Integrand size = 13, antiderivative size = 81 \[ \int \left (-b+a x^4\right )^{3/4} \, dx=\frac {1}{4} x \left (-b+a x^4\right )^{3/4}-\frac {3 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [4]{a}} \]
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Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {201, 246, 218, 212, 209} \[ \int \left (-b+a x^4\right )^{3/4} \, dx=-\frac {3 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{8 \sqrt [4]{a}}+\frac {1}{4} x \left (a x^4-b\right )^{3/4} \]
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Rule 201
Rule 209
Rule 212
Rule 218
Rule 246
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (-b+a x^4\right )^{3/4}-\frac {1}{4} (3 b) \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx \\ & = \frac {1}{4} x \left (-b+a x^4\right )^{3/4}-\frac {1}{4} (3 b) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = \frac {1}{4} x \left (-b+a x^4\right )^{3/4}-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = \frac {1}{4} x \left (-b+a x^4\right )^{3/4}-\frac {3 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [4]{a}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \left (-b+a x^4\right )^{3/4} \, dx=\frac {1}{4} x \left (-b+a x^4\right )^{3/4}-\frac {3 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [4]{a}} \]
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Time = 1.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14
method | result | size |
pseudoelliptic | \(\frac {4 \left (a \,x^{4}-b \right )^{\frac {3}{4}} x \,a^{\frac {1}{4}}+6 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b -3 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) b}{16 a^{\frac {1}{4}}}\) | \(92\) |
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.44 \[ \int \left (-b+a x^4\right )^{3/4} \, dx=\frac {1}{4} \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x - \frac {3}{16} \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \log \left (\frac {27 \, {\left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{3} + \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a x\right )}}{x}\right ) + \frac {3}{16} i \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \log \left (\frac {27 \, {\left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{3} + i \, \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a x\right )}}{x}\right ) - \frac {3}{16} i \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \log \left (\frac {27 \, {\left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{3} - i \, \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a x\right )}}{x}\right ) + \frac {3}{16} \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \log \left (\frac {27 \, {\left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{3} - \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a x\right )}}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.51 \[ \int \left (-b+a x^4\right )^{3/4} \, dx=- \frac {b^{\frac {3}{4}} x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int \left (-b+a x^4\right )^{3/4} \, dx=\frac {3}{16} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} + \frac {{\left (a x^{4} - b\right )}^{\frac {3}{4}} b}{4 \, {\left (a - \frac {a x^{4} - b}{x^{4}}\right )} x^{3}} \]
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\[ \int \left (-b+a x^4\right )^{3/4} \, dx=\int { {\left (a x^{4} - b\right )}^{\frac {3}{4}} \,d x } \]
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Time = 5.75 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.48 \[ \int \left (-b+a x^4\right )^{3/4} \, dx=\frac {x\,{\left (a\,x^4-b\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {a\,x^4}{b}\right )}{{\left (1-\frac {a\,x^4}{b}\right )}^{3/4}} \]
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