3.1695 \(\int \frac{\sqrt{d+e x}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=112 \[ \frac{2 (a+b x) \sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.164595, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 (a+b x) \sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0536314, size = 81, normalized size = 0.72 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{d+e x}-\sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{b^{3/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 104, normalized size = 0.9 \[ 2\,{\frac{bx+a}{\sqrt{ \left ( bx+a \right ) ^{2}}b\sqrt{b \left ( ae-bd \right ) }} \left ( -\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) ae+\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) bd+\sqrt{ex+d}\sqrt{b \left ( ae-bd \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220214, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt{e x + d}}{b}, -\frac{2 \,{\left (\sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) - \sqrt{e x + d}\right )}}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt((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]

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Sympy [A]  time = 14.9637, size = 95, normalized size = 0.85 \[ \sqrt{- \frac{a e}{b^{3}} + \frac{d}{b^{2}}} \log{\left (- b \sqrt{- \frac{a e}{b^{3}} + \frac{d}{b^{2}}} + \sqrt{d + e x} \right )} - \sqrt{- \frac{a e}{b^{3}} + \frac{d}{b^{2}}} \log{\left (b \sqrt{- \frac{a e}{b^{3}} + \frac{d}{b^{2}}} + \sqrt{d + e x} \right )} + \frac{2 \sqrt{d + e x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214394, size = 99, normalized size = 0.88 \[ 2 \,{\left (\frac{{\left (b d - a e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b} + \frac{\sqrt{x e + d}}{b}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt((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) + sqrt(x*e + d)/b)*sign(b*x + a)