\(\int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx\) [599]

Optimal result

Integrand size = 16, antiderivative size = 20 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=\frac {2 \sqrt {x}}{a \sqrt {a-b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, 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} \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=\frac {2 \sqrt {x}}{a \sqrt {a-b x}} \]

[In]

Int[1/(Sqrt[x]*(a - b*x)^(3/2)),x]

[Out]

(2*Sqrt[x])/(a*Sqrt[a - 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 {2 \sqrt {x}}{a \sqrt {a-b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=\frac {2 \sqrt {x}}{a \sqrt {a-b x}} \]

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a - b*x])

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
gosper	\(\frac {2 \sqrt {x}}{a \sqrt {-b x +a}}\)	\(17\)
default	\(\frac {2 \sqrt {x}}{a \sqrt {-b x +a}}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=-\frac {2 \, \sqrt {-b x + a} \sqrt {x}}{a b x - a^{2}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=\begin {cases} \frac {2}{a \sqrt {b} \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i}{a \sqrt {b} \sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=\frac {2 \, \sqrt {x}}{\sqrt {-b x + a} a} \]

[In]

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

[Out]

2*sqrt(x)/(sqrt(-b*x + a)*a)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (16) = 32\).

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.65 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=-\frac {4 \, \sqrt {-b} b}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx=\frac {2\,\sqrt {x}\,\sqrt {a-b\,x}}{a^2-a\,b\,x} \]

[In]

int(1/(x^(1/2)*(a - b*x)^(3/2)),x)

[Out]

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