3.221 \(\int \frac{d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}} \]

[Out]

-(ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]]/(Sqrt[c]*Sqrt[e]*Sqrt[c*d - b*e])
)

_______________________________________________________________________________________

Rubi [A]  time = 0.0524903, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]

[Out]

-(ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]]/(Sqrt[c]*Sqrt[e]*Sqrt[c*d - b*e])
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.3808, size = 42, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e - c d}} \right )}}{\sqrt{c} \sqrt{e} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

atan(sqrt(c)*sqrt(e)*x/sqrt(b*e - c*d))/(sqrt(c)*sqrt(e)*sqrt(b*e - c*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.020428, size = 48, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{\sqrt{c} \sqrt{e} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]

[Out]

ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]]/(Sqrt[c]*Sqrt[e]*Sqrt[-(c*d) + b*
e])

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 33, normalized size = 0.7 \[{1\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

1/((b*e-c*d)*c*e)^(1/2)*arctan(x*c*e/((b*e-c*d)*c*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((e*x^2 + d)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.268387, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \,{\left (c^{2} d e - b c e^{2}\right )} x - \sqrt{c^{2} d e - b c e^{2}}{\left (c e x^{2} + c d - b e\right )}}{c e x^{2} - c d + b e}\right )}{2 \, \sqrt{c^{2} d e - b c e^{2}}}, \frac{\arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right )}{\sqrt{-c^{2} d e + b c e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="fricas")

[Out]

[1/2*log(-(2*(c^2*d*e - b*c*e^2)*x - sqrt(c^2*d*e - b*c*e^2)*(c*e*x^2 + c*d - b*
e))/(c*e*x^2 - c*d + b*e))/sqrt(c^2*d*e - b*c*e^2), arctan(-sqrt(-c^2*d*e + b*c*
e^2)*x/(c*d - b*e))/sqrt(-c^2*d*e + b*c*e^2)]

_______________________________________________________________________________________

Sympy [A]  time = 0.564659, size = 124, normalized size = 2.53 \[ - \frac{\sqrt{- \frac{1}{c e \left (b e - c d\right )}} \log{\left (- b e \sqrt{- \frac{1}{c e \left (b e - c d\right )}} + c d \sqrt{- \frac{1}{c e \left (b e - c d\right )}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c e \left (b e - c d\right )}} \log{\left (b e \sqrt{- \frac{1}{c e \left (b e - c d\right )}} - c d \sqrt{- \frac{1}{c e \left (b e - c d\right )}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

-sqrt(-1/(c*e*(b*e - c*d)))*log(-b*e*sqrt(-1/(c*e*(b*e - c*d))) + c*d*sqrt(-1/(c
*e*(b*e - c*d))) + x)/2 + sqrt(-1/(c*e*(b*e - c*d)))*log(b*e*sqrt(-1/(c*e*(b*e -
c*d))) - c*d*sqrt(-1/(c*e*(b*e - c*d))) + x)/2

_______________________________________________________________________________________

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((e*x^2 + d)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="giac")

[Out]

Timed out