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]

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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ -\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

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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fricas [A]  time = 0.44, size = 23, normalized size = 1.05 \[ \frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a}}{3 \, a x^{\frac {3}{2}}} \]

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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giac [B]  time = 1.40, size = 42, normalized size = 1.91 \[ \frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a} b^{4}}{3 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {3}{2}} a {\left | b \right |}} \]

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))

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maple [A]  time = 0.00, size = 17, normalized size = 0.77 \[ -\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}} \]

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 = 1.31, size = 16, normalized size = 0.73 \[ -\frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \]

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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mupad [B]  time = 0.24, size = 21, normalized size = 0.95 \[ \frac {\left (\frac {2\,b\,x}{3\,a}-\frac {2}{3}\right )\,\sqrt {a-b\,x}}{x^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x)^(1/2)/x^(5/2),x)

[Out]

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

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sympy [B]  time = 1.55, size = 88, normalized size = 4.00 \[ \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 x} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {2 i b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a} & \text {otherwise} \end {cases} \]

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/(b*x)) > 1), (-2*I*s
qrt(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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