3.1638 \(\int \frac{\sqrt{d+e x}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x}}{b (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0949649, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x}}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.7222, size = 56, normalized size = 0.8 \[ - \frac{\sqrt{d + e x}}{b \left (a + b x\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{3}{2}} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d + e*x)/(b*(a + b*x)) + e*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(b*
*(3/2)*sqrt(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0821236, size = 70, normalized size = 1. \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x}}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 64, normalized size = 0.9 \[ -{\frac{e}{b \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{e}{b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.219162, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b e x + a e\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right ) - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{2 \, \sqrt{b^{2} d - a b e}{\left (b^{2} x + a b\right )}}, -\frac{{\left (b e x + a e\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ) + \sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{\sqrt{-b^{2} d + a b e}{\left (b^{2} x + a b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 11.1732, size = 675, normalized size = 9.64 \[ - \frac{2 a e^{2} \sqrt{d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} + \frac{a e^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} \log{\left (- a^{2} e^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + \sqrt{d + e x} \right )}}{2 b} - \frac{a e^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} \log{\left (a^{2} e^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + \sqrt{d + e x} \right )}}{2 b} - \frac{d e \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} \log{\left (- a^{2} e^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + \sqrt{d + e x} \right )}}{2} + \frac{d e \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} \log{\left (a^{2} e^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt{- \frac{1}{b \left (a e - b d\right )^{3}}} + \sqrt{d + e x} \right )}}{2} + \frac{2 d e \sqrt{d + e x}}{2 a^{2} e^{2} - 2 a b d e + 2 a b e^{2} x - 2 b^{2} d e x} + \frac{2 e \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e}{b} - d}} \right )}}{b \sqrt{\frac{a e}{b} - d}} & \text{for}\: \frac{a e}{b} - d > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{- \frac{a e}{b} + d}} \right )}}{b \sqrt{- \frac{a e}{b} + d}} & \text{for}\: d + e x > - \frac{a e}{b} + d \wedge \frac{a e}{b} - d < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{- \frac{a e}{b} + d}} \right )}}{b \sqrt{- \frac{a e}{b} + d}} & \text{for}\: d + e x < - \frac{a e}{b} + d \wedge \frac{a e}{b} - d < 0 \end{cases}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3
*d*e*x) + a*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*
d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*
d)**3)) + sqrt(d + e*x))/(2*b) - a*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**
2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**
2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - d*e*sqrt(-1/(b*(a*e - b*d
)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e -
b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + d*e*sqrt(
-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqr
t(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x)
)/2 + 2*d*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**2*x - 2*b**2*d*e*x
) + 2*e*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/
b - d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b
- d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*
sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211229, size = 108, normalized size = 1.54 \[ \frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{\sqrt{-b^{2} d + a b e} b} - \frac{\sqrt{x e + d} e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")

[Out]

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