Optimal. Leaf size=204 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{x^3}{4 a \left (a+c x^4\right )} \]
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Rubi [A] time = 0.119539, antiderivative size = 204, 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 = {290, 297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{x^3}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 290
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+c x^4\right )^2} \, dx &=\frac{x^3}{4 a \left (a+c x^4\right )}+\frac{\int \frac{x^2}{a+c x^4} \, dx}{4 a}\\ &=\frac{x^3}{4 a \left (a+c x^4\right )}-\frac{\int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{8 a \sqrt{c}}+\frac{\int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{8 a \sqrt{c}}\\ &=\frac{x^3}{4 a \left (a+c x^4\right )}+\frac{\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a c}+\frac{\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a c}+\frac{\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}{16 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\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}{16 \sqrt{2} a^{5/4} c^{3/4}}\\ &=\frac{x^3}{4 a \left (a+c x^4\right )}+\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}\\ &=\frac{x^3}{4 a \left (a+c x^4\right )}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0983371, size = 184, normalized size = 0.9 \[ \frac{\frac{\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 \sqrt [4]{a} x^3}{a+c x^4}}{32 a^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 154, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{\sqrt{2}}{32\,ac}\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{\sqrt{2}}{16\,ac}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{\sqrt{2}}{16\,ac}\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 [A] time = 1.96515, size = 441, normalized size = 2.16 \begin{align*} \frac{4 \, x^{3} - 4 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} \arctan \left (-a c x \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} + \sqrt{-a^{3} c \sqrt{-\frac{1}{a^{5} c^{3}}} + x^{2}} a c \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}}\right ) +{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + x\right ) -{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (-a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + x\right )}{16 \,{\left (a c x^{4} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.654603, size = 46, normalized size = 0.23 \begin{align*} \frac{x^{3}}{4 a^{2} + 4 a c x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{5} c^{3} + 1, \left ( t \mapsto t \log{\left (4096 t^{3} a^{4} c^{2} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13392, size = 265, normalized size = 1.3 \begin{align*} \frac{x^{3}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{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 )}{16 \, a^{2} c^{3}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{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 )}{16 \, a^{2} c^{3}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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