3.249 \(\int \frac{1}{x \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}-\frac{c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}+\frac{\log (x)}{a^2 d}+\frac{c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )^2} \]

[Out]

(c*(d - e*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) - (Sqrt[c]*e^3*ArcTan[(Sqrt[c]
*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^2 + a*e^2)^2) - (Sqrt[c]*e*ArcTan[(Sqrt[c]*x^2)/
Sqrt[a]])/(4*a^(3/2)*(c*d^2 + a*e^2)) + Log[x]/(a^2*d) - (e^4*Log[d + e*x^2])/(2
*d*(c*d^2 + a*e^2)^2) - (c*d*(c*d^2 + 2*a*e^2)*Log[a + c*x^4])/(4*a^2*(c*d^2 + a
*e^2)^2)

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Rubi [A]  time = 0.485347, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}-\frac{c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}+\frac{\log (x)}{a^2 d}+\frac{c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(c*(d - e*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) - (Sqrt[c]*e^3*ArcTan[(Sqrt[c]
*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^2 + a*e^2)^2) - (Sqrt[c]*e*ArcTan[(Sqrt[c]*x^2)/
Sqrt[a]])/(4*a^(3/2)*(c*d^2 + a*e^2)) + Log[x]/(a^2*d) - (e^4*Log[d + e*x^2])/(2
*d*(c*d^2 + a*e^2)^2) - (c*d*(c*d^2 + 2*a*e^2)*Log[a + c*x^4])/(4*a^2*(c*d^2 + a
*e^2)^2)

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Rubi in Sympy [A]  time = 74.1124, size = 189, normalized size = 0.9 \[ - \frac{e^{4} \log{\left (d + e x^{2} \right )}}{2 d \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \left (d - e x^{2}\right )}{4 a \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} - \frac{c d \left (2 a e^{2} + c d^{2}\right ) \log{\left (a + c x^{4} \right )}}{4 a^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{\log{\left (x^{2} \right )}}{2 a^{2} d} - \frac{\sqrt{c} e^{3} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{\sqrt{c} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

-e**4*log(d + e*x**2)/(2*d*(a*e**2 + c*d**2)**2) + c*(d - e*x**2)/(4*a*(a + c*x*
*4)*(a*e**2 + c*d**2)) - c*d*(2*a*e**2 + c*d**2)*log(a + c*x**4)/(4*a**2*(a*e**2
+ c*d**2)**2) + log(x**2)/(2*a**2*d) - sqrt(c)*e**3*atan(sqrt(c)*x**2/sqrt(a))/
(2*sqrt(a)*(a*e**2 + c*d**2)**2) - sqrt(c)*e*atan(sqrt(c)*x**2/sqrt(a))/(4*a**(3
/2)*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.320713, size = 241, normalized size = 1.15 \[ \frac{-2 a^2 e^4 \left (a+c x^4\right ) \log \left (d+e x^2\right )+4 \log (x) \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2-c d^2 \left (a+c x^4\right ) \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )+\sqrt{a} \sqrt{c} d e \left (a+c x^4\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt{a} \sqrt{c} d e \left (a+c x^4\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+a c d \left (d-e x^2\right ) \left (a e^2+c d^2\right )}{4 a^2 d \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(a*c*d*(c*d^2 + a*e^2)*(d - e*x^2) + Sqrt[a]*Sqrt[c]*d*e*(c*d^2 + 3*a*e^2)*(a +
c*x^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[a]*Sqrt[c]*d*e*(c*d^2 + 3*
a*e^2)*(a + c*x^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 4*(c*d^2 + a*e^2)^2
*(a + c*x^4)*Log[x] - 2*a^2*e^4*(a + c*x^4)*Log[d + e*x^2] - c*d^2*(c*d^2 + 2*a*
e^2)*(a + c*x^4)*Log[a + c*x^4])/(4*a^2*d*(c*d^2 + a*e^2)^2*(a + c*x^4))

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Maple [A]  time = 0.029, size = 309, normalized size = 1.5 \[{\frac{\ln \left ( x \right ) }{{a}^{2}d}}-{\frac{c{x}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{2}{x}^{2}{d}^{2}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}+{\frac{cd{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}-{\frac{c\ln \left ( c{x}^{4}+a \right ) d{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}}-{\frac{{c}^{2}\ln \left ( c{x}^{4}+a \right ){d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}}-{\frac{3\,c{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{2}e{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{e}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,d \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

ln(x)/a^2/d-1/4/(a*e^2+c*d^2)^2*c/(c*x^4+a)*x^2*e^3-1/4/(a*e^2+c*d^2)^2*c^2/a/(c
*x^4+a)*x^2*d^2*e+1/4/(a*e^2+c*d^2)^2*c/(c*x^4+a)*d*e^2+1/4/(a*e^2+c*d^2)^2*c^2/
a/(c*x^4+a)*d^3-1/2/(a*e^2+c*d^2)^2*c/a*ln(c*x^4+a)*d*e^2-1/4/(a*e^2+c*d^2)^2*c^
2/a^2*ln(c*x^4+a)*d^3-3/4/(a*e^2+c*d^2)^2*c/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2)
)*e^3-1/4/(a*e^2+c*d^2)^2*c^2/a/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))*d^2*e-1/2*
e^4*ln(e*x^2+d)/d/(a*e^2+c*d^2)^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(1/((c*x^4 + a)^2*(e*x^2 + d)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 130.493, size = 1, normalized size = 0. \[ \left [\frac{2 \, a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} - 2 \,{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} +{\left (a^{2} c d^{3} e + 3 \, a^{3} d e^{3} +{\left (a c^{2} d^{3} e + 3 \, a^{2} c d e^{3}\right )} x^{4}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - 2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} x^{4}\right )} \log \left (c x^{4} + a\right ) - 4 \,{\left (a^{2} c e^{4} x^{4} + a^{3} e^{4}\right )} \log \left (e x^{2} + d\right ) + 8 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \log \left (x\right )}{8 \,{\left (a^{3} c^{2} d^{5} + 2 \, a^{4} c d^{3} e^{2} + a^{5} d e^{4} +{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )} x^{4}\right )}}, \frac{a c^{2} d^{4} + a^{2} c d^{2} e^{2} -{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} +{\left (a^{2} c d^{3} e + 3 \, a^{3} d e^{3} +{\left (a c^{2} d^{3} e + 3 \, a^{2} c d e^{3}\right )} x^{4}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) -{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} x^{4}\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (a^{2} c e^{4} x^{4} + a^{3} e^{4}\right )} \log \left (e x^{2} + d\right ) + 4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{3} c^{2} d^{5} + 2 \, a^{4} c d^{3} e^{2} + a^{5} d e^{4} +{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )} x^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x),x, algorithm="fricas")

[Out]

[1/8*(2*a*c^2*d^4 + 2*a^2*c*d^2*e^2 - 2*(a*c^2*d^3*e + a^2*c*d*e^3)*x^2 + (a^2*c
*d^3*e + 3*a^3*d*e^3 + (a*c^2*d^3*e + 3*a^2*c*d*e^3)*x^4)*sqrt(-c/a)*log((c*x^4
- 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) - 2*(a*c^2*d^4 + 2*a^2*c*d^2*e^2 + (c^3*d
^4 + 2*a*c^2*d^2*e^2)*x^4)*log(c*x^4 + a) - 4*(a^2*c*e^4*x^4 + a^3*e^4)*log(e*x^
2 + d) + 8*(a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2 +
a^2*c*e^4)*x^4)*log(x))/(a^3*c^2*d^5 + 2*a^4*c*d^3*e^2 + a^5*d*e^4 + (a^2*c^3*d
^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)*x^4), 1/4*(a*c^2*d^4 + a^2*c*d^2*e^2 - (a*
c^2*d^3*e + a^2*c*d*e^3)*x^2 + (a^2*c*d^3*e + 3*a^3*d*e^3 + (a*c^2*d^3*e + 3*a^2
*c*d*e^3)*x^4)*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) - (a*c^2*d^4 + 2*a^2*c*d^2*
e^2 + (c^3*d^4 + 2*a*c^2*d^2*e^2)*x^4)*log(c*x^4 + a) - 2*(a^2*c*e^4*x^4 + a^3*e
^4)*log(e*x^2 + d) + 4*(a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c
^2*d^2*e^2 + a^2*c*e^4)*x^4)*log(x))/(a^3*c^2*d^5 + 2*a^4*c*d^3*e^2 + a^5*d*e^4
+ (a^2*c^3*d^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)*x^4)]

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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(1/x/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276621, size = 377, normalized size = 1.8 \[ -\frac{{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} - \frac{e^{5}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )}} - \frac{{\left (c^{2} d^{2} e + 3 \, a c e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} + \frac{c^{3} d^{3} x^{4} + 2 \, a c^{2} d x^{4} e^{2} - a c^{2} d^{2} x^{2} e + 2 \, a c^{2} d^{3} - a^{2} c x^{2} e^{3} + 3 \, a^{2} c d e^{2}}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}{\left (c x^{4} + a\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x),x, algorithm="giac")

[Out]

-1/4*(c^2*d^3 + 2*a*c*d*e^2)*ln(c*x^4 + a)/(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*
e^4) - 1/2*e^5*ln(abs(x^2*e + d))/(c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5) - 1/4*
(c^2*d^2*e + 3*a*c*e^3)*arctan(c*x^2/sqrt(a*c))/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 +
a^3*e^4)*sqrt(a*c)) + 1/4*(c^3*d^3*x^4 + 2*a*c^2*d*x^4*e^2 - a*c^2*d^2*x^2*e + 2
*a*c^2*d^3 - a^2*c*x^2*e^3 + 3*a^2*c*d*e^2)/((a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^
4*e^4)*(c*x^4 + a)) + 1/2*ln(x^2)/(a^2*d)