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

Optimal. Leaf size=191 \[ -\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d e}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e}-\frac{a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 \sqrt{d} e} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.376297, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d e}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e}-\frac{a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 \sqrt{d} e} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.412064, size = 139, normalized size = 0.73 \[ \frac{-\frac{3 a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x}+2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}-\frac{3 a f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

((-3*a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(e*x + f*Sqrt[a + (e^2*x^2
)/f^2]) + 2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2) - (3*a*f^2*ArcTanh[Sqrt[
d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/Sqrt[d])/(6*e)

_______________________________________________________________________________________

Maple [F]  time = 0.01, size = 0, normalized size = 0. \[ \int \sqrt{d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.337377, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{d} f^{2} \log \left (\frac{\sqrt{d} f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} +{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} d}{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}\right ) + 2 \,{\left (5 \, d e x - d f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{12 \, d e}, -\frac{3 \, a \sqrt{-d} f^{2} \arctan \left (\frac{d}{\sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt{-d}}\right ) -{\left (5 \, d e x - d f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{6 \, d e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*a*sqrt(d)*f^2*log((sqrt(d)*f*sqrt((e^2*x^2 + a*f^2)/f^2) + (e*x + 2*d)*
sqrt(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*(5*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*e), -1/6*(3*a*sqrt(-d)*f^2*arctan(
d/(sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)*sqrt(-d))) - (5*d*e*x - d*f*sqr
t((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*e)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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