Optimal. Leaf size=253 \[ \frac{5 x^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{16 a^2}-\frac{157 x^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{320 a^3}+\frac{557 x \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{640 a^4}-\frac{237 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{237 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1}{5} x^5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{11 x^4 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{40 a} \]
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Rubi [A] time = 0.14026, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6171, 99, 151, 12, 93, 212, 206, 203} \[ \frac{5 x^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{16 a^2}-\frac{157 x^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{320 a^3}+\frac{557 x \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{640 a^4}-\frac{237 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{237 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1}{5} x^5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{11 x^4 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{40 a} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 99
Rule 151
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int e^{-\frac{3}{2} \coth ^{-1}(a x)} x^4 \, dx &=-\operatorname{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} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5-\frac{1}{5} \operatorname{Subst}\left (\int \frac{-\frac{11}{2 a}+\frac{4 x}{a^2}}{x^5 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5+\frac{1}{20} \operatorname{Subst}\left (\int \frac{-\frac{75}{4 a^2}+\frac{33 x}{2 a^3}}{x^4 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5-\frac{1}{60} \operatorname{Subst}\left (\int \frac{-\frac{471}{8 a^3}+\frac{75 x}{2 a^4}}{x^3 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{157 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{320 a^3}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5+\frac{1}{120} \operatorname{Subst}\left (\int \frac{-\frac{1671}{16 a^4}+\frac{471 x}{8 a^5}}{x^2 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{557 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{640 a^4}-\frac{157 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{320 a^3}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5-\frac{1}{120} \operatorname{Subst}\left (\int -\frac{3555}{32 a^5 x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{557 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{640 a^4}-\frac{157 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{320 a^3}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5+\frac{237 \operatorname{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{557 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{640 a^4}-\frac{157 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{320 a^3}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5+\frac{237 \operatorname{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{557 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{640 a^4}-\frac{157 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{320 a^3}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5-\frac{237 \operatorname{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 \operatorname{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 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{640 a^4}-\frac{157 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{320 a^3}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{16 a^2}-\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a}+\frac{1}{5} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^5-\frac{237 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{237 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}\\ \end{align*}
Mathematica [A] time = 5.31345, size = 173, normalized size = 0.68 \[ \frac{\frac{5500 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{14032 e^{\frac{5}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac{23936 e^{\frac{9}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}-\frac{22016 e^{\frac{13}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}+\frac{8192 e^{\frac{17}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^5}+1185 \log \left (1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )-1185 \log \left (e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right )+2370 \tan ^{-1}\left (e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )}{1280 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50333, size = 350, normalized size = 1.38 \begin{align*} -\frac{1}{1280} \, a{\left (\frac{4 \,{\left (1375 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{19}{4}} - 1992 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{4}} + 3710 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{4}} - 1440 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{4}} + 395 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64321, size = 331, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (128 \, a^{5} x^{5} - 48 \, a^{4} x^{4} + 24 \, a^{3} x^{3} - 114 \, a^{2} x^{2} + 243 \, a x + 557\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2517, size = 316, normalized size = 1.25 \begin{align*} \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{1440 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} - \frac{3710 \,{\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{1992 \,{\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{1375 \,{\left (a x - 1\right )}^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{4}} - 395 \, \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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