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

Optimal. Leaf size=24 \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{a+b x}{c-b x}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0600706, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1961, 12, 203} \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{a+b x}{c-b x}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1961

Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n -
1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^(1/n + 1), x], x], x, ((e*(a + b*x
^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/
n] && IntegerQ[r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [B]  time = 0.200211, size = 93, normalized size = 3.88 \[ \frac{2 b \sqrt{c-b x} \sqrt{\frac{a+b x}{c-b x}} \sin ^{-1}\left (\frac{b \sqrt{c-b x}}{\sqrt{-b} \sqrt{-b (a+c)}}\right )}{(-b)^{3/2} \sqrt{-b (a+c)} \sqrt{\frac{a+b x}{a+c}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*Sqrt[c - b*x]*Sqrt[(a + b*x)/(c - b*x)]*ArcSin[(b*Sqrt[c - b*x])/(Sqrt[-b]*Sqrt[-(b*(a + c))])])/((-b)^(3
/2)*Sqrt[-(b*(a + c))]*Sqrt[(a + b*x)/(a + c)])

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Maple [B]  time = 0.029, size = 85, normalized size = 3.5 \begin{align*} -{(bx-c)\arctan \left ({\frac{2\,bx+a-c}{2\,b}\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( bx+a \right ) \left ( bx-c \right ) }}}} \right ) \sqrt{-{\frac{bx+a}{bx-c}}}{\frac{1}{\sqrt{{b}^{2}}}}{\frac{1}{\sqrt{- \left ( bx+a \right ) \left ( bx-c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45854, size = 32, normalized size = 1.33 \begin{align*} \frac{2 \, \arctan \left (\sqrt{-\frac{b x + a}{b x - c}}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71513, size = 54, normalized size = 2.25 \begin{align*} \frac{2 \, \arctan \left (\sqrt{-\frac{b x + a}{b x - c}}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.20831, size = 55, normalized size = 2.29 \begin{align*} -\frac{\arcsin \left (-\frac{2 \, b x + a - c}{a + c}\right ) \mathrm{sgn}\left (-a b - b c\right ) \mathrm{sgn}\left (b x - c\right )}{{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(-(2*b*x + a - c)/(a + c))*sgn(-a*b - b*c)*sgn(b*x - c)/abs(b)