3.2070 \(\int \frac{a+b x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.052506, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.7754, size = 41, normalized size = 0.87 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\sqrt{b} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0325717, size = 47, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 37, normalized size = 0.8 \[ 2\,{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.294633, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{\sqrt{b^{2} d - a b e}}, -\frac{2 \, \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{\sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 6.21392, size = 189, normalized size = 4.02 \[ - 2 \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{b}{a e - b d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{1}{d + e x} > - \frac{b}{a e - b d} \wedge \frac{b}{a e - b d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{b}{a e - b d} < 0 \wedge \frac{1}{d + e x} < - \frac{b}{a e - b d} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.283704, size = 55, normalized size = 1.17 \[ \frac{2 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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