3.1704 \(\int \frac{\sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=168 \[ -\frac{e \sqrt{d+e x}}{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]

[Out]

-(e*Sqrt[d + e*x])/(4*b*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e*
x]/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^2*(a + b*x)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(3/2)*(b*d - a*e)^(3/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])

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Rubi [A]  time = 0.256376, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{e \sqrt{d+e x}}{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-(e*Sqrt[d + e*x])/(4*b*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e*
x]/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^2*(a + b*x)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(3/2)*(b*d - a*e)^(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**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.226696, size = 120, normalized size = 0.71 \[ \frac{e^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )+\sqrt{b} \sqrt{d+e x} \sqrt{b d-a e} (a e-b (2 d+e x))}{4 b^{3/2} (a+b x) \sqrt{(a+b x)^2} (b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(Sqrt[b]*Sqrt[b*d - a*e]*Sqrt[d + e*x]*(a*e - b*(2*d + e*x)) + e^2*(a + b*x)^2*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(3/2)*(b*d - a*e)^(3/2)*(a
+ b*x)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.021, size = 200, normalized size = 1.2 \[{\frac{bx+a}{ \left ( 4\,ae-4\,bd \right ) b} \left ( \arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){x}^{2}{b}^{2}{e}^{2}+2\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xab{e}^{2}+\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}b+\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){a}^{2}{e}^{2}-\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae+\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

1/4*(arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^2*e^2+2*arctan((e*x+d)^(1
/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b*e^2+(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b+arctan(
(e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*e^2-(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a
*e+(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b*d)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(a*e-b*d)/
b/((b*x+a)^2)^(3/2)

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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)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21843, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} \sqrt{e x + d} +{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (a^{2} b^{2} d - a^{3} b e +{\left (b^{4} d - a b^{3} e\right )} x^{2} + 2 \,{\left (a b^{3} d - a^{2} b^{2} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (b e x + 2 \, b d - a e\right )} \sqrt{e x + d} -{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} b^{2} d - a^{3} b e +{\left (b^{4} d - a b^{3} e\right )} x^{2} + 2 \,{\left (a b^{3} d - a^{2} b^{2} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e)*sqrt(e*x + d) + (b^2*e^2*x^2
+ 2*a*b*e^2*x + a^2*e^2)*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)))/((a^2*b^2*d - a^3*b*e + (b^4*d - a*b^3*e)
*x^2 + 2*(a*b^3*d - a^2*b^2*e)*x)*sqrt(b^2*d - a*b*e)), -1/4*(sqrt(-b^2*d + a*b*
e)*(b*e*x + 2*b*d - a*e)*sqrt(e*x + d) - (b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*a
rctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^2*b^2*d - a^3*b*e
+ (b^4*d - a*b^3*e)*x^2 + 2*(a*b^3*d - a^2*b^2*e)*x)*sqrt(-b^2*d + a*b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(sqrt(d + e*x)/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.227741, size = 302, normalized size = 1.8 \[ \frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{2} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} + \sqrt{x e + d} b d e^{2} - \sqrt{x e + d} a e^{3}}{4 \,{\left (b^{2} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

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)^(3/2),x, algorithm="giac")

[Out]

1/4*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((b^2*d*sign(-(x*e + d)*b*e
+ b*d*e - a*e^2) - a*b*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(-b^2*d + a*
b*e)) + 1/4*((x*e + d)^(3/2)*b*e^2 + sqrt(x*e + d)*b*d*e^2 - sqrt(x*e + d)*a*e^3
)/((b^2*d*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a*b*e*sign(-(x*e + d)*b*e + b*d
*e - a*e^2))*((x*e + d)*b - b*d + a*e)^2)