\(\int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx\) [502]

Optimal result

Integrand size = 16, antiderivative size = 22 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=-\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37} \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=-\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

[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*} \text {integral}& = -\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=-\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

[In]

Integrate[Sqrt[a - b*x]/x^(5/2),x]

[Out]

(-2*(a - b*x)^(3/2))/(3*a*x^(3/2))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
gosper	\(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\)	\(17\)
risch	\(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {\sqrt {-b x +a}}{x^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{2}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=\frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a}}{3 \, a x^{\frac {3}{2}}} \]

[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))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=\begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 x} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {2 i b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=-\frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \]

[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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (16) = 32\).

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=\frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a} b^{4}}{3 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {3}{2}} a {\left | b \right |}} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx=\frac {\left (\frac {2\,b\,x}{3\,a}-\frac {2}{3}\right )\,\sqrt {a-b\,x}}{x^{3/2}} \]

[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)