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

Optimal. Leaf size=76 \[ \frac{(c+d x) \sqrt{\frac{a+b x}{c+d x}}}{d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.039146, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1959, 288, 208} \[ \frac{(c+d x) \sqrt{\frac{a+b x}{c+d x}}}{d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} d^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(a + b*x)/(c + d*x)],x]

[Out]

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

Rule 1959

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, 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 \sqrt{\frac{a+b x}{c+d x}} \, dx &=(2 (b c-a d)) \operatorname{Subst}\left (\int \frac{x^2}{\left (b-d x^2\right )^2} \, dx,x,\sqrt{\frac{a+b x}{c+d x}}\right )\\ &=\frac{\sqrt{\frac{a+b x}{c+d x}} (c+d x)}{d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{b-d x^2} \, dx,x,\sqrt{\frac{a+b x}{c+d x}}\right )}{d}\\ &=\frac{\sqrt{\frac{a+b x}{c+d x}} (c+d x)}{d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.273589, size = 123, normalized size = 1.62 \[ \frac{\sqrt{\frac{a+b x}{c+d x}} \left (b \sqrt{d} (a+b x) (c+d x)-\sqrt{a+b x} (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{b d^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 152, normalized size = 2. \begin{align*}{\frac{dx+c}{2\,d}\sqrt{{\frac{bx+a}{dx+c}}} \left ( \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 ) ad-\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 ) bc+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78148, size = 424, normalized size = 5.58 \begin{align*} \left [-\frac{{\left (b c - a d\right )} \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 ) - 2 \,{\left (b d^{2} x + b c d\right )} \sqrt{\frac{b x + a}{d x + c}}}{2 \, b d^{2}}, \frac{{\left (b c - a d\right )} \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 ) +{\left (b d^{2} x + b c d\right )} \sqrt{\frac{b x + a}{d x + c}}}{b d^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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),x)

[Out]

Timed out

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Giac [A]  time = 1.21165, size = 161, normalized size = 2.12 \begin{align*} \frac{\sqrt{b d x^{2} + b c x + a d x + a c} \mathrm{sgn}\left (d x + c\right )}{d} + \frac{{\left (b c \mathrm{sgn}\left (d x + c\right ) - a d \mathrm{sgn}\left (d x + c\right )\right )} \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 )}{2 \, b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*d*x^2 + b*c*x + a*d*x + a*c)*sgn(d*x + c)/d + 1/2*(b*c*sgn(d*x + c) - a*d*sgn(d*x + c))*sqrt(b*d)*log(a
bs(-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))/(b*d^2)