3.2069 \(\int \frac{(a+b x) \sqrt{d+e x}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=62 \[ \frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0769908, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.0296, size = 53, normalized size = 0.85 \[ \frac{2 \sqrt{d + e x}}{b} - \frac{2 \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0574987, size = 62, normalized size = 1. \[ \frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 92, normalized size = 1.5 \[ 2\,{\frac{\sqrt{ex+d}}{b}}-2\,{\frac{ae}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+2\,{\frac{d}{\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.299231, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt{e x + d}}{b}, -\frac{2 \,{\left (\sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) - \sqrt{e x + d}\right )}}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 16.4864, size = 178, normalized size = 2.87 \[ \frac{2 \left (\frac{e \sqrt{d + e x}}{b} - \frac{e \left (a e - b d\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.302078, size = 90, normalized size = 1.45 \[ \frac{2 \,{\left (b d - a e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b} + \frac{2 \, \sqrt{x e + d}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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