Optimal. Leaf size=250 \[ \frac{113 x^2 \sqrt [4]{1-\frac{1}{a x}}}{96 a^2 \sqrt [4]{\frac{1}{a x}+1}}-\frac{521 x \sqrt [4]{1-\frac{1}{a x}}}{192 a^3 \sqrt [4]{\frac{1}{a x}+1}}-\frac{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{\frac{1}{a x}+1}}-\frac{475 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{475 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{x^4 \sqrt [4]{1-\frac{1}{a x}}}{4 \sqrt [4]{\frac{1}{a x}+1}}-\frac{17 x^3 \sqrt [4]{1-\frac{1}{a x}}}{24 a \sqrt [4]{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.139989, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6171, 98, 151, 155, 12, 93, 298, 203, 206} \[ \frac{113 x^2 \sqrt [4]{1-\frac{1}{a x}}}{96 a^2 \sqrt [4]{\frac{1}{a x}+1}}-\frac{521 x \sqrt [4]{1-\frac{1}{a x}}}{192 a^3 \sqrt [4]{\frac{1}{a x}+1}}-\frac{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{\frac{1}{a x}+1}}-\frac{475 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{475 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{x^4 \sqrt [4]{1-\frac{1}{a x}}}{4 \sqrt [4]{\frac{1}{a x}+1}}-\frac{17 x^3 \sqrt [4]{1-\frac{1}{a x}}}{24 a \sqrt [4]{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/4}}{x^5 \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{\frac{17}{2 a}-\frac{8 x}{a^2}}{x^4 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1}{12} \operatorname{Subst}\left (\int \frac{\frac{113}{4 a^2}-\frac{51 x}{2 a^3}}{x^3 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{24} \operatorname{Subst}\left (\int \frac{\frac{521}{8 a^3}-\frac{113 x}{2 a^4}}{x^2 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{521 \sqrt [4]{1-\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{\frac{1425}{16 a^4}-\frac{521 x}{8 a^5}}{x \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{521 \sqrt [4]{1-\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1}{12} a \operatorname{Subst}\left (\int \frac{1425}{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{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{521 \sqrt [4]{1-\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{475 \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 )}{128 a^4}\\ &=-\frac{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{521 \sqrt [4]{1-\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{475 \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 )}{32 a^4}\\ &=-\frac{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{521 \sqrt [4]{1-\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}+\frac{475 \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 )}{64 a^4}-\frac{475 \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 )}{64 a^4}\\ &=-\frac{2467 \sqrt [4]{1-\frac{1}{a x}}}{192 a^4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{521 \sqrt [4]{1-\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{113 \sqrt [4]{1-\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{17 \sqrt [4]{1-\frac{1}{a x}} x^3}{24 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^4}{4 \sqrt [4]{1+\frac{1}{a x}}}-\frac{475 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{475 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}\\ \end{align*}
Mathematica [A] time = 5.30586, size = 161, normalized size = 0.64 \[ \frac{-3072 e^{-\frac{1}{2} \coth ^{-1}(a x)}-\frac{6292 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{7376 e^{\frac{7}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}-\frac{5248 e^{\frac{11}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+\frac{1536 e^{\frac{15}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}-1425 \log \left (1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )+1425 \log \left (e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right )+2850 \tan ^{-1}\left (e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )}{384 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.329, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50864, size = 329, normalized size = 1.32 \begin{align*} -\frac{1}{384} \, a{\left (\frac{4 \,{\left (1573 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} - 2875 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 2343 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} - 657 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{4 \,{\left (a x - 1\right )} a^{5}}{a x + 1} - \frac{6 \,{\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} - \frac{2850 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{5}} - \frac{1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{5}} + \frac{1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{5}} + \frac{3072 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7037, size = 313, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} - 521 \, a x - 2467\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 2850 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - 1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{384 \, a^{4}} \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 = 1.21674, size = 301, normalized size = 1.2 \begin{align*} \frac{1}{384} \, a{\left (\frac{2850 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{5}} + \frac{1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{5}} - \frac{1425 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{5}} - \frac{3072 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{5}} + \frac{4 \,{\left (\frac{2343 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \frac{2875 \,{\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{1573 \,{\left (a x - 1\right )}^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{3}} - 657 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{a^{5}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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