3.2050 \(\int \frac{1}{\sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\sqrt{e} \sqrt{c d^2-a e^2}} \]

[Out]

(2*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*
e^2]*Sqrt[d + e*x])])/(Sqrt[e]*Sqrt[c*d^2 - a*e^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.134768, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\sqrt{e} \sqrt{c d^2-a e^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*
e^2]*Sqrt[d + e*x])])/(Sqrt[e]*Sqrt[c*d^2 - a*e^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.0431, size = 78, normalized size = 0.93 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{\sqrt{e} \sqrt{a e^{2} - c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(d + e*x)*s
qrt(a*e**2 - c*d**2)))/(sqrt(e)*sqrt(a*e**2 - c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0784675, size = 97, normalized size = 1.15 \[ -\frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\sqrt{e} \sqrt{a e^2-c d^2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 91, normalized size = 1.1 \[ -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}{\sqrt{ex+d}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(e*x+d)^(1/2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(c*d*x+a*e)^(1/2)/((a*e
^2-c*d^2)*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.222211, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d} +{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{\sqrt{-c d^{2} e + a e^{3}}}, -\frac{2 \, \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right )}{\sqrt{c d^{2} e - a e^{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[log(-(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2*e - a*e^3)*sqrt(e*x
+ d) + (c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2)*sqrt(-c*d^2*e + a*e^3))/(e^
2*x^2 + 2*d*e*x + d^2))/sqrt(-c*d^2*e + a*e^3), -2*arctan(sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*e^2
+ (c*d^2*e + a*e^3)*x))/sqrt(c*d^2*e - a*e^3)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out