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

Optimal. Leaf size=105 \[ \frac{a e \log \left (a+c x^4\right )}{4 c \left (a e^2+c d^2\right )}+\frac{d^2 \log \left (d+e x^2\right )}{2 e \left (a e^2+c d^2\right )}-\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{c} \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.250944, antiderivative size = 105, 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 \[ \frac{a e \log \left (a+c x^4\right )}{4 c \left (a e^2+c d^2\right )}+\frac{d^2 \log \left (d+e x^2\right )}{2 e \left (a e^2+c d^2\right )}-\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{c} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.6023, size = 88, normalized size = 0.84 \[ - \frac{\sqrt{a} d \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{c} \left (a e^{2} + c d^{2}\right )} + \frac{a e \log{\left (a + c x^{4} \right )}}{4 c \left (a e^{2} + c d^{2}\right )} + \frac{d^{2} \log{\left (d + e x^{2} \right )}}{2 e \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0580545, size = 77, normalized size = 0.73 \[ \frac{-2 \sqrt{a} \sqrt{c} d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )+a e^2 \log \left (a+c x^4\right )+2 c d^2 \log \left (d+e x^2\right )}{4 a c e^3+4 c^2 d^2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 92, normalized size = 0.9 \[{\frac{ae\ln \left ( c{x}^{4}+a \right ) }{4\,c \left ( a{e}^{2}+c{d}^{2} \right ) }}-{\frac{ad}{2\,a{e}^{2}+2\,c{d}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{2}\ln \left ( e{x}^{2}+d \right ) }{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(e*x^2+d)/(c*x^4+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08229, size = 1, normalized size = 0.01 \[ \left [\frac{c d e \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} - 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) + a e^{2} \log \left (c x^{4} + a\right ) + 2 \, c d^{2} \log \left (e x^{2} + d\right )}{4 \,{\left (c^{2} d^{2} e + a c e^{3}\right )}}, -\frac{2 \, c d e \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right ) - a e^{2} \log \left (c x^{4} + a\right ) - 2 \, c d^{2} \log \left (e x^{2} + d\right )}{4 \,{\left (c^{2} d^{2} e + a c e^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275688, size = 122, normalized size = 1.16 \[ \frac{a e{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{2} + a c e^{2}\right )}} + \frac{d^{2}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e + a e^{3}\right )}} - \frac{a d \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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