3.518 \(\int \frac{\sqrt{2-b x}}{x^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{(2-b x)^{3/2}}{3 x^{3/2}} \]

[Out]

-(2 - b*x)^(3/2)/(3*x^(3/2))

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Rubi [A]  time = 0.0014399, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {37} \[ -\frac{(2-b x)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - b*x]/x^(5/2),x]

[Out]

-(2 - b*x)^(3/2)/(3*x^(3/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{2-b x}}{x^{5/2}} \, dx &=-\frac{(2-b x)^{3/2}}{3 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0053656, size = 19, normalized size = 1. \[ -\frac{(2-b x)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 - b*x]/x^(5/2),x]

[Out]

-(2 - b*x)^(3/2)/(3*x^(3/2))

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \begin{align*} -{\frac{1}{3} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x+2)^(1/2)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.02419, size = 18, normalized size = 0.95 \begin{align*} -\frac{{\left (-b x + 2\right )}^{\frac{3}{2}}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(5/2),x, algorithm="maxima")

[Out]

-1/3*(-b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 1.6011, size = 51, normalized size = 2.68 \begin{align*} \frac{{\left (b x - 2\right )} \sqrt{-b x + 2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(5/2),x, algorithm="fricas")

[Out]

1/3*(b*x - 2)*sqrt(-b*x + 2)/x^(3/2)

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Sympy [B]  time = 2.32574, size = 82, normalized size = 4.32 \begin{align*} \begin{cases} \frac{b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{3} - \frac{2 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{3 x} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\\frac{i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{3} - \frac{2 i \sqrt{b} \sqrt{1 - \frac{2}{b x}}}{3 x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)**(1/2)/x**(5/2),x)

[Out]

Piecewise((b**(3/2)*sqrt(-1 + 2/(b*x))/3 - 2*sqrt(b)*sqrt(-1 + 2/(b*x))/(3*x), 2/Abs(b*x) > 1), (I*b**(3/2)*sq
rt(1 - 2/(b*x))/3 - 2*I*sqrt(b)*sqrt(1 - 2/(b*x))/(3*x), True))

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Giac [B]  time = 1.28805, size = 47, normalized size = 2.47 \begin{align*} \frac{{\left (b x - 2\right )} \sqrt{-b x + 2} b^{4}}{3 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{3}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(5/2),x, algorithm="giac")

[Out]

1/3*(b*x - 2)*sqrt(-b*x + 2)*b^4/(((b*x - 2)*b + 2*b)^(3/2)*abs(b))