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

Optimal. Leaf size=22 \[ -\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

[Out]

(-2*(a - b*x)^(3/2))/(3*a*x^(3/2))

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Rubi [A]  time = 0.0017392, antiderivative size = 22, 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} \[ -\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - b*x]/x^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0063745, size = 22, normalized size = 1. \[ -\frac{2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - b*x]/x^(5/2),x]

[Out]

(-2*(a - b*x)^(3/2))/(3*a*x^(3/2))

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,a} \left ( -bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02236, size = 22, normalized size = 1. \begin{align*} -\frac{2 \,{\left (-b x + a\right )}^{\frac{3}{2}}}{3 \, a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-b*x + a)^(3/2)/(a*x^(3/2))

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Fricas [A]  time = 1.59402, size = 57, normalized size = 2.59 \begin{align*} \frac{2 \,{\left (b x - a\right )} \sqrt{-b x + a}}{3 \, a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.13464, size = 92, normalized size = 4.18 \begin{align*} \begin{cases} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 x} + \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3 a} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{3 x} + \frac{2 i b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{3 a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.24095, size = 57, normalized size = 2.59 \begin{align*} \frac{2 \,{\left (b x - a\right )} \sqrt{-b x + a} b^{4}}{3 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{3}{2}} a{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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