Integrand size = 14, antiderivative size = 287 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1003 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \]
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Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6306, 100, 156, 160, 12, 95, 218, 212, 209} \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1003 \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {26111 \sqrt [4]{\frac {1}{a x}+1}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 x \sqrt [4]{\frac {1}{a x}+1}}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 x^2 \sqrt [4]{\frac {1}{a x}+1}}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 x^3 \sqrt [4]{\frac {1}{a x}+1}}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 x^4 \sqrt [4]{\frac {1}{a x}+1}}{40 a \sqrt [4]{1-\frac {1}{a x}}} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 160
Rule 209
Rule 212
Rule 218
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/4}}{x^6 \left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{5} \text {Subst}\left (\int \frac {-\frac {21}{2 a}-\frac {10 x}{a^2}}{x^5 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{20} \text {Subst}\left (\int \frac {\frac {181}{4 a^2}+\frac {42 x}{a^3}}{x^4 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{60} \text {Subst}\left (\int \frac {-\frac {1189}{8 a^3}-\frac {543 x}{4 a^4}}{x^3 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{120} \text {Subst}\left (\int \frac {\frac {5533}{16 a^4}+\frac {1189 x}{4 a^5}}{x^2 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{120} \text {Subst}\left (\int \frac {-\frac {15045}{32 a^5}-\frac {5533 x}{16 a^6}}{x \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{60} a \text {Subst}\left (\int \frac {15045}{64 a^6 x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1003 \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{256 a^5} \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1003 \text {Subst}\left (\int \frac {1}{-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 {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1003 \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 {1003 \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 {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1003 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \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 = 5.18 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.69 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {-8 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {32 e^{\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}+\frac {122 e^{\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {233 e^{\frac {1}{2} \coth ^{-1}(a x)}}{6 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {1661 e^{\frac {1}{2} \coth ^{-1}(a x)}}{48 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {4117 e^{\frac {1}{2} \coth ^{-1}(a x)}}{192 \left (-1+e^{2 \coth ^{-1}(a x)}\right )}+\frac {1003}{128} \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-\frac {1003}{256} \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+\frac {1003}{256} \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{a^5} \]
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\[\int \frac {x^{4}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}d x\]
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Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.53 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {30090 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 15045 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 15045 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (384 \, a^{6} x^{6} + 1392 \, a^{5} x^{5} + 2456 \, a^{4} x^{4} + 3826 \, a^{3} x^{3} + 7911 \, a^{2} x^{2} - 20578 \, a x - 26111\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{3840 \, {\left (a^{6} x - a^{5}\right )}} \]
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\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\int \frac {x^{4}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.96 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {4 \, {\left (\frac {58985 \, {\left (a x - 1\right )}}{a x + 1} - \frac {125920 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {137930 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {72216 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + \frac {15045 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} - 7680\right )}}{a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {21}{4}} - 5 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} + 10 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} - 10 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 5 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \]
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Time = 0.35 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.89 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {15045 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} + \frac {30720}{a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {4 \, {\left (\frac {49120 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {61130 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {33816 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {7365 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{4}} - 20585 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \]
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Time = 4.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.86 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1003\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {1003\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {\frac {787\,{\left (a\,x-1\right )}^2}{6\,{\left (a\,x+1\right )}^2}-\frac {13793\,{\left (a\,x-1\right )}^3}{96\,{\left (a\,x+1\right )}^3}+\frac {3009\,{\left (a\,x-1\right )}^4}{40\,{\left (a\,x+1\right )}^4}-\frac {1003\,{\left (a\,x-1\right )}^5}{64\,{\left (a\,x+1\right )}^5}-\frac {11797\,\left (a\,x-1\right )}{192\,\left (a\,x+1\right )}+8}{a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-5\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+10\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}-10\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}+5\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}-a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{21/4}} \]
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