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

Optimal. Leaf size=61 \[ \frac{\tanh ^{-1}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (a d+b c)+a c+b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0254436, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1981, 621, 206} \[ \frac{\tanh ^{-1}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (a d+b c)+a c+b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a + b*x)*(c + d*x)],x]

[Out]

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

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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) (c+d x)}} \, dx &=\int \frac{1}{\sqrt{a c+(b c+a d) x+b d x^2}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1}{4 b d-x^2} \, dx,x,\frac{b c+a d+2 b d x}{\sqrt{a c+(b c+a d) x+b d x^2}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{b c+a d+2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{a c+(b c+a d) x+b d x^2}}\right )}{\sqrt{b} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.0628196, size = 95, normalized size = 1.56 \[ \frac{2 \sqrt{a+b x} \sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b \sqrt{d} \sqrt{(a+b x) (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 49, normalized size = 0.8 \begin{align*}{\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{ac+ \left ( ad+bc \right ) x+bd{x}^{2}} \right ){\frac{1}{\sqrt{bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54732, size = 435, normalized size = 7.13 \begin{align*} \left [\frac{\sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, \sqrt{b d x^{2} + a c +{\left (b c + a d\right )} x}{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )}{2 \, b d}, -\frac{\sqrt{-b d} \arctan \left (\frac{\sqrt{b d x^{2} + a c +{\left (b c + a d\right )} x}{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*sqrt(b*d*x^2 + a*c + (b*c + a*d)*x)*(2*b*
d*x + b*c + a*d)*sqrt(b*d) + 8*(b^2*c*d + a*b*d^2)*x)/(b*d), -sqrt(-b*d)*arctan(1/2*sqrt(b*d*x^2 + a*c + (b*c
+ a*d)*x)*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x))/(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(1/((b*x+a)*(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27431, size = 92, normalized size = 1.51 \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 )}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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