Optimal. Leaf size=214 \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4}}-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )} \]
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Rubi [A] time = 0.136988, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {290, 325, 211, 1165, 628, 1162, 617, 204} \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4}}-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+c x^4\right )^2} \, dx &=\frac{1}{4 a x^3 \left (a+c x^4\right )}+\frac{7 \int \frac{1}{x^4 \left (a+c x^4\right )} \, dx}{4 a}\\ &=-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )}-\frac{(7 c) \int \frac{1}{a+c x^4} \, dx}{4 a^2}\\ &=-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )}-\frac{(7 c) \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{8 a^{5/2}}-\frac{(7 c) \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{8 a^{5/2}}\\ &=-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )}-\frac{\left (7 \sqrt{c}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{5/2}}-\frac{\left (7 \sqrt{c}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{5/2}}+\frac{\left (7 c^{3/4}\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}{16 \sqrt{2} a^{11/4}}+\frac{\left (7 c^{3/4}\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}{16 \sqrt{2} a^{11/4}}\\ &=-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )}+\frac{7 c^{3/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{\left (7 c^{3/4}\right ) \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^{11/4}}+\frac{\left (7 c^{3/4}\right ) \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^{11/4}}\\ &=-\frac{7}{12 a^2 x^3}+\frac{1}{4 a x^3 \left (a+c x^4\right )}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4}}+\frac{7 c^{3/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4}}\\ \end{align*}
Mathematica [A] time = 0.132402, size = 194, normalized size = 0.91 \[ \frac{-\frac{24 a^{3/4} c x}{a+c x^4}-\frac{32 a^{3/4}}{x^3}+21 \sqrt{2} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-21 \sqrt{2} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+42 \sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-42 \sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{96 a^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 155, normalized size = 0.7 \begin{align*} -{\frac{cx}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{7\,c\sqrt{2}}{32\,{a}^{3}}\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{7\,c\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{7\,c\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{1}{3\,{x}^{3}{a}^{2}}} \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 = 1.8176, size = 489, normalized size = 2.29 \begin{align*} -\frac{28 \, c x^{4} + 84 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{8} x \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{4}} - a^{8} \sqrt{\frac{a^{6} \sqrt{-\frac{c^{3}}{a^{11}}} + c^{2} x^{2}}{c^{2}}} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{4}}}{c^{2}}\right ) + 21 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, c x\right ) - 21 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, c x\right ) + 16 \, a}{48 \,{\left (a^{2} c x^{7} + a^{3} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.30136, size = 56, normalized size = 0.26 \begin{align*} - \frac{4 a + 7 c x^{4}}{12 a^{3} x^{3} + 12 a^{2} c x^{7}} + \operatorname{RootSum}{\left (65536 t^{4} a^{11} + 2401 c^{3}, \left ( t \mapsto t \log{\left (- \frac{16 t a^{3}}{7 c} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32951, size = 258, normalized size = 1.21 \begin{align*} -\frac{c x}{4 \,{\left (c x^{4} + a\right )} a^{2}} - \frac{7 \, \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 )}{16 \, a^{3}} - \frac{7 \, \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 )}{16 \, a^{3}} - \frac{7 \, \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 )}{32 \, a^{3}} + \frac{7 \, \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 )}{32 \, a^{3}} - \frac{1}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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