Integrand size = 42, antiderivative size = 133 \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^6}}{-\sqrt {c} x^2+\sqrt {-b+a x^6}}\right )}{\sqrt {2} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \]
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\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {c}{a \left (-b+a x^6\right )^{3/4}}+\frac {x^2}{\left (-b+a x^6\right )^{3/4}}-\frac {b c-3 a b x^2-c^2 x^4}{a \left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )}\right ) \, dx \\ & = -\frac {\int \frac {b c-3 a b x^2-c^2 x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx}{a}-\frac {c \int \frac {1}{\left (-b+a x^6\right )^{3/4}} \, dx}{a}+\int \frac {x^2}{\left (-b+a x^6\right )^{3/4}} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (-b+a x^2\right )^{3/4}} \, dx,x,x^3\right )-\frac {\int \left (-\frac {b c}{\left (b-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}-\frac {3 a b x^2}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )}-\frac {c^2 x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )}\right ) \, dx}{a}-\frac {\left (c \left (1-\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {a x^6}{b}\right )^{3/4}} \, dx}{a \left (-b+a x^6\right )^{3/4}} \\ & = -\frac {c x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+(3 b) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx+\frac {(b c) \int \frac {1}{\left (b-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}+\frac {c^2 \int \frac {x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx}{a}+\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a x^3} \\ & = \frac {\sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a \sqrt [4]{b} x^3}-\frac {c x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+(3 b) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx+\frac {(b c) \int \frac {1}{\left (b-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}+\frac {c^2 \int \frac {x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx}{a} \\ \end{align*}
Time = 8.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^6}}{\sqrt {c} x^2-\sqrt {-b+a x^6}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \]
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Time = 1.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.16
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{6}-b}}{\sqrt {a \,x^{6}-b}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}-b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}-b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )\right )}{4 c^{\frac {3}{4}}}\) | \(154\) |
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Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} + c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} + c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\int \frac {x^2\,\left (a\,x^6+2\,b\right )}{{\left (a\,x^6-b\right )}^{3/4}\,\left (a\,x^6+c\,x^4-b\right )} \,d x \]
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