3.1148 \(\int \frac{1}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{c (a-b x)}}\right )}{b \sqrt{c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0199325, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {63, 217, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{c (a-b x)}}\right )}{b \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]

[Out]

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

Rule 63

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 217

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]

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.01539, size = 48, normalized size = 1.26 \[ -\frac{2 \sqrt{a-b x} \tan ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right )}{b \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 71, normalized size = 1.9 \begin{align*}{\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.57401, size = 252, normalized size = 6.63 \begin{align*} \left [-\frac{\sqrt{-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right )}{2 \, b c}, -\frac{\arctan \left (\frac{\sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{c} x}{b^{2} c x^{2} - a^{2} c}\right )}{b \sqrt{c}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-c)*log(2*b^2*c*x^2 - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c)/(b*c), -arctan(sqrt(
-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c))/(b*sqrt(c))]

________________________________________________________________________________________

Sympy [C]  time = 3.42973, size = 90, normalized size = 2.37 \begin{align*} - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), a**2/(b**2*x**2))/(4*pi**(3/2)*b*sq
rt(c)) + meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), a**2*exp_polar(-2*I*pi)/(
b**2*x**2))/(4*pi**(3/2)*b*sqrt(c))

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out