3.236 \(\int \frac{1}{x^5 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=156 \[ \frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )}+\frac{c^2 d \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )}-\frac{\log (x) \left (c d^2-a e^2\right )}{a^2 d^3}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )}+\frac{e}{2 a d^2 x^2}-\frac{1}{4 a d x^4} \]

[Out]

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

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Rubi [A]  time = 0.374493, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )}+\frac{c^2 d \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )}-\frac{\log (x) \left (c d^2-a e^2\right )}{a^2 d^3}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )}+\frac{e}{2 a d^2 x^2}-\frac{1}{4 a d x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(d + e*x^2)*(a + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 54.4211, size = 139, normalized size = 0.89 \[ - \frac{e^{4} \log{\left (d + e x^{2} \right )}}{2 d^{3} \left (a e^{2} + c d^{2}\right )} - \frac{1}{4 a d x^{4}} + \frac{e}{2 a d^{2} x^{2}} + \frac{c^{2} d \log{\left (a + c x^{4} \right )}}{4 a^{2} \left (a e^{2} + c d^{2}\right )} + \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (x^{2} \right )}}{2 a^{2} d^{3}} + \frac{c^{\frac{3}{2}} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(e*x**2+d)/(c*x**4+a),x)

[Out]

-e**4*log(d + e*x**2)/(2*d**3*(a*e**2 + c*d**2)) - 1/(4*a*d*x**4) + e/(2*a*d**2*
x**2) + c**2*d*log(a + c*x**4)/(4*a**2*(a*e**2 + c*d**2)) + (a*e**2 - c*d**2)*lo
g(x**2)/(2*a**2*d**3) + c**(3/2)*e*atan(sqrt(c)*x**2/sqrt(a))/(2*a**(3/2)*(a*e**
2 + c*d**2))

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Mathematica [A]  time = 0.161162, size = 209, normalized size = 1.34 \[ -\frac{a^2 d^2 e^2+2 a^2 e^4 x^4 \log \left (d+e x^2\right )-2 a^2 d e^3 x^2-4 a^2 e^4 x^4 \log (x)+2 \sqrt{a} c^{3/2} d^3 e x^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt{a} c^{3/2} d^3 e x^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-c^2 d^4 x^4 \log \left (a+c x^4\right )+a c d^4-2 a c d^3 e x^2+4 c^2 d^4 x^4 \log (x)}{4 a^2 d^3 x^4 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(d + e*x^2)*(a + c*x^4)),x]

[Out]

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

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Maple [A]  time = 0.017, size = 145, normalized size = 0.9 \[ -{\frac{1}{4\,ad{x}^{4}}}+{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{3}a}}-{\frac{\ln \left ( x \right ) c}{{a}^{2}d}}+{\frac{e}{2\,a{d}^{2}{x}^{2}}}+{\frac{{c}^{2}d\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ){a}^{2}}}+{\frac{{c}^{2}e}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{e}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(e*x^2+d)/(c*x^4+a),x)

[Out]

-1/4/a/d/x^4+1/d^3/a*ln(x)*e^2-1/d/a^2*ln(x)*c+1/2*e/a/d^2/x^2+1/4*c^2*d*ln(c*x^
4+a)/a^2/(a*e^2+c*d^2)+1/2*c^2/(a*e^2+c*d^2)/a*e/(a*c)^(1/2)*arctan(c*x^2/(a*c)^
(1/2))-1/2*e^4*ln(e*x^2+d)/d^3/(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(1/((c*x^4 + a)*(e*x^2 + d)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 87.5516, size = 1, normalized size = 0.01 \[ \left [\frac{a c d^{3} e x^{4} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} + 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) + c^{2} d^{4} x^{4} \log \left (c x^{4} + a\right ) - 2 \, a^{2} e^{4} x^{4} \log \left (e x^{2} + d\right ) - a c d^{4} - a^{2} d^{2} e^{2} - 4 \,{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x^{4} \log \left (x\right ) + 2 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2}}{4 \,{\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{4}}, -\frac{2 \, a c d^{3} e x^{4} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) - c^{2} d^{4} x^{4} \log \left (c x^{4} + a\right ) + 2 \, a^{2} e^{4} x^{4} \log \left (e x^{2} + d\right ) + a c d^{4} + a^{2} d^{2} e^{2} + 4 \,{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x^{4} \log \left (x\right ) - 2 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2}}{4 \,{\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*(e*x^2 + d)*x^5),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276662, size = 227, normalized size = 1.46 \[ \frac{c^{2} d{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (a^{2} c d^{2} + a^{3} e^{2}\right )}} + \frac{c^{2} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right ) e}{2 \,{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt{a c}} - \frac{e^{5}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{5} e + a d^{3} e^{3}\right )}} - \frac{{\left (c d^{2} - a e^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2} d^{3}} + \frac{3 \, c d^{2} x^{4} - 3 \, a x^{4} e^{2} + 2 \, a d x^{2} e - a d^{2}}{4 \, a^{2} d^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*(e*x^2 + d)*x^5),x, algorithm="giac")

[Out]

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