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

Optimal. Leaf size=96 \[ -\frac{e \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.138898, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{e \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.0017, size = 85, normalized size = 0.89 \[ - \frac{e \log{\left (a + c x^{4} \right )}}{4 \left (a e^{2} + c d^{2}\right )} + \frac{e \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{c} d \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0604976, size = 67, normalized size = 0.7 \[ \frac{\frac{2 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a}}-e \log \left (a+c x^4\right )+2 e \log \left (d+e x^2\right )}{4 a e^2+4 c d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 83, normalized size = 0.9 \[ -{\frac{e\ln \left ( c{x}^{4}+a \right ) }{4\,a{e}^{2}+4\,c{d}^{2}}}+{\frac{cd}{2\,a{e}^{2}+2\,c{d}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{e\ln \left ( e{x}^{2}+d \right ) }{2\,a{e}^{2}+2\,c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(e*x^2+d)/(c*x^4+a),x)

[Out]

-1/4*e*ln(c*x^4+a)/(a*e^2+c*d^2)+1/2*c/(a*e^2+c*d^2)*d/(a*c)^(1/2)*arctan(c*x^2/
(a*c)^(1/2))+1/2*e*ln(e*x^2+d)/(a*e^2+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 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.555051, size = 1, normalized size = 0.01 \[ \left [\frac{d \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} + 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - e \log \left (c x^{4} + a\right ) + 2 \, e \log \left (e x^{2} + d\right )}{4 \,{\left (c d^{2} + a e^{2}\right )}}, -\frac{2 \, d \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) + e \log \left (c x^{4} + a\right ) - 2 \, e \log \left (e x^{2} + d\right )}{4 \,{\left (c d^{2} + a e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274714, size = 115, normalized size = 1.2 \[ \frac{c d \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} - \frac{e{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c d^{2} + a e^{2}\right )}} + \frac{e^{2}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e + a e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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