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

Optimal. Leaf size=82 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-b}+2 e x}{\sqrt{b+2 d e}}\right )}{\sqrt{b+2 d e}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-b}-2 e x}{\sqrt{b+2 d e}}\right )}{\sqrt{b+2 d e}} \]

[Out]

-(ArcTan[(Sqrt[-b + 2*d*e] - 2*e*x)/Sqrt[b + 2*d*e]]/Sqrt[b + 2*d*e]) + ArcTan[(
Sqrt[-b + 2*d*e] + 2*e*x)/Sqrt[b + 2*d*e]]/Sqrt[b + 2*d*e]

_______________________________________________________________________________________

Rubi [A]  time = 0.187703, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-b}+2 e x}{\sqrt{b+2 d e}}\right )}{\sqrt{b+2 d e}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-b}-2 e x}{\sqrt{b+2 d e}}\right )}{\sqrt{b+2 d e}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(Sqrt[-b + 2*d*e] - 2*e*x)/Sqrt[b + 2*d*e]]/Sqrt[b + 2*d*e]) + ArcTan[(
Sqrt[-b + 2*d*e] + 2*e*x)/Sqrt[b + 2*d*e]]/Sqrt[b + 2*d*e]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.7482, size = 73, normalized size = 0.89 \[ \frac{\operatorname{atan}{\left (\frac{2 e x - \sqrt{- b + 2 d e}}{\sqrt{b + 2 d e}} \right )}}{\sqrt{b + 2 d e}} + \frac{\operatorname{atan}{\left (\frac{2 e x + \sqrt{- b + 2 d e}}{\sqrt{b + 2 d e}} \right )}}{\sqrt{b + 2 d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan((2*e*x - sqrt(-b + 2*d*e))/sqrt(b + 2*d*e))/sqrt(b + 2*d*e) + atan((2*e*x +
sqrt(-b + 2*d*e))/sqrt(b + 2*d*e))/sqrt(b + 2*d*e)

_______________________________________________________________________________________

Mathematica [B]  time = 0.182985, size = 181, normalized size = 2.21 \[ \frac{\frac{\left (\sqrt{b^2-4 d^2 e^2}-b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}\right )}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}+\frac{\left (\sqrt{b^2-4 d^2 e^2}+b-2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}\right )}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}} \]

Antiderivative was successfully verified.

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

[Out]

(((-b + 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b - Sqrt[b^2 -
4*d^2*e^2]]])/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]] + ((b - 2*d*e + Sqrt[b^2 - 4*d^2*e
^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b + Sqrt[b^2 -
4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])

_______________________________________________________________________________________

Maple [A]  time = 0.037, size = 71, normalized size = 0.9 \[ -{1\arctan \left ({1 \left ( -2\,ex+\sqrt{2\,de-b} \right ){\frac{1}{\sqrt{2\,de+b}}}} \right ){\frac{1}{\sqrt{2\,de+b}}}}+{1\arctan \left ({1 \left ( 2\,ex+\sqrt{2\,de-b} \right ){\frac{1}{\sqrt{2\,de+b}}}} \right ){\frac{1}{\sqrt{2\,de+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan((-2*e*x+(2*d*e-b)^(1/2))/(2*d*e+b)^(1/2))/(2*d*e+b)^(1/2)+arctan((2*e*x+
(2*d*e-b)^(1/2))/(2*d*e+b)^(1/2))/(2*d*e+b)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{e^{2} x^{4} + b x^{2} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^2 + d)/(e^2*x^4 + b*x^2 + d^2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.28728, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (\frac{2 \,{\left (2 \, d e^{2} + b e\right )} x^{3} - 2 \,{\left (2 \, d^{2} e + b d\right )} x +{\left (e^{2} x^{4} -{\left (4 \, d e + b\right )} x^{2} + d^{2}\right )} \sqrt{-2 \, d e - b}}{e^{2} x^{4} + b x^{2} + d^{2}}\right )}{2 \, \sqrt{-2 \, d e - b}}, \frac{\arctan \left (\frac{e x}{\sqrt{2 \, d e + b}}\right ) + \arctan \left (\frac{e^{2} x^{3} +{\left (d e + b\right )} x}{\sqrt{2 \, d e + b} d}\right )}{\sqrt{2 \, d e + b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log((2*(2*d*e^2 + b*e)*x^3 - 2*(2*d^2*e + b*d)*x + (e^2*x^4 - (4*d*e + b)*x
^2 + d^2)*sqrt(-2*d*e - b))/(e^2*x^4 + b*x^2 + d^2))/sqrt(-2*d*e - b), (arctan(e
*x/sqrt(2*d*e + b)) + arctan((e^2*x^3 + (d*e + b)*x)/(sqrt(2*d*e + b)*d)))/sqrt(
2*d*e + b)]

_______________________________________________________________________________________

Sympy [A]  time = 1.31954, size = 122, normalized size = 1.49 \[ - \frac{\sqrt{- \frac{1}{b + 2 d e}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (- b \sqrt{- \frac{1}{b + 2 d e}} - 2 d e \sqrt{- \frac{1}{b + 2 d e}}\right )}{e} \right )}}{2} + \frac{\sqrt{- \frac{1}{b + 2 d e}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (b \sqrt{- \frac{1}{b + 2 d e}} + 2 d e \sqrt{- \frac{1}{b + 2 d e}}\right )}{e} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.453956, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done