Integrand size = 57, antiderivative size = 514 \[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\frac {\left (\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}\right )^{3/4} \left (32 a^2 c^3-96 a^2 c^3 x^2+36 a b c^3 x^3-36 a^2 c^2 d x^3+96 a^2 c^3 x^4-72 a b c^3 x^5+72 a^2 c^2 d x^5+64 a^2 c^3 x^6+45 b^2 c^3 x^6-42 a b c^2 d x^6-3 a^2 c d^2 x^6+36 a b c^3 x^7-36 a^2 c^2 d x^7-96 a^2 c^3 x^8-45 b^2 c^3 x^8+42 a b c^2 d x^8+3 a^2 c d^2 x^8-96 a^2 c^2 d x^9-45 b^2 c^2 d x^9+6 a b c d^2 x^9+7 a^2 d^3 x^9\right )}{96 a^3 c^2 x^9}+\frac {\left (-32 a^2 b c^3-15 b^3 c^3+32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {\left (32 a^2 b c^3+15 b^3 c^3-32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}} \]
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\[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-a+a x^2+b x^3} \int \frac {\left (-3+x^2\right ) \sqrt [4]{-c+c x^2+d x^3} \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}} \\ & = \frac {\sqrt [4]{-a+a x^2+b x^3} \int \left (-\frac {3 \sqrt [4]{-c+c x^2+d x^3}}{x^{10} \sqrt [4]{-a+a x^2+b x^3}}+\frac {7 \sqrt [4]{-c+c x^2+d x^3}}{x^8 \sqrt [4]{-a+a x^2+b x^3}}-\frac {5 \sqrt [4]{-c+c x^2+d x^3}}{x^6 \sqrt [4]{-a+a x^2+b x^3}}-\frac {2 \sqrt [4]{-c+c x^2+d x^3}}{x^4 \sqrt [4]{-a+a x^2+b x^3}}+\frac {\sqrt [4]{-c+c x^2+d x^3}}{x^2 \sqrt [4]{-a+a x^2+b x^3}}\right ) \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}} \\ & = \frac {\sqrt [4]{-a+a x^2+b x^3} \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^2 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}-\frac {\left (2 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^4 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}-\frac {\left (3 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^{10} \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}-\frac {\left (5 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^6 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}+\frac {\left (7 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^8 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}} \\ \end{align*}
Time = 16.59 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.58 \[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=-\frac {\left (\frac {b x^3+a \left (-1+x^2\right )}{d x^3+c \left (-1+x^2\right )}\right )^{3/4} \left (d x^3+c \left (-1+x^2\right )\right ) \left (45 b^2 c^2 x^6-6 a b c x^3 \left (d x^3+6 c \left (-1+x^2\right )\right )+a^2 \left (-7 d^2 x^6+4 c d x^3 \left (-1+x^2\right )+32 c^2 \left (1-2 x^2+x^4+3 x^6\right )\right )\right )}{96 a^3 c^2 x^9}+\frac {\left (-15 b^3 c^3+5 a b^2 c^2 d+a^2 b \left (-32 c^3+3 c d^2\right )+a^3 \left (32 c^2 d+7 d^3\right )\right ) \left (\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b x^3+a \left (-1+x^2\right )}{d x^3+c \left (-1+x^2\right )}}}{\sqrt [4]{a}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b x^3+a \left (-1+x^2\right )}{d x^3+c \left (-1+x^2\right )}}}{\sqrt [4]{a}}\right )\right )}{64 a^{13/4} c^{11/4}} \]
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\[\int \frac {\left (x^{2}-3\right ) \left (x^{6}+x^{4}-2 x^{2}+1\right )}{x^{10} \left (\frac {b \,x^{3}+a \,x^{2}-a}{d \,x^{3}+c \,x^{2}-c}\right )^{\frac {1}{4}}}d x\]
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Timed out. \[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\int { \frac {{\left (x^{6} + x^{4} - 2 \, x^{2} + 1\right )} {\left (x^{2} - 3\right )}}{x^{10} \left (\frac {b x^{3} + a x^{2} - a}{d x^{3} + c x^{2} - c}\right )^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\int { \frac {{\left (x^{6} + x^{4} - 2 \, x^{2} + 1\right )} {\left (x^{2} - 3\right )}}{x^{10} \left (\frac {b x^{3} + a x^{2} - a}{d x^{3} + c x^{2} - c}\right )^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx=\text {Hanged} \]
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