Integrand size = 15, antiderivative size = 92 \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=\frac {\left (b+a x^4\right )^{3/4} \left (3 b x+4 a x^5\right )}{32 a}-\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}} \]
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Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 327, 246, 218, 212, 209} \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=-\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{64 a^{5/4}}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{64 a^{5/4}}+\frac {3 b x \left (a x^4+b\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (a x^4+b\right )^{3/4} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 285
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}+\frac {1}{8} (3 b) \int \frac {x^4}{\sqrt [4]{b+a x^4}} \, dx \\ & = \frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx}{32 a} \\ & = \frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{32 a} \\ & = \frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {\left (3 b^2\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}-\frac {\left (3 b^2\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} \\ & = \frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.95 \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=\frac {2 \sqrt [4]{a} x \left (b+a x^4\right )^{3/4} \left (3 b+4 a x^4\right )-3 b^2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}} \]
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Time = 1.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(\frac {16 \left (a \,x^{4}+b \right )^{\frac {3}{4}} a^{\frac {5}{4}} x^{5}+12 b x \left (a \,x^{4}+b \right )^{\frac {3}{4}} a^{\frac {1}{4}}+6 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{2}-3 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) b^{2}}{128 a^{\frac {5}{4}}}\) | \(106\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.34 \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=-\frac {3 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{6} + \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4} x\right )}}{x}\right ) - 3 i \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{6} + i \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4} x\right )}}{x}\right ) + 3 i \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{6} - i \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4} x\right )}}{x}\right ) - 3 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{6} - \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4} x\right )}}{x}\right ) - 4 \, {\left (4 \, a x^{5} + 3 \, b x\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{128 \, a} \]
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Result contains complex when optimal does not.
Time = 1.51 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.42 \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=\frac {b^{\frac {3}{4}} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (72) = 144\).
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.61 \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=\frac {3 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{128 \, a} + \frac {\frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} a b^{2}}{x^{3}} + \frac {3 \, {\left (a x^{4} + b\right )}^{\frac {7}{4}} b^{2}}{x^{7}}}{32 \, {\left (a^{3} - \frac {2 \, {\left (a x^{4} + b\right )} a^{2}}{x^{4}} + \frac {{\left (a x^{4} + b\right )}^{2} a}{x^{8}}\right )}} \]
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\[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=\int { {\left (a x^{4} + b\right )}^{\frac {3}{4}} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (b+a x^4\right )^{3/4} \, dx=\int x^4\,{\left (a\,x^4+b\right )}^{3/4} \,d x \]
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