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

Optimal. Leaf size=64 \[ \frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{3/2} \sqrt{e} \sqrt{c d-b e}} \]

[Out]

x/c - ((2*c*d - b*e)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(3/2)*Sqrt
[e]*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.121866, antiderivative size = 64, 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 \[ \frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{3/2} \sqrt{e} \sqrt{c d-b e}} \]

Antiderivative was successfully verified.

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

[Out]

x/c - ((2*c*d - b*e)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(3/2)*Sqrt
[e]*Sqrt[c*d - b*e])

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Rubi in Sympy [A]  time = 27.18, size = 54, normalized size = 0.84 \[ \frac{x}{c} - \frac{\left (b e - 2 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e - c d}} \right )}}{c^{\frac{3}{2}} \sqrt{e} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/c - (b*e - 2*c*d)*atan(sqrt(c)*sqrt(e)*x/sqrt(b*e - c*d))/(c**(3/2)*sqrt(e)*sq
rt(b*e - c*d))

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Mathematica [A]  time = 0.0925596, size = 63, normalized size = 0.98 \[ \frac{x}{c}-\frac{(b e-2 c d) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{3/2} \sqrt{e} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

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

[Out]

x/c - ((-2*c*d + b*e)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(3/2)*S
qrt[e]*Sqrt[-(c*d) + b*e])

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Maple [A]  time = 0.003, size = 79, normalized size = 1.2 \[{\frac{x}{c}}-{\frac{be}{c}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}+2\,{\frac{d}{\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276516, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, c d - b e\right )} \log \left (\frac{2 \,{\left (c^{2} d e - b c e^{2}\right )} x + \sqrt{c^{2} d e - b c e^{2}}{\left (c e x^{2} + c d - b e\right )}}{c e x^{2} - c d + b e}\right ) - 2 \, \sqrt{c^{2} d e - b c e^{2}} x}{2 \, \sqrt{c^{2} d e - b c e^{2}} c}, \frac{{\left (2 \, c d - b e\right )} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) + \sqrt{-c^{2} d e + b c e^{2}} x}{\sqrt{-c^{2} d e + b c e^{2}} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.95221, size = 212, normalized size = 3.31 \[ \frac{\sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) \log{\left (x + \frac{- b c e \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) + c^{2} d \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right )}{b e - 2 c d} \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) \log{\left (x + \frac{b c e \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) - c^{2} d \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right )}{b e - 2 c d} \right )}}{2} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d)*log(x + (-b*c*e*sqrt(-1/(c**3*e*(b*e
- c*d)))*(b*e - 2*c*d) + c**2*d*sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d))/(b
*e - 2*c*d))/2 - sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d)*log(x + (b*c*e*sqrt
(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d) - c**2*d*sqrt(-1/(c**3*e*(b*e - c*d)))*(
b*e - 2*c*d))/(b*e - 2*c*d))/2 + x/c

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

[Out]

Timed out