Optimal. Leaf size=253 \[ \frac{5 x^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{16 a^2}+\frac{157 x^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{320 a^3}+\frac{557 x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{640 a^4}-\frac{237 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{237 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1}{5} x^5 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{11 x^4 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{40 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.139706, 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, 298, 203, 206} \[ \frac{5 x^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{16 a^2}+\frac{157 x^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{320 a^3}+\frac{557 x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{640 a^4}-\frac{237 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{237 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1}{5} x^5 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{11 x^4 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{40 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6171
Rule 99
Rule 151
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
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} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5-\frac{1}{5} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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 \operatorname{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 \operatorname{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 \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 \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 \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 = 5.23589, size = 173, normalized size = 0.68 \[ \frac{\frac{5500 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{14032 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac{23936 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+\frac{22016 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}+\frac{8192 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^5}-1185 \log \left (1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+1185 \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )-2370 \tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{1280 a^5} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.158, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47139, size = 350, normalized size = 1.38 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68208, size = 333, normalized size = 1.32 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21739, 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{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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]