\(\int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx\) [594]

Optimal result

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

[Out]

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

[In]

Int[1/(x^(3/2)*Sqrt[a - b*x]),x]

[Out]

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

Mathematica [A] (verified)

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

[In]

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

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

Maple [A] (verified)

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

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx=-\frac {2 \, \sqrt {-b x + a}}{a \sqrt {x}} \]

[In]

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx=\begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{a} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx=-\frac {2 \, \sqrt {-b x + a} b^{2}}{\sqrt {{\left (b x - a\right )} b + a b} a {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx=-\frac {2\,\sqrt {a-b\,x}}{a\,\sqrt {x}} \]

[In]

int(1/(x^(3/2)*(a - b*x)^(1/2)),x)

[Out]

-(2*(a - b*x)^(1/2))/(a*x^(1/2))