Optimal. Leaf size=394 \[ \frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{d}{3 a x^3} \]
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Rubi [A] time = 0.625247, 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, Rules used = {1504, 1422, 212, 208, 205} \[ \frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{d}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 1504
Rule 1422
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{d+e x^4}{x^4 \left (a+b x^4+c x^8\right )} \, dx &=-\frac{d}{3 a x^3}-\frac{\int \frac{3 (b d-a e)+3 c d x^4}{a+b x^4+c x^8} \, dx}{3 a}\\ &=-\frac{d}{3 a x^3}-\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx}{2 a}-\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx}{2 a}\\ &=-\frac{d}{3 a x^3}+\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b-\sqrt{b^2-4 a c}}}+\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b-\sqrt{b^2-4 a c}}}+\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b+\sqrt{b^2-4 a c}}}+\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b+\sqrt{b^2-4 a c}}}\\ &=-\frac{d}{3 a x^3}+\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]{-b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}+\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]{-b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b+\sqrt{b^2-4 a c}\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]{-b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b-\sqrt{b^2-4 a c}\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]{-b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0735016, size = 86, normalized size = 0.22 \[ -\frac{3 \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 c}\& \right ]+\frac{4 d}{x^3}}{12 a} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 68, normalized size = 0.2 \begin{align*} -{\frac{d}{3\,a{x}^{3}}}+{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\mathit{sage}_{2}}{a} - \frac{d}{3 \, a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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