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

Optimal. Leaf size=394 \[ \frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{d}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 1.38581, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{d}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 153.716, size = 394, normalized size = 1. \[ - \frac{2^{\frac{3}{4}} c^{\frac{3}{4}} \left (2 a e - b d - d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{3}{4}} c^{\frac{3}{4}} \left (2 a e - b d - d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} c^{\frac{3}{4}} \left (2 a e - b d + d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} c^{\frac{3}{4}} \left (2 a e - b d + d \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} - \frac{d}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(3/4)*c**(3/4)*(2*a*e - b*d - d*sqrt(-4*a*c + b**2))*atan(2**(1/4)*c**(1/4)*
x/(-b + sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b + sqrt(-4*a*c + b**2))**(3/4)*sqrt
(-4*a*c + b**2)) - 2**(3/4)*c**(3/4)*(2*a*e - b*d - d*sqrt(-4*a*c + b**2))*atanh
(2**(1/4)*c**(1/4)*x/(-b + sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b + sqrt(-4*a*c +
b**2))**(3/4)*sqrt(-4*a*c + b**2)) + 2**(3/4)*c**(3/4)*(2*a*e - b*d + d*sqrt(-4
*a*c + b**2))*atan(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c + b**2))**(1/4))/(4*a*(
-b - sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) + 2**(3/4)*c**(3/4)*(2*a*e
- b*d + d*sqrt(-4*a*c + b**2))*atanh(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c + b*
*2))**(1/4))/(4*a*(-b - sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) - d/(3*
a*x**3)

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Mathematica [C]  time = 0.0993131, size = 86, normalized size = 0.22 \[ -\frac{3 \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]+\frac{4 d}{x^3}}{12 a} \]

Antiderivative was successfully verified.

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

[Out]

-((4*d)/x^3 + 3*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*#1^3 + 2*c*#1^7) & ])/(12*a)

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Maple [C]  time = 0.009, size = 68, normalized size = 0.2 \[ -{\frac{d}{3\,a{x}^{3}}}+{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( -cd{{\it \_R}}^{4}+ae-bd \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*d/a/x^3+1/4/a*sum((-_R^4*c*d+a*e-b*d)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(
_Z^8*c+_Z^4*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{c d x^{4} + b d - a e}{c x^{8} + b x^{4} + a}\,{d x}}{a} - \frac{d}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((c*d*x^4 + b*d - a*e)/(c*x^8 + b*x^4 + a), x)/a - 1/3*d/(a*x^3)

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Fricas [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)/((c*x^8 + b*x^4 + a)*x^4),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{4} + d}{{\left (c x^{8} + b x^{4} + a\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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