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

Optimal. Leaf size=151 \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x^2}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.363493, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x^2}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 58.7762, size = 133, normalized size = 0.88 \[ - \frac{e^{3} \log{\left (a + c x^{4} \right )}}{4 \left (a e^{2} + c d^{2}\right )^{2}} + \frac{e^{3} \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x^{2}}{4 a \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{c} d \left (3 a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.249135, size = 117, normalized size = 0.77 \[ \frac{\frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\left (a e^2+c d^2\right ) \left (a e+c d x^2\right )}{a \left (a+c x^4\right )}-e^3 \log \left (a+c x^4\right )+2 e^3 \log \left (d+e x^2\right )}{4 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 257, normalized size = 1.7 \[{\frac{c{x}^{2}{e}^{2}d}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{2}{d}^{3}{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) a}}+{\frac{{e}^{3}a}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{d}^{2}ec}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{e}^{3}\ln \left ( a \left ( c{x}^{4}+a \right ) \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{3\,cd{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{c}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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)^2*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 7.24755, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a c d^{2} e + 2 \, a^{2} e^{3} + 2 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} +{\left (a c d^{3} + 3 \, a^{2} d e^{2} +{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{4}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} + 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - 2 \,{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (c x^{4} + a\right ) + 4 \,{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (e x^{2} + d\right )}{8 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{4}\right )}}, \frac{a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} -{\left (a c d^{3} + 3 \, a^{2} d e^{2} +{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{4}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) -{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (c x^{4} + a\right ) + 2 \,{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.275941, size = 269, normalized size = 1.78 \[ -\frac{e^{3}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac{{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} + \frac{a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} + a^{2} e^{3}}{4 \,{\left (c x^{4} + a\right )}{\left (c d^{2} + a e^{2}\right )}^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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