Integrand size = 29, antiderivative size = 122 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {\left (-b+a x^4\right )^{3/4} \left (-8 a^2 x+5 b x+4 a x^5\right )}{16 a^2}+\frac {\left (-24 a^2 b+5 b^2\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}+\frac {\left (-24 a^2 b+5 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \]
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Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1425, 396, 246, 218, 212, 209} \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=-\frac {1}{16} x \left (8-\frac {5 b}{a^2}\right ) \left (a x^4-b\right )^{3/4}-\frac {b \left (24 a^2-5 b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}-\frac {b \left (24 a^2-5 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}+\frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 396
Rule 1425
Rubi steps \begin{align*} \text {integral}& = \frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}+\frac {\int \frac {-8 a b-\left (16 a^2-10 b\right ) x^4}{\sqrt [4]{-b+a x^4}} \, dx}{8 a} \\ & = -\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx}{32 a^2} \\ & = -\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^2} \\ & = -\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{64 a^2}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{64 a^2} \\ & = -\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (24 a^2-5 b\right ) b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}-\frac {\left (24 a^2-5 b\right ) b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {2 \sqrt [4]{a} x \left (-b+a x^4\right )^{3/4} \left (-8 a^2+5 b+4 a x^4\right )-\left (24 a^2-5 b\right ) b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )-\left (24 a^2-5 b\right ) b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \]
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Time = 0.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.63
method | result | size |
pseudoelliptic | \(\frac {-32 a^{\frac {9}{4}} x \left (a \,x^{4}-b \right )^{\frac {3}{4}}+16 a^{\frac {5}{4}} x^{5} \left (a \,x^{4}-b \right )^{\frac {3}{4}}+20 b x \,a^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {3}{4}}+48 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2} b -24 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) a^{2} b -10 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{2}+5 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) b^{2}}{64 a^{\frac {9}{4}}}\) | \(199\) |
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 627, normalized size of antiderivative = 5.14 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=-\frac {a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} + {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + i \, a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - i \, a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (4 \, a x^{5} - {\left (8 \, a^{2} - 5 \, b\right )} x\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}}{64 \, a^{2}} \]
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Result contains complex when optimal does not.
Time = 2.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {a x^{5} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {9}{4}\right )} - \frac {b^{\frac {3}{4}} x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {x^{9} e^{- \frac {i \pi }{4}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {13}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (102) = 204\).
Time = 0.29 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.96 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {1}{8} \, a {\left (\frac {b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{a} - \frac {4 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} - b\right )} a}{x^{4}}\right )} x^{3}}\right )} + \frac {1}{4} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} - \frac {5 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{64 \, a^{2}} + \frac {\frac {9 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} a b^{2}}{x^{3}} - \frac {5 \, {\left (a x^{4} - b\right )}^{\frac {7}{4}} b^{2}}{x^{7}}}{16 \, {\left (a^{4} - \frac {2 \, {\left (a x^{4} - b\right )} a^{3}}{x^{4}} + \frac {{\left (a x^{4} - b\right )}^{2} a^{2}}{x^{8}}\right )}} \]
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\[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\int { \frac {2 \, x^{8} - 2 \, a x^{4} - b}{{\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\int -\frac {-2\,x^8+2\,a\,x^4+b}{{\left (a\,x^4-b\right )}^{1/4}} \,d x \]
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