Integrand size = 14, antiderivative size = 253 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {237 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {237 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \]
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Time = 0.10 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6306, 101, 156, 12, 95, 304, 209, 212} \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {237 \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {237 \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {557 x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{640 a^4}+\frac {157 x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{320 a^3}+\frac {5 x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{16 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}+\frac {11 x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{40 a} \]
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 304
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/4}}{x^6 \left (1-\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {1}{5} \text {Subst}\left (\int \frac {\frac {11}{2 a}+\frac {4 x}{a^2}}{x^5 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5+\frac {1}{20} \text {Subst}\left (\int \frac {-\frac {75}{4 a^2}-\frac {33 x}{2 a^3}}{x^4 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {1}{60} \text {Subst}\left (\int \frac {\frac {471}{8 a^3}+\frac {75 x}{2 a^4}}{x^3 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5+\frac {1}{120} \text {Subst}\left (\int \frac {-\frac {1671}{16 a^4}-\frac {471 x}{8 a^5}}{x^2 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {1}{120} \text {Subst}\left (\int \frac {3555}{32 a^5 x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {237 \text {Subst}\left (\int \frac {1}{x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{256 a^5} \\ & = \frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {237 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^5} \\ & = \frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5+\frac {237 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {237 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \\ & = \frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {237 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {237 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \\ \end{align*}
Time = 5.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.68 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {8192 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}+\frac {22016 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {23936 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {14032 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {5500 e^{\frac {3}{2} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}-2370 \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-1185 \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+1185 \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{1280 a^5} \]
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\[\int \frac {x^{4}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}}d x\]
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Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.47 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {2 \, {\left (128 \, a^{5} x^{5} + 304 \, a^{4} x^{4} + 376 \, a^{3} x^{3} + 514 \, a^{2} x^{2} + 871 \, a x + 557\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 2370 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{1280 \, a^{5}} \]
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\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\int \frac {x^{4}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{1280} \, a {\left (\frac {4 \, {\left (395 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 1440 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 3710 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 1992 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 1375 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{6}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} - \frac {2370 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \]
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Time = 0.41 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1}{1280} \, a {\left (\frac {2370 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {1185 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} + \frac {4 \, {\left (\frac {1992 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {3710 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {1440 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {395 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{4}} - 1375 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \]
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Time = 0.18 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.91 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {275\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{64}-\frac {249\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{40}+\frac {371\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{32}-\frac {9\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}}{2}+\frac {79\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}}{64}}{a^5+\frac {10\,a^5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a^5\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a^5\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a^5\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {5\,a^5\,\left (a\,x-1\right )}{a\,x+1}}+\frac {237\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}+\frac {237\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5} \]
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