/(b*x))/(5*x**2), 2*Abs(1/(b*x)) > 1), (I*b**(9/2)*x**2*sq
rt(1 - 2/(b*x))/(15*b**2*x**2 - 30*b*x) - I*b**(7/2)*x*sqrt(1 - 2/(b*x))/(15*b**
2*x**2 - 30*b*x) - 8*I*b**(5/2)*sqrt(1 - 2/(b*x))/(15*b**2*x**2 - 30*b*x) + 12*I
*b**(3/2)*sqrt(1 - 2/(b*x))/(x*(15*b**2*x**2 - 30*b*x)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221332, size = 65, normalized size = 1.62 \[ \frac{{\left ({\left (b x - 2\right )} b^{5} + 5 \, b^{5}\right )}{\left (b x - 2\right )} \sqrt{-b x + 2} b}{15 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{5}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x + 2)/x^(7/2),x, algorithm="giac")

[Out]

1/15*((b*x - 2)*b^5 + 5*b^5)*(b*x - 2)*sqrt(-b*x + 2)*b/(((b*x - 2)*b + 2*b)^(5/
2)*abs(b))