3.1993 \(\int \frac{\sqrt{d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.108999, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}} \]

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),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.9183, size = 58, normalized size = 0.89 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\sqrt{c} \sqrt{d} \sqrt{a e^{2} - c d^{2}}} \]

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),x)

[Out]

2*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))/(sqrt(c)*sqrt(d)*sqr
t(a*e**2 - c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0427814, size = 65, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}} \]

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),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 48, normalized size = 0.7 \[ 2\,{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

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),x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229901, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right )}{\sqrt{c^{2} d^{3} - a c d e^{2}}}, -\frac{2 \, \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right )}{\sqrt{-c^{2} d^{3} + a c d e^{2}}}\right ] \]

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),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.6051, size = 214, normalized size = 3.29 \[ 2 \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e^{2} - c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{a e^{2} - c d^{2}}{c d}}} & \text{for}\: \frac{a e^{2} - c d^{2}}{c d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} & \text{for}\: d + e x > \frac{- a e^{2} + c d^{2}}{c d} \wedge \frac{a e^{2} - c d^{2}}{c d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} & \text{for}\: \frac{a e^{2} - c d^{2}}{c d} < 0 \wedge d + e x < \frac{- a e^{2} + c d^{2}}{c d} \end{cases}\right ) \]

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),x)

[Out]

2*Piecewise((atan(sqrt(d + e*x)/sqrt((a*e**2 - c*d**2)/(c*d)))/(c*d*sqrt((a*e**2
- c*d**2)/(c*d))), (a*e**2 - c*d**2)/(c*d) > 0), (-acoth(sqrt(d + e*x)/sqrt((-a
*e**2 + c*d**2)/(c*d)))/(c*d*sqrt((-a*e**2 + c*d**2)/(c*d))), ((a*e**2 - c*d**2)
/(c*d) < 0) & (d + e*x > (-a*e**2 + c*d**2)/(c*d))), (-atanh(sqrt(d + e*x)/sqrt(
(-a*e**2 + c*d**2)/(c*d)))/(c*d*sqrt((-a*e**2 + c*d**2)/(c*d))), ((a*e**2 - c*d*
*2)/(c*d) < 0) & (d + e*x < (-a*e**2 + c*d**2)/(c*d))))

_______________________________________________________________________________________

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

[Out]

Timed out