Integrand size = 16, antiderivative size = 233 \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}-\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}}+\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}}-\frac {a \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}}+\frac {a \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}} \]
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Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {327, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=-\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}}+\frac {a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt {2} b^{5/4}}-\frac {a \log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{16 \sqrt {2} b^{5/4}}+\frac {a \log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{16 \sqrt {2} b^{5/4}}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b} \]
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Rule 210
Rule 217
Rule 246
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (a-b x^4\right )^{3/4}}{4 b}+\frac {a \int \frac {1}{\sqrt [4]{a-b x^4}} \, dx}{4 b} \\ & = -\frac {x \left (a-b x^4\right )^{3/4}}{4 b}+\frac {a \text {Subst}\left (\int \frac {1}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{4 b} \\ & = -\frac {x \left (a-b x^4\right )^{3/4}}{4 b}+\frac {a \text {Subst}\left (\int \frac {1-\sqrt {b} x^2}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{8 b}+\frac {a \text {Subst}\left (\int \frac {1+\sqrt {b} x^2}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{8 b} \\ & = -\frac {x \left (a-b x^4\right )^{3/4}}{4 b}+\frac {a \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {b}}-\frac {\sqrt {2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{16 b^{3/2}}+\frac {a \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {b}}+\frac {\sqrt {2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{16 b^{3/2}}-\frac {a \text {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{b}}+2 x}{-\frac {1}{\sqrt {b}}-\frac {\sqrt {2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}}-\frac {a \text {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{b}}-2 x}{-\frac {1}{\sqrt {b}}+\frac {\sqrt {2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}} \\ & = -\frac {x \left (a-b x^4\right )^{3/4}}{4 b}-\frac {a \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}}+\frac {a \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}}+\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}}-\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}} \\ & = -\frac {x \left (a-b x^4\right )^{3/4}}{4 b}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}}+\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{5/4}}-\frac {a \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}}+\frac {a \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt {2} b^{5/4}} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.62 \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=-\frac {4 \sqrt [4]{b} x \left (a-b x^4\right )^{3/4}+\sqrt {2} a \arctan \left (\frac {-\sqrt {b} x^2+\sqrt {a-b x^4}}{\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}\right )-\sqrt {2} a \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}{\sqrt {b} x^2+\sqrt {a-b x^4}}\right )}{16 b^{5/4}} \]
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Time = 4.59 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {b^{\frac {1}{4}} x -\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) \sqrt {2}\, a -\arctan \left (\frac {b^{\frac {1}{4}} x +\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) \sqrt {2}\, a -\frac {\sqrt {2}\, \ln \left (\frac {-b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}{b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}\right ) a}{2}-4 x \left (-b \,x^{4}+a \right )^{\frac {3}{4}} b^{\frac {1}{4}}}{16 b^{\frac {5}{4}}}\) | \(174\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=-\frac {b \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (\frac {b^{4} x \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{x}\right ) - b \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{4} x \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{x}\right ) - i \, b \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{4} x \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{x}\right ) + i \, b \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{4} x \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{x}\right ) + 4 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} x}{16 \, b} \]
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Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.17 \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {9}{4}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {1}{4}}}\right )} a}{32 \, b} - \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}} a}{4 \, {\left (b^{2} - \frac {{\left (b x^{4} - a\right )} b}{x^{4}}\right )} x^{3}} \]
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\[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=\int { \frac {x^{4}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx=\int \frac {x^4}{{\left (a-b\,x^4\right )}^{1/4}} \,d x \]
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