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

Optimal. Leaf size=133 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{e \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((2*c*d - b*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*Sqrt[b^2 - 4*a*c]*(
c*d^2 - b*d*e + a*e^2)) + (e*Log[d + e*x^2])/(2*(c*d^2 - b*d*e + a*e^2)) - (e*Lo
g[a + b*x^2 + c*x^4])/(4*(c*d^2 - b*d*e + a*e^2))

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Rubi [A]  time = 0.25077, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{e \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((2*c*d - b*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*Sqrt[b^2 - 4*a*c]*(
c*d^2 - b*d*e + a*e^2)) + (e*Log[d + e*x^2])/(2*(c*d^2 - b*d*e + a*e^2)) - (e*Lo
g[a + b*x^2 + c*x^4])/(4*(c*d^2 - b*d*e + a*e^2))

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Rubi in Sympy [A]  time = 47.2823, size = 124, normalized size = 0.93 \[ \frac{e \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )} - \frac{e \log{\left (a + b x^{2} + c x^{4} \right )}}{4 \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 \sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*log(d + e*x**2)/(2*(a*e**2 - b*d*e + c*d**2)) - e*log(a + b*x**2 + c*x**4)/(4*
(a*e**2 - b*d*e + c*d**2)) + (b*e - 2*c*d)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b*
*2))/(2*sqrt(-4*a*c + b**2)*(a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 0.11464, size = 112, normalized size = 0.84 \[ \frac{(2 b e-4 c d) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )+e \sqrt{4 a c-b^2} \left (\log \left (a+b x^2+c x^4\right )-2 \log \left (d+e x^2\right )\right )}{4 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((-4*c*d + 2*b*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*
e*(-2*Log[d + e*x^2] + Log[a + b*x^2 + c*x^4]))/(4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2)
+ e*(b*d - a*e)))

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Maple [A]  time = 0.011, size = 176, normalized size = 1.3 \[ -{\frac{e\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a{e}^{2}-4\,bde+4\,c{d}^{2}}}-{\frac{be}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{cd}{a{e}^{2}-bde+c{d}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{e\ln \left ( e{x}^{2}+d \right ) }{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(e*x^2+d)/(c*x^4+b*x^2+a),x)

[Out]

-1/4*e*ln(c*x^4+b*x^2+a)/(a*e^2-b*d*e+c*d^2)-1/2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)
^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*e+1/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^
2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c*d+1/2*e*ln(e*x^2+d)/(a*e^2-b*d*
e+c*d^2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 12.9931, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, c d - b e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (e \log \left (c x^{4} + b x^{2} + a\right ) - 2 \, e \log \left (e x^{2} + d\right )\right )}}{4 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (2 \, c d - b e\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (e \log \left (c x^{4} + b x^{2} + a\right ) - 2 \, e \log \left (e x^{2} + d\right )\right )}}{4 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.301574, size = 181, normalized size = 1.36 \[ -\frac{e{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (c d^{2} - b d e + a e^{2}\right )}} + \frac{e^{2}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )}} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

-1/4*e*ln(c*x^4 + b*x^2 + a)/(c*d^2 - b*d*e + a*e^2) + 1/2*e^2*ln(abs(x^2*e + d)
)/(c*d^2*e - b*d*e^2 + a*e^3) + 1/2*(2*c*d - b*e)*arctan((2*c*x^2 + b)/sqrt(-b^2
+ 4*a*c))/((c*d^2 - b*d*e + a*e^2)*sqrt(-b^2 + 4*a*c))