\(\int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\) [593]

Optimal result

Integrand size = 16, antiderivative size = 29 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {65, 223, 209} \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \]

[In]

Int[1/(Sqrt[x]*Sqrt[a - b*x]),x]

[Out]

(2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/Sqrt[b]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right ) \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=\frac {4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

(4*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/Sqrt[b]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(21)=42\).

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76

method	result	size
default	\(\frac {\sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{\sqrt {x}\, \sqrt {-b x +a}\, \sqrt {b}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=\left [-\frac {\sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right )}{b}, -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}\right ] \]

[In]

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

[Out]

[-sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a)/b, -2*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/
sqrt(b)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=\begin {cases} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).

Time = 77.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=\frac {2 \, b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {\sqrt {a-b\,x}-\sqrt {a}}{\sqrt {b}\,\sqrt {x}}\right )}{\sqrt {b}} \]

[In]

int(1/(x^(1/2)*(a - b*x)^(1/2)),x)

[Out]

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