Integrand size = 52, antiderivative size = 128 \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \]
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\[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^{13/4} \sqrt [4]{-b+a x^2} \left (-b+a x^2+x^3\right )} \, dx}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (-3 b+a x^8\right ) \left (b-a x^8+x^{12}\right )}{x^{10} \sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {3 b}{x^{10} \sqrt [4]{-b+a x^8}}-\frac {a}{x^2 \sqrt [4]{-b+a x^8}}+\frac {2 x^2 \left (3 b-a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (3 b-a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {3 b x^2}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )}+\frac {a x^{10}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = -\frac {4 b \sqrt [4]{1-\frac {a x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{8},\frac {1}{4},-\frac {1}{8},\frac {a x^2}{b}\right )}{3 x^2 \sqrt [4]{-b x+a x^3}}+\frac {4 a \sqrt [4]{1-\frac {a x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {1}{4},\frac {7}{8},\frac {a x^2}{b}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (24 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ \end{align*}
Time = 12.78 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {-b x+a x^3}}{\sqrt {2} x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \]
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\[\int \frac {\left (a \,x^{2}-3 b \right ) \left (-a \,x^{2}+x^{3}+b \right )}{x^{3} \left (a \,x^{2}+x^{3}-b \right ) \left (a \,x^{3}-b x \right )^{\frac {1}{4}}}d x\]
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Timed out. \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\int { -\frac {{\left (a x^{2} - x^{3} - b\right )} {\left (a x^{2} - 3 \, b\right )}}{{\left (a x^{3} - b x\right )}^{\frac {1}{4}} {\left (a x^{2} + x^{3} - b\right )} x^{3}} \,d x } \]
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\[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\int { -\frac {{\left (a x^{2} - x^{3} - b\right )} {\left (a x^{2} - 3 \, b\right )}}{{\left (a x^{3} - b x\right )}^{\frac {1}{4}} {\left (a x^{2} + x^{3} - b\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\int -\frac {\left (3\,b-a\,x^2\right )\,\left (x^3-a\,x^2+b\right )}{x^3\,{\left (a\,x^3-b\,x\right )}^{1/4}\,\left (x^3+a\,x^2-b\right )} \,d x \]
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