3.1419 \(\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.060896, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.75618, size = 41, normalized size = 0.87 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\sqrt{b} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0352816, size = 47, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 37, normalized size = 0.8 \[ 2\,{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.228043, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{\sqrt{b^{2} c - a b d}}, -\frac{2 \, \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{\sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.18598, size = 189, normalized size = 4.02 \[ - 2 \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{\frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{b}{a d - b c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{- \frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{1}{c + d x} > - \frac{b}{a d - b c} \wedge \frac{b}{a d - b c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{- \frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{b}{a d - b c} < 0 \wedge \frac{1}{c + d x} < - \frac{b}{a d - b c} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)/(d*x+c)**(1/2),x)

[Out]

-2*Piecewise((atan(1/(sqrt(b/(a*d - b*c))*sqrt(c + d*x)))/(sqrt(b/(a*d - b*c))*(
a*d - b*c)), b/(a*d - b*c) > 0), (-acoth(1/(sqrt(-b/(a*d - b*c))*sqrt(c + d*x)))
/(sqrt(-b/(a*d - b*c))*(a*d - b*c)), (b/(a*d - b*c) < 0) & (1/(c + d*x) > -b/(a*
d - b*c))), (-atanh(1/(sqrt(-b/(a*d - b*c))*sqrt(c + d*x)))/(sqrt(-b/(a*d - b*c)
)*(a*d - b*c)), (b/(a*d - b*c) < 0) & (1/(c + d*x) < -b/(a*d - b*c))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217547, size = 51, normalized size = 1.09 \[ \frac{2 \, \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*sqrt(d*x + c)),x, algorithm="giac")

[Out]

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