3.740 \(\int \frac{\sqrt{\frac{a+b x}{c+d x}}}{a+b x} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.066614, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1961, 12, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 208

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

Rubi steps

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

Mathematica [B]  time = 0.0733484, size = 97, normalized size = 2.37 \[ \frac{2 \sqrt{b c-a d} \sqrt{\frac{a+b x}{c+d x}} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b \sqrt{d} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*c - a*d]*Sqrt[(a + b*x)/(c + d*x)]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[b*c - a*d]])/(b*Sqrt[d]*Sqrt[a + b*x])

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Maple [B]  time = 0.02, size = 80, normalized size = 2. \begin{align*}{(dx+c)\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) \sqrt{{\frac{bx+a}{dx+c}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72651, size = 251, normalized size = 6.12 \begin{align*} \left [\frac{\sqrt{b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt{b d}{\left (d x + c\right )} \sqrt{\frac{b x + a}{d x + c}}\right )}{b d}, -\frac{2 \, \sqrt{-b d} \arctan \left (\frac{\sqrt{-b d}{\left (d x + c\right )} \sqrt{\frac{b x + a}{d x + c}}}{b d x + a d}\right )}{b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(b*d)*log(2*b*d*x + b*c + a*d + 2*sqrt(b*d)*(d*x + c)*sqrt((b*x + a)/(d*x + c)))/(b*d), -2*sqrt(-b*d)*arc
tan(sqrt(-b*d)*(d*x + c)*sqrt((b*x + a)/(d*x + c))/(b*d*x + a*d))/(b*d)]

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

[Out]

Timed out

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Giac [B]  time = 1.25758, size = 100, normalized size = 2.44 \begin{align*} -\frac{\sqrt{b d} \log \left ({\left | -2 \,{\left (\sqrt{b d} x - \sqrt{b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt{b d} b c - \sqrt{b d} a d \right |}\right ) \mathrm{sgn}\left (d x + c\right )}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*d)*log(abs(-2*(sqrt(b*d)*x - sqrt(b*d*x^2 + b*c*x + a*d*x + a*c))*b*d - sqrt(b*d)*b*c - sqrt(b*d)*a*d)
)*sgn(d*x + c)/(b*d)