3.215 \(\int \frac{(d+e x^2)^3}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)

Optimal. Leaf size=86 \[ \frac{x (3 c d-b e)}{c^2}-\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{e x^3}{3 c} \]

[Out]

((3*c*d - b*e)*x)/c^2 + (e*x^3)/(3*c) - ((2*c*d - b*e)^2*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(5/2
)*Sqrt[e]*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.107268, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1149, 390, 208} \[ \frac{x (3 c d-b e)}{c^2}-\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]

[Out]

((3*c*d - b*e)*x)/c^2 + (e*x^3)/(3*c) - ((2*c*d - b*e)^2*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(5/2
)*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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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 )^3}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac{\left (d+e x^2\right )^2}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\int \left (\frac{3 c d-b e}{c^2}+\frac{e x^2}{c}+\frac{4 c^2 d^2-4 b c d e+b^2 e^2}{c^2 \left (-c d+b e+c e x^2\right )}\right ) \, dx\\ &=\frac{(3 c d-b e) x}{c^2}+\frac{e x^3}{3 c}+\frac{(2 c d-b e)^2 \int \frac{1}{-c d+b e+c e x^2} \, dx}{c^2}\\ &=\frac{(3 c d-b e) x}{c^2}+\frac{e x^3}{3 c}-\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}\\ \end{align*}

Mathematica [A]  time = 0.0464509, size = 84, normalized size = 0.98 \[ -\frac{x (b e-3 c d)}{c^2}+\frac{(b e-2 c d)^2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{5/2} \sqrt{e} \sqrt{b e-c d}}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]

[Out]

-(((-3*c*d + b*e)*x)/c^2) + (e*x^3)/(3*c) + ((-2*c*d + b*e)^2*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/
(c^(5/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])

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Maple [A]  time = 0.004, size = 142, normalized size = 1.7 \begin{align*}{\frac{e{x}^{3}}{3\,c}}-{\frac{bex}{{c}^{2}}}+3\,{\frac{dx}{c}}+{\frac{{b}^{2}{e}^{2}}{{c}^{2}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}-4\,{\frac{bde}{c\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }+4\,{\frac{{d}^{2}}{\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)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

1/3*e*x^3/c-1/c^2*b*e*x+3/c*d*x+1/c^2/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))*b^2*e^2-4/c/((
b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))*b*d*e+4/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*
c*e)^(1/2))*d^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)^3/(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 = 1.9537, size = 632, normalized size = 7.35 \begin{align*} \left [\frac{2 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 3 \,{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{c^{2} d e - b c e^{2}} \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 ) + 6 \,{\left (3 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x}{6 \,{\left (c^{4} d e - b c^{3} e^{2}\right )}}, \frac{{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} - 3 \,{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \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 ) + 3 \,{\left (3 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x}{3 \,{\left (c^{4} d e - b c^{3} e^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[1/6*(2*(c^3*d*e^2 - b*c^2*e^3)*x^3 + 3*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sqrt(c^2*d*e - b*c*e^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)) + 6*(3*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^
3)*x)/(c^4*d*e - b*c^3*e^2), 1/3*((c^3*d*e^2 - b*c^2*e^3)*x^3 - 3*(4*c^2*d^2 - 4*b*c*d*e + b^2*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)) + 3*(3*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*x
)/(c^4*d*e - b*c^3*e^2)]

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Sympy [B]  time = 0.739891, size = 275, normalized size = 3.2 \begin{align*} - \frac{\sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log{\left (x + \frac{- b c^{2} e \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} + c^{3} d \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac{\sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log{\left (x + \frac{b c^{2} e \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} - c^{3} d \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac{e x^{3}}{3 c} - \frac{x \left (b e - 3 c d\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

-sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2*log(x + (-b*c**2*e*sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)
**2 + c**3*d*sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2)/(b**2*e**2 - 4*b*c*d*e + 4*c**2*d**2))/2 + sqrt(-
1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2*log(x + (b*c**2*e*sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2 - c*
*3*d*sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2)/(b**2*e**2 - 4*b*c*d*e + 4*c**2*d**2))/2 + e*x**3/(3*c) -
x*(b*e - 3*c*d)/c**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)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

Timed out