Integrand size = 37, antiderivative size = 114 \[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \left (3 b-17 a x^3\right ) \left (-b x+a x^4\right )^{3/4}}{63 x^6}+\frac {4}{3} a^{7/4} \arctan \left (\frac {\sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )+\frac {4}{3} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {2077, 2036, 335, 281, 246, 218, 212, 209, 2041, 2039} \[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {4 b \left (a x^4-b x\right )^{3/4}}{21 x^6}-\frac {68 a \left (a x^4-b x\right )^{3/4}}{63 x^3} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 281
Rule 335
Rule 2036
Rule 2039
Rule 2041
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 a^2}{\sqrt [4]{-b x+a x^4}}+\frac {b^2}{x^6 \sqrt [4]{-b x+a x^4}}-\frac {3 a b}{x^3 \sqrt [4]{-b x+a x^4}}\right ) \, dx \\ & = \left (2 a^2\right ) \int \frac {1}{\sqrt [4]{-b x+a x^4}} \, dx-(3 a b) \int \frac {1}{x^3 \sqrt [4]{-b x+a x^4}} \, dx+b^2 \int \frac {1}{x^6 \sqrt [4]{-b x+a x^4}} \, dx \\ & = \frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {4 a \left (-b x+a x^4\right )^{3/4}}{3 x^3}+\frac {1}{7} (4 a b) \int \frac {1}{x^3 \sqrt [4]{-b x+a x^4}} \, dx+\frac {\left (2 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-b+a x^3}} \, dx}{\sqrt [4]{-b x+a x^4}} \\ & = \frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^4}} \\ & = \frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (4 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}+\frac {\left (4 a^2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {4 b \left (-b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (-b x+a x^4\right )^{3/4}}{63 x^3}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 5.00 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.62 \[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \left (-b x+a x^4\right )^{3/4} \left (3 \left (b-a x^3\right )-\frac {14 a x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {3}{4},\frac {1}{4},\frac {a x^3}{b}\right )}{\left (1-\frac {a x^3}{b}\right )^{3/4}}\right )}{63 x^6} \]
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Time = 0.41 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11
method | result | size |
pseudoelliptic | \(-\frac {2 \left (42 \arctan \left (\frac {{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{x \,a^{\frac {1}{4}}}\right ) a^{\frac {7}{4}} x^{6}-21 \ln \left (\frac {-a^{\frac {1}{4}} x -{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x -{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}\right ) a^{\frac {7}{4}} x^{6}+34 a {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {3}{4}} x^{3}-6 b {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {3}{4}}\right )}{63 x^{6}}\) | \(126\) |
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Timed out. \[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {\left (a x^{3} - b\right ) \left (2 a x^{3} - b\right )}{x^{6} \sqrt [4]{x \left (a x^{3} - b\right )}}\, dx \]
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\[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {{\left (2 \, a x^{3} - b\right )} {\left (a x^{3} - b\right )}}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (94) = 188\).
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.83 \[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=\frac {2}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {2}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) - \frac {4}{21} \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}} - \frac {8}{9} \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a \]
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Time = 6.94 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-b+a x^3\right ) \left (-b+2 a x^3\right )}{x^6 \sqrt [4]{-b x+a x^4}} \, dx=-\frac {4\,\left (3\,b^2+17\,a^2\,x^6-20\,a\,b\,x^3-42\,a^2\,x^6\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {a\,x^3}{b}\right )\right )}{63\,x^5\,{\left (a\,x^4-b\,x\right )}^{1/4}} \]
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