Optimal. Leaf size=190 \[ \frac{\sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}+\frac{\sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} c^{5/4}}+\frac{x}{c} \]
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Rubi [A] time = 0.14224, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {321, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}+\frac{\sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} c^{5/4}}+\frac{x}{c} \]
Antiderivative was successfully verified.
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Rule 321
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^4}{a+c x^4} \, dx &=\frac{x}{c}-\frac{a \int \frac{1}{a+c x^4} \, dx}{c}\\ &=\frac{x}{c}-\frac{\sqrt{a} \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{2 c}-\frac{\sqrt{a} \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{2 c}\\ &=\frac{x}{c}-\frac{\sqrt{a} \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c^{3/2}}-\frac{\sqrt{a} \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c^{3/2}}+\frac{\sqrt [4]{a} \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} c^{5/4}}+\frac{\sqrt [4]{a} \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} c^{5/4}}\\ &=\frac{x}{c}+\frac{\sqrt [4]{a} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \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} c^{5/4}}+\frac{\sqrt [4]{a} \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} c^{5/4}}\\ &=\frac{x}{c}+\frac{\sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{5/4}}+\frac{\sqrt [4]{a} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.031357, size = 173, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+8 \sqrt [4]{c} x}{8 c^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 133, normalized size = 0.7 \begin{align*}{\frac{x}{c}}-{\frac{\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{\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \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 [A] time = 1.79231, size = 284, normalized size = 1.49 \begin{align*} -\frac{4 \, c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{c^{4} x \left (-\frac{a}{c^{5}}\right )^{\frac{3}{4}} - \sqrt{c^{2} \sqrt{-\frac{a}{c^{5}}} + x^{2}} c^{4} \left (-\frac{a}{c^{5}}\right )^{\frac{3}{4}}}{a}\right ) + c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} \log \left (c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} + x\right ) - c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} \log \left (-c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} + x\right ) - 4 \, x}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.375381, size = 22, normalized size = 0.12 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} c^{5} + a, \left ( t \mapsto t \log{\left (- 4 t c + x \right )} \right )\right )} + \frac{x}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14575, size = 232, normalized size = 1.22 \begin{align*} \frac{x}{c} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \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 \, c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \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 \, c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, c^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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