3.302 \(\int \frac{1}{\sqrt{d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}}} \, dx\)

Optimal. Leaf size=147 \[ \frac{a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 d^{3/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e} \]

[Out]

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

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Rubi [A]  time = 0.24196, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 d^{3/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.671794, size = 141, normalized size = 0.96 \[ \frac{\frac{a f^2 \tanh ^{-1}\left (\frac{\sqrt{d}}{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}\right )}{2 d^{3/2}}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(e*x + sqrt(e^2*x^2/f^2 + a)*f + d),x, algorithm="maxima")

[Out]

integrate(1/sqrt(e*x + sqrt(e^2*x^2/f^2 + a)*f + d), x)

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Fricas [A]  time = 0.33397, size = 1, normalized size = 0.01 \[ \left [\frac{a \sqrt{d} f^{2} \log \left (\frac{\sqrt{d} f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} +{\left (e x + 2 \, d\right )} \sqrt{d} + 2 \, \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} d}{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}\right ) + 2 \,{\left (d e x - d f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{4 \, d^{2} e}, \frac{a \sqrt{-d} f^{2} \arctan \left (\frac{d}{\sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt{-d}}\right ) +{\left (d e x - d f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{2 \, d^{2} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(e*x + sqrt(e^2*x^2/f^2 + a)*f + d),x, algorithm="fricas")

[Out]

[1/4*(a*sqrt(d)*f^2*log((sqrt(d)*f*sqrt((e^2*x^2 + a*f^2)/f^2) + (e*x + 2*d)*sqr
t(d) + 2*sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)*d)/(e*x + f*sqrt((e^2*x^2
+ a*f^2)/f^2))) + 2*(d*e*x - d*f*sqrt((e^2*x^2 + a*f^2)/f^2) + 2*d^2)*sqrt(e*x
+ f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/(d^2*e), 1/2*(a*sqrt(-d)*f^2*arctan(d/(sqr
t(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)*sqrt(-d))) + (d*e*x - d*f*sqrt((e^2*x
^2 + a*f^2)/f^2) + 2*d^2)*sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/(d^2*e)
]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(e*x + sqrt(e^2*x^2/f^2 + a)*f + d),x, algorithm="giac")

[Out]

integrate(1/sqrt(e*x + sqrt(e^2*x^2/f^2 + a)*f + d), x)