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

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])

________________________________________________________________________________________

Rubi [A]  time = 0.078468, 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, Rules used = {1149, 388, 208} \[ \frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{3/2} \sqrt{e} \sqrt{c d-b e}} \]

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])

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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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

Mathematica [A]  time = 0.0551712, 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)*Sqrt[e]*Sqrt[-(c*d) + b*e])

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 79, normalized size = 1.2 \begin{align*}{\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 ) } \end{align*}

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.00839, size = 424, normalized size = 6.62 \begin{align*} \left [-\frac{\sqrt{c^{2} d e - b c e^{2}}{\left (2 \, c d - b e\right )} \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 \,{\left (c^{2} d e - b c e^{2}\right )} x}{2 \,{\left (c^{3} d e - b c^{2} e^{2}\right )}}, -\frac{\sqrt{-c^{2} d e + b c e^{2}}{\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 ) -{\left (c^{2} d e - b c e^{2}\right )} x}{c^{3} d e - b c^{2} e^{2}}\right ] \end{align*}

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, algorithm="fricas")

[Out]

[-1/2*(sqrt(c^2*d*e - b*c*e^2)*(2*c*d - b*e)*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)) - 2*(c^2*d*e - b*c*e^2)*x)/(c^3*d*e - b*c^2*e^2), -(sqrt(-c^2*d*e + b*c*e^2)*(2*c*d - b*e)*arcta
n(-sqrt(-c^2*d*e + b*c*e^2)*x/(c*d - b*e)) - (c^2*d*e - b*c*e^2)*x)/(c^3*d*e - b*c^2*e^2)]

________________________________________________________________________________________

Sympy [B]  time = 0.568459, size = 212, normalized size = 3.31 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

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

[Out]

Timed out