3.48 \(\int \frac{d+e x^4}{x \left (a+b x^4+c x^8\right )} \, dx\)

Optimal. Leaf size=78 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 a \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b x^4+c x^8\right )}{8 a}+\frac{d \log (x)}{a} \]

[Out]

((b*d - 2*a*e)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*a*Sqrt[b^2 - 4*a*c])
+ (d*Log[x])/a - (d*Log[a + b*x^4 + c*x^8])/(8*a)

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Rubi [A]  time = 0.277635, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 a \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b x^4+c x^8\right )}{8 a}+\frac{d \log (x)}{a} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^4)/(x*(a + b*x^4 + c*x^8)),x]

[Out]

((b*d - 2*a*e)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*a*Sqrt[b^2 - 4*a*c])
+ (d*Log[x])/a - (d*Log[a + b*x^4 + c*x^8])/(8*a)

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Rubi in Sympy [A]  time = 36.7427, size = 73, normalized size = 0.94 \[ \frac{d \log{\left (x^{4} \right )}}{4 a} - \frac{d \log{\left (a + b x^{4} + c x^{8} \right )}}{8 a} - \frac{\left (2 a e - b d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{4}}{\sqrt{- 4 a c + b^{2}}} \right )}}{4 a \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**4+d)/x/(c*x**8+b*x**4+a),x)

[Out]

d*log(x**4)/(4*a) - d*log(a + b*x**4 + c*x**8)/(8*a) - (2*a*e - b*d)*atanh((b +
2*c*x**4)/sqrt(-4*a*c + b**2))/(4*a*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0567099, size = 80, normalized size = 1.03 \[ \frac{d \log (x)}{a}-\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right ]}{4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^4)/(x*(a + b*x^4 + c*x^8)),x]

[Out]

(d*Log[x])/a - RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1
] + c*d*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a)

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Maple [A]  time = 0.009, size = 106, normalized size = 1.4 \[ -{\frac{d\ln \left ( c{x}^{8}+b{x}^{4}+a \right ) }{8\,a}}+{\frac{e}{2}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bd}{4\,a}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{d\ln \left ( x \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^4+d)/x/(c*x^8+b*x^4+a),x)

[Out]

-1/8*d*ln(c*x^8+b*x^4+a)/a+1/2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^
(1/2))*e-1/4/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2))*b*d+d*ln(
x)/a

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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((e*x^4 + d)/((c*x^8 + b*x^4 + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.861789, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d - 2 \, a e\right )} \log \left (-\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + b^{3} - 4 \, a b c -{\left (2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (d \log \left (c x^{8} + b x^{4} + a\right ) - 8 \, d \log \left (x\right )\right )}}{8 \, \sqrt{b^{2} - 4 \, a c} a}, -\frac{2 \,{\left (b d - 2 \, a e\right )} \arctan \left (-\frac{{\left (2 \, c x^{4} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (d \log \left (c x^{8} + b x^{4} + a\right ) - 8 \, d \log \left (x\right )\right )}}{8 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/((c*x^8 + b*x^4 + a)*x),x, algorithm="fricas")

[Out]

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

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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((e*x**4+d)/x/(c*x**8+b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282423, size = 105, normalized size = 1.35 \[ -\frac{d{\rm ln}\left (c x^{8} + b x^{4} + a\right )}{8 \, a} + \frac{d{\rm ln}\left (x^{4}\right )}{4 \, a} - \frac{{\left (b d - 2 \, a e\right )} \arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/((c*x^8 + b*x^4 + a)*x),x, algorithm="giac")

[Out]

-1/8*d*ln(c*x^8 + b*x^4 + a)/a + 1/4*d*ln(x^4)/a - 1/4*(b*d - 2*a*e)*arctan((2*c
*x^4 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a)