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

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]))

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Rubi [A]  time = 0.0288674, 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, Rules used = {1149, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}} \]

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]))

Rule 1149

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac{1}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx\\ &=-\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}}\\ \end{align*}

Mathematica [A]  time = 0.0125817, 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])

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Maple [A]  time = 0.003, size = 33, normalized size = 0.7 \begin{align*}{\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60306, size = 279, normalized size = 5.69 \begin{align*} \left [\frac{\log \left (\frac{c e x^{2} + c d - b e - 2 \, \sqrt{c^{2} d e - b c e^{2}} x}{c e x^{2} - c d + b e}\right )}{2 \, \sqrt{c^{2} d e - b c e^{2}}}, -\frac{\sqrt{-c^{2} d e + b c e^{2}} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right )}{c^{2} d e - b c e^{2}}\right ] \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.218389, size = 124, normalized size = 2.53 \begin{align*} - \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} \end{align*}

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="giac")

[Out]

Timed out