\(\int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx\) [638]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=\frac {\sqrt {x}}{\sqrt {2-b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, 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} \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=\frac {\sqrt {x}}{\sqrt {2-b x}} \]

[In]

Int[1/(Sqrt[x]*(2 - b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 - b*x]

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 {\sqrt {x}}{\sqrt {2-b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=\frac {\sqrt {x}}{\sqrt {2-b x}} \]

[In]

Integrate[1/(Sqrt[x]*(2 - b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 - b*x]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {\sqrt {x}}{\sqrt {-b x +2}}\)	\(13\)
default	\(\frac {\sqrt {x}}{\sqrt {-b x +2}}\)	\(13\)
meijerg	\(\frac {\sqrt {x}\, \sqrt {2}}{2 \sqrt {-\frac {b x}{2}+1}}\)	\(17\)

[In]

int(1/(-b*x+2)^(3/2)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=-\frac {\sqrt {-b x + 2} \sqrt {x}}{b x - 2} \]

[In]

integrate(1/(-b*x+2)^(3/2)/x^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-b*x + 2)*sqrt(x)/(b*x - 2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=\begin {cases} \frac {1}{\sqrt {b} \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {i}{\sqrt {b} \sqrt {1 - \frac {2}{b x}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(-b*x+2)**(3/2)/x**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=\frac {\sqrt {x}}{\sqrt {-b x + 2}} \]

[In]

integrate(1/(-b*x+2)^(3/2)/x^(1/2),x, algorithm="maxima")

[Out]

sqrt(x)/sqrt(-b*x + 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=-\frac {4 \, \sqrt {-b} b}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} {\left | b \right |}} \]

[In]

integrate(1/(-b*x+2)^(3/2)/x^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx=\frac {\sqrt {x}}{\sqrt {2-b\,x}} \]

[In]

int(1/(x^(1/2)*(2 - b*x)^(3/2)),x)

[Out]

x^(1/2)/(2 - b*x)^(1/2)