3.2057 \(\int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{2 \sqrt{d+e x}}{\left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.27173, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 \sqrt{d+e x}}{\left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.056, size = 126, normalized size = 0.91 \[ - \frac{2 \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} + \frac{2 \sqrt{d + e x}}{\left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

-2*sqrt(e)*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(d
+ e*x)*sqrt(a*e**2 - c*d**2)))/(a*e**2 - c*d**2)**(3/2) + 2*sqrt(d + e*x)/((a*e*
*2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.173922, size = 116, normalized size = 0.83 \[ \frac{2 \sqrt{d+e x} \left (\sqrt{a e^2-c d^2}-\sqrt{e} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )\right )}{\left (a e^2-c d^2\right )^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(Sqrt[-(c*d^2) + a*e^2] - Sqrt[e]*Sqrt[a*e + c*d*x]*ArcTanh[(Sq
rt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]]))/((-(c*d^2) + a*e^2)^(3/2)*Sqr
t[(a*e + c*d*x)*(d + e*x)])

_______________________________________________________________________________________

Maple [A]  time = 0.029, size = 136, normalized size = 1. \[ -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}{\sqrt{ex+d} \left ( cdx+ae \right ) \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}} \left ( e{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}-\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-2*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(e*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^
2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)-((a*e^2-c*d^2)*e)^(1/2))/(e*x+d)^(1/2)/(c*d
*x+a*e)/(a*e^2-c*d^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(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229729, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-\frac{e}{c d^{2} - a e^{2}}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{a c d^{3} e - a^{2} d e^{3} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x^{2} +{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x}, \frac{2 \,{\left ({\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{\frac{e}{c d^{2} - a e^{2}}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d^{2} - a e^{2}}}}\right ) - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}\right )}}{a c d^{3} e - a^{2} d e^{3} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x^{2} +{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="fricas")

[Out]

[-((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^
2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x
+ d^2)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c*d^3
*e - a^2*d*e^3 + (c^2*d^3*e - a*c*d*e^3)*x^2 + (c^2*d^4 - a^2*e^4)*x), 2*((c*d*e
*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(sqrt(e*x + d)/(
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2)))) - sqrt(c*d
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c*d^3*e - a^2*d*e^3 + (c^2
*d^3*e - a*c*d*e^3)*x^2 + (c^2*d^4 - a^2*e^4)*x)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.580764, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x