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

Optimal. Leaf size=315 \[ \frac{f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 e^2 \left (2 d e-b f^2\right )}-\frac{f^2 \left (4 a e-\frac{b^2 f^2}{e}\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{2 \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{4 \sqrt{2} e^{5/2} \sqrt{2 d e-b f^2}}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e} \]

[Out]

(f^2*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/(4*e^2
*(2*d*e - b*f^2)) + (d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(3/2)/(3*e) - (f
^2*(4*a*e - (b^2*f^2)/e)*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(3/2))/(2*(
2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) - (f^2
*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + b*x + (e
^2*x^2)/f^2]])/Sqrt[2*d*e - b*f^2]])/(4*Sqrt[2]*e^(5/2)*Sqrt[2*d*e - b*f^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.86417, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{f^2 \left (4 a e^2-b^2 f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 e^2 \left (2 d e-b f^2\right )}-\frac{f^2 \left (4 a e-\frac{b^2 f^2}{e}\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{2 \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{4 \sqrt{2} e^{5/2} \sqrt{2 d e-b f^2}}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(f^2*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/(4*e^2
*(2*d*e - b*f^2)) + (d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(3/2)/(3*e) - (f
^2*(4*a*e - (b^2*f^2)/e)*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(3/2))/(2*(
2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) - (f^2
*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + b*x + (e
^2*x^2)/f^2]])/Sqrt[2*d*e - b*f^2]])/(4*Sqrt[2]*e^(5/2)*Sqrt[2*d*e - b*f^2])

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.732814, size = 222, normalized size = 0.7 \[ \frac{\left (b^2 f^4-4 a e^2 f^2\right ) \sqrt{f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x}}{4 e^2 \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x}}{\sqrt{b f^2-2 d e}}\right )}{4 e^{5/2} \sqrt{2 b f^2-4 d e}}+\frac{\left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(3/2)/(3*e) + ((-4*a*e^2*f^2 + b^2*f
^4)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(4*e^2*(b*f^2 + 2*e*(e*x +
f*Sqrt[a + x*(b + (e^2*x)/f^2)]))) + (f^2*(4*a*e^2 - b^2*f^2)*ArcTan[(Sqrt[2]*Sq
rt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[-2*d*e + b*f^2]])/(4
*e^(5/2)*Sqrt[-4*d*e + 2*b*f^2])

_______________________________________________________________________________________

Maple [F]  time = 0.012, size = 0, normalized size = 0. \[ \int \sqrt{d+ex+f\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.502121, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (b^{2} f^{4} - 4 \, a e^{2} f^{2}\right )} \sqrt{-2 \, b e f^{2} + 4 \, d e^{2}} \log \left (\frac{2 \, \sqrt{-2 \, b e f^{2} + 4 \, d e^{2}} e f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} - \sqrt{-2 \, b e f^{2} + 4 \, d e^{2}}{\left (b f^{2} - 2 \, e^{2} x - 4 \, d e\right )} - 4 \,{\left (b e f^{2} - 2 \, d e^{2}\right )} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{b f^{2} + 2 \, e^{2} x + 2 \, e f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}\right ) - 4 \,{\left (3 \, b^{2} e f^{4} - 2 \, b d e^{2} f^{2} - 8 \, d^{2} e^{3} + 10 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )} x - 2 \,{\left (b e^{2} f^{3} - 2 \, d e^{3} f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{48 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )}}, \frac{3 \,{\left (b^{2} f^{4} - 4 \, a e^{2} f^{2}\right )} \sqrt{2 \, b e f^{2} - 4 \, d e^{2}} \arctan \left (\frac{b f^{2} - 2 \, d e}{\sqrt{2 \, b e f^{2} - 4 \, d e^{2}} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}\right ) + 2 \,{\left (3 \, b^{2} e f^{4} - 2 \, b d e^{2} f^{2} - 8 \, d^{2} e^{3} + 10 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )} x - 2 \,{\left (b e^{2} f^{3} - 2 \, d e^{3} f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{24 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(3*(b^2*f^4 - 4*a*e^2*f^2)*sqrt(-2*b*e*f^2 + 4*d*e^2)*log((2*sqrt(-2*b*e*
f^2 + 4*d*e^2)*e*f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) - sqrt(-2*b*e*f^2 + 4*d
*e^2)*(b*f^2 - 2*e^2*x - 4*d*e) - 4*(b*e*f^2 - 2*d*e^2)*sqrt(e*x + f*sqrt((b*f^2
*x + e^2*x^2 + a*f^2)/f^2) + d))/(b*f^2 + 2*e^2*x + 2*e*f*sqrt((b*f^2*x + e^2*x^
2 + a*f^2)/f^2))) - 4*(3*b^2*e*f^4 - 2*b*d*e^2*f^2 - 8*d^2*e^3 + 10*(b*e^3*f^2 -
2*d*e^4)*x - 2*(b*e^2*f^3 - 2*d*e^3*f)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))*s
qrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d))/(b*e^3*f^2 - 2*d*e^4), 1
/24*(3*(b^2*f^4 - 4*a*e^2*f^2)*sqrt(2*b*e*f^2 - 4*d*e^2)*arctan((b*f^2 - 2*d*e)/
(sqrt(2*b*e*f^2 - 4*d*e^2)*sqrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) +
d))) + 2*(3*b^2*e*f^4 - 2*b*d*e^2*f^2 - 8*d^2*e^3 + 10*(b*e^3*f^2 - 2*d*e^4)*x -
2*(b*e^2*f^3 - 2*d*e^3*f)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))*sqrt(e*x + f*s
qrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d))/(b*e^3*f^2 - 2*d*e^4)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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