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

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

[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 = 0.275817, 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 \[ \frac{x \left (b^2 e^2-5 b c d e+7 c^2 d^2\right )}{c^3}-\frac{(2 c d-b e)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{7/2} \sqrt{e} \sqrt{c d-b e}}+\frac{e x^3 (4 c d-b e)}{3 c^2}+\frac{e^2 x^5}{5 c} \]

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])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \left (b^{2} e^{2} - 5 b c d e + 7 c^{2} d^{2}\right ) \int \frac{1}{c^{3}}\, dx + \frac{e^{2} x^{5}}{5 c} - \frac{e x^{3} \left (b e - 4 c d\right )}{3 c^{2}} - \frac{\left (b e - 2 c d\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e - c d}} \right )}}{c^{\frac{7}{2}} \sqrt{e} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b**2*e**2 - 5*b*c*d*e + 7*c**2*d**2)*Integral(c**(-3), x) + e**2*x**5/(5*c) - e
*x**3*(b*e - 4*c*d)/(3*c**2) - (b*e - 2*c*d)**3*atan(sqrt(c)*sqrt(e)*x/sqrt(b*e
- c*d))/(c**(7/2)*sqrt(e)*sqrt(b*e - c*d))

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Mathematica [A]  time = 0.123384, size = 121, normalized size = 1. \[ -\frac{x \left (-b^2 e^2+5 b c d e-7 c^2 d^2\right )}{c^3}-\frac{(b e-2 c d)^3 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{7/2} \sqrt{e} \sqrt{b e-c d}}-\frac{e x^3 (b e-4 c d)}{3 c^2}+\frac{e^2 x^5}{5 c} \]

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 = 0.014, size = 226, normalized size = 1.9 \[{\frac{{e}^{2}{x}^{5}}{5\,c}}-{\frac{b{x}^{3}{e}^{2}}{3\,{c}^{2}}}+{\frac{4\,de{x}^{3}}{3\,c}}+{\frac{{b}^{2}{e}^{2}x}{{c}^{3}}}-5\,{\frac{bdex}{{c}^{2}}}+7\,{\frac{{d}^{2}x}{c}}-{\frac{{b}^{3}{e}^{3}}{{c}^{3}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}+6\,{\frac{{b}^{2}d{e}^{2}}{{c}^{2}\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }-12\,{\frac{b{d}^{2}e}{c\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }+8\,{\frac{{d}^{3}}{\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) } \]

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(x*c*e/((b*e-c*d)*c*e)^(1/2))*b^3*e^3+6/
c^2/((b*e-c*d)*c*e)^(1/2)*arctan(x*c*e/((b*e-c*d)*c*e)^(1/2))*b^2*d*e^2-12/c/((b
*e-c*d)*c*e)^(1/2)*arctan(x*c*e/((b*e-c*d)*c*e)^(1/2))*b*d^2*e+8/((b*e-c*d)*c*e)
^(1/2)*arctan(x*c*e/((b*e-c*d)*c*e)^(1/2))*d^3

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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)^4/(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 = 0.281362, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (\frac{2 \,{\left (c^{2} d e - b c e^{2}\right )} x + \sqrt{c^{2} d e - b c e^{2}}{\left (c e x^{2} + c d - b e\right )}}{c e x^{2} - c d + b e}\right ) - 2 \,{\left (3 \, c^{2} e^{2} x^{5} + 5 \,{\left (4 \, c^{2} d e - b c e^{2}\right )} x^{3} + 15 \,{\left (7 \, c^{2} d^{2} - 5 \, b c d e + b^{2} e^{2}\right )} x\right )} \sqrt{c^{2} d e - b c e^{2}}}{30 \, \sqrt{c^{2} d e - b c e^{2}} c^{3}}, \frac{15 \,{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) +{\left (3 \, c^{2} e^{2} x^{5} + 5 \,{\left (4 \, c^{2} d e - b c e^{2}\right )} x^{3} + 15 \,{\left (7 \, c^{2} d^{2} - 5 \, b c d e + b^{2} e^{2}\right )} x\right )} \sqrt{-c^{2} d e + b c e^{2}}}{15 \, \sqrt{-c^{2} d e + b c e^{2}} c^{3}}\right ] \]

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

[Out]

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

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Sympy [A]  time = 3.15263, size = 343, normalized size = 2.83 \[ \frac{\sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log{\left (x + \frac{- b c^{3} e \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} + c^{4} d \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log{\left (x + \frac{b c^{3} e \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} - c^{4} d \sqrt{- \frac{1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} + \frac{e^{2} x^{5}}{5 c} - \frac{x^{3} \left (b e^{2} - 4 c d e\right )}{3 c^{2}} + \frac{x \left (b^{2} e^{2} - 5 b c d e + 7 c^{2} d^{2}\right )}{c^{3}} \]

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

[Out]

Timed out