3.1639 \(\int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=76 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.133347, antiderivative size = 76, 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 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.0637, size = 61, normalized size = 0.8 \[ \frac{\sqrt{d + e x}}{\left (a + b x\right ) \left (a e - b d\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\sqrt{b} \left (a e - b d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.124003, size = 77, normalized size = 1.01 \[ \frac{\frac{\sqrt{d+e x}}{a+b x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}}}{a e-b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 77, normalized size = 1. \[{\frac{e}{ \left ( ae-bd \right ) \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{e}{ae-bd}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220418, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b e x + a e\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right ) + 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{2 \,{\left (a b d - a^{2} e +{\left (b^{2} d - a b e\right )} x\right )} \sqrt{b^{2} d - a b e}}, \frac{{\left (b e x + a e\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ) - \sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{{\left (a b d - a^{2} e +{\left (b^{2} d - a b e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{2} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x)**2*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.210896, size = 131, normalized size = 1.72 \[ -\frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} - \frac{\sqrt{x e + d} e}{{\left ({\left (x e + d\right )} b - b d + a e\right )}{\left (b d - a e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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