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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]