3.1169 \(\int \frac{1}{\sqrt{a+b x} \sqrt{-a d+b d x}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(a*d) + b*d*x]])/(b*Sqrt[d])

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Rubi [A]  time = 0.024387, antiderivative size = 39, 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, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(a*d) + b*d*x]])/(b*Sqrt[d])

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 206

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

Rubi steps

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

Mathematica [A]  time = 0.0213767, size = 39, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(a*d) + b*d*x]])/(b*Sqrt[d])

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Maple [B]  time = 0.007, size = 76, normalized size = 2. \begin{align*}{\sqrt{ \left ( bx+a \right ) \left ( bdx-ad \right ) }\ln \left ({{b}^{2}dx{\frac{1}{\sqrt{{b}^{2}d}}}}+\sqrt{{b}^{2}d{x}^{2}-{a}^{2}d} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bdx-ad}}}{\frac{1}{\sqrt{{b}^{2}d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*d*x-a*d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58296, size = 248, normalized size = 6.36 \begin{align*} \left [\frac{\log \left (2 \, b^{2} d x^{2} + 2 \, \sqrt{b d x - a d} \sqrt{b x + a} b \sqrt{d} x - a^{2} d\right )}{2 \, b \sqrt{d}}, -\frac{\sqrt{-d} \arctan \left (\frac{\sqrt{b d x - a d} \sqrt{b x + a} b \sqrt{-d} x}{b^{2} d x^{2} - a^{2} d}\right )}{b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 3.35799, size = 88, normalized size = 2.26 \begin{align*} \frac{{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{d}} - \frac{i{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{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*sqrt(
d)) - I*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(d))

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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*d*x-a*d)^(1/2),x, algorithm="giac")

[Out]

Timed out