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

Optimal. Leaf size=121 \[ \frac{x \left (b^2 e^2-5 b c d e+7 c^2 d^2\right )}{c^3}+\frac{e x^3 (4 c d-b e)}{3 c^2}-\frac{(2 c d-b e)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{7/2} \sqrt{e} \sqrt{c d-b e}}+\frac{e^2 x^5}{5 c} \]

[Out]

((7*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*x)/c^3 + (e*(4*c*d - b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((2*c*d - b*e)^3
*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(7/2)*Sqrt[e]*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.159732, antiderivative size = 121, 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} \[ \frac{x \left (b^2 e^2-5 b c d e+7 c^2 d^2\right )}{c^3}+\frac{e x^3 (4 c d-b e)}{3 c^2}-\frac{(2 c d-b e)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{7/2} \sqrt{e} \sqrt{c d-b e}}+\frac{e^2 x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*x)/c^3 + (e*(4*c*d - b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((2*c*d - b*e)^3
*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(7/2)*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 )^4}{-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 )^3}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\int \left (\frac{7 c^2 d^2-5 b c d e+b^2 e^2}{c^3}+\frac{e (4 c d-b e) x^2}{c^2}+\frac{e^2 x^4}{c}+\frac{8 c^3 d^3-12 b c^2 d^2 e+6 b^2 c d e^2-b^3 e^3}{c^3 \left (-c d+b e+c e x^2\right )}\right ) \, dx\\ &=\frac{\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac{e (4 c d-b e) x^3}{3 c^2}+\frac{e^2 x^5}{5 c}+\frac{(2 c d-b e)^3 \int \frac{1}{-c d+b e+c e x^2} \, dx}{c^3}\\ &=\frac{\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac{e (4 c d-b e) x^3}{3 c^2}+\frac{e^2 x^5}{5 c}-\frac{(2 c d-b e)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{7/2} \sqrt{e} \sqrt{c d-b e}}\\ \end{align*}

Mathematica [A]  time = 0.0766953, size = 121, normalized size = 1. \[ -\frac{x \left (-b^2 e^2+5 b c d e-7 c^2 d^2\right )}{c^3}-\frac{e x^3 (b e-4 c d)}{3 c^2}-\frac{(b e-2 c d)^3 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{7/2} \sqrt{e} \sqrt{b e-c d}}+\frac{e^2 x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((-7*c^2*d^2 + 5*b*c*d*e - b^2*e^2)*x)/c^3) - (e*(-4*c*d + b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((-2*c*d +
b*e)^3*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(7/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])

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Maple [B]  time = 0.01, size = 226, normalized size = 1.9 \begin{align*}{\frac{{e}^{2}{x}^{5}}{5\,c}}-{\frac{b{x}^{3}{e}^{2}}{3\,{c}^{2}}}+{\frac{4\,de{x}^{3}}{3\,c}}+{\frac{{b}^{2}{e}^{2}x}{{c}^{3}}}-5\,{\frac{bdex}{{c}^{2}}}+7\,{\frac{{d}^{2}x}{c}}-{\frac{{b}^{3}{e}^{3}}{{c}^{3}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}+6\,{\frac{{b}^{2}d{e}^{2}}{{c}^{2}\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }-12\,{\frac{b{d}^{2}e}{c\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }+8\,{\frac{{d}^{3}}{\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)^4/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

1/5*e^2*x^5/c-1/3/c^2*x^3*b*e^2+4/3/c*x^3*d*e+1/c^3*b^2*e^2*x-5/c^2*b*d*e*x+7/c*d^2*x-1/c^3/((b*e-c*d)*c*e)^(1
/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))*b^3*e^3+6/c^2/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2)
)*b^2*d*e^2-12/c/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))*b*d^2*e+8/((b*e-c*d)*c*e)^(1/2)*arc
tan(c*e*x/((b*e-c*d)*c*e)^(1/2))*d^3

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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)^4/(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 [B]  time = 1.94085, size = 903, normalized size = 7.46 \begin{align*} \left [\frac{6 \,{\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{5} + 10 \,{\left (4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} - 15 \,{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\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 ) + 30 \,{\left (7 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x}{30 \,{\left (c^{5} d e - b c^{4} e^{2}\right )}}, \frac{3 \,{\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{5} + 5 \,{\left (4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} - 15 \,{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\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 ) + 15 \,{\left (7 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x}{15 \,{\left (c^{5} d e - b c^{4} e^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(6*(c^4*d*e^3 - b*c^3*e^4)*x^5 + 10*(4*c^4*d^2*e^2 - 5*b*c^3*d*e^3 + b^2*c^2*e^4)*x^3 - 15*(8*c^3*d^3 -
12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*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)) + 30*(7*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*x)/(c^5
*d*e - b*c^4*e^2), 1/15*(3*(c^4*d*e^3 - b*c^3*e^4)*x^5 + 5*(4*c^4*d^2*e^2 - 5*b*c^3*d*e^3 + b^2*c^2*e^4)*x^3 -
15*(8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*sqrt(-c^2*d*e + b*c*e^2)*arctan(-sqrt(-c^2*d*e + b*
c*e^2)*x/(c*d - b*e)) + 15*(7*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*x)/(c^5*d*e - b*c^4*
e^2)]

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Sympy [B]  time = 0.954936, size = 343, normalized size = 2.83 \begin{align*} \frac{\sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log{\left (x + \frac{- b c^{3} e \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} + c^{4} d \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log{\left (x + \frac{b c^{3} e \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} - c^{4} d \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} + \frac{e^{2} x^{5}}{5 c} - \frac{x^{3} \left (b e^{2} - 4 c d e\right )}{3 c^{2}} + \frac{x \left (b^{2} e^{2} - 5 b c d e + 7 c^{2} d^{2}\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3*log(x + (-b*c**3*e*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)*
*3 + c**4*d*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3)/(b**3*e**3 - 6*b**2*c*d*e**2 + 12*b*c**2*d**2*e -
8*c**3*d**3))/2 - sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3*log(x + (b*c**3*e*sqrt(-1/(c**7*e*(b*e - c*d)
))*(b*e - 2*c*d)**3 - c**4*d*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3)/(b**3*e**3 - 6*b**2*c*d*e**2 + 12
*b*c**2*d**2*e - 8*c**3*d**3))/2 + e**2*x**5/(5*c) - x**3*(b*e**2 - 4*c*d*e)/(3*c**2) + x*(b**2*e**2 - 5*b*c*d
*e + 7*c**2*d**2)/c**3

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

[Out]

Timed out