Optimal. Leaf size=253 \[ \frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{d x}{c} \]
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Rubi [A] time = 0.210931, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {1394, 1280, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{d x}{c} \]
Antiderivative was successfully verified.
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Rule 1394
Rule 1280
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{d+\frac{e}{x^2}}{c+\frac{a}{x^4}} \, dx &=\int \frac{x^2 \left (e+d x^2\right )}{a+c x^4} \, dx\\ &=\frac{d x}{c}-\frac{\int \frac{a d-c e x^2}{a+c x^4} \, dx}{c}\\ &=\frac{d x}{c}-\frac{\left (\frac{\sqrt{a} d}{\sqrt{c}}-e\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 c}-\frac{\left (\frac{\sqrt{a} d}{\sqrt{c}}+e\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 c}\\ &=\frac{d x}{c}-\frac{\left (\frac{\sqrt{a} d}{\sqrt{c}}-e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}-\frac{\left (\frac{\sqrt{a} d}{\sqrt{c}}-e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}+\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}\\ &=\frac{d x}{c}+\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}\\ &=\frac{d x}{c}+\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.106166, size = 293, normalized size = 1.16 \[ \frac{\left (a^{5/4} \sqrt{c} d+a^{3/4} c e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a c^{7/4}}-\frac{\left (a^{5/4} \sqrt{c} d+a^{3/4} c e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a c^{7/4}}+\frac{\left (a^{3/4} c e-a^{5/4} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{2 \sqrt{2} a c^{7/4}}+\frac{\left (a^{3/4} c e-a^{5/4} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt{2} \sqrt [4]{a}}\right )}{2 \sqrt{2} a c^{7/4}}+\frac{d x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 266, normalized size = 1.1 \begin{align*}{\frac{dx}{c}}-{\frac{d\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{d\sqrt{2}}{8\,c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{d\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{e\sqrt{2}}{8\,c}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{e\sqrt{2}}{4\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{e\sqrt{2}}{4\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36182, size = 1489, normalized size = 5.89 \begin{align*} \frac{c \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x +{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x -{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x +{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + c \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x -{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + 4 \, d x}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.838989, size = 109, normalized size = 0.43 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a c^{5} - 64 t^{2} a c^{3} d e + a^{2} d^{4} + 2 a c d^{2} e^{2} + c^{2} e^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a c^{4} e - 4 t a^{2} c d^{3} + 12 t a c^{2} d e^{2}}{a^{2} d^{4} - c^{2} e^{4}} \right )} \right )\right )} + \frac{d x}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10218, size = 347, normalized size = 1.37 \begin{align*} \frac{d x}{c} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c d - \left (a c^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c d + \left (a c^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{3} d - \left (a c^{3}\right )^{\frac{3}{4}} c^{2} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{5}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{3} d + \left (a c^{3}\right )^{\frac{3}{4}} c^{2} e\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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