Optimal. Leaf size=255 \[ -\frac{\sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a^2}-\frac{3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{4 a^2}+\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a^2} \]
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Rubi [A] time = 0.168539, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6126, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a^2}-\frac{3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{4 a^2}+\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a^2} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 80
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{3}{2} \tanh ^{-1}(a x)} x \, dx &=\int \frac{x (1+a x)^{3/4}}{(1-a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}+\frac{3 \int \frac{(1+a x)^{3/4}}{(1-a x)^{3/4}} \, dx}{4 a}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}+\frac{9 \int \frac{1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{8 a}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{2 a^2}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a^2}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^2}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}+\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}\\ &=-\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac{\sqrt [4]{1-a x} (1+a x)^{7/4}}{2 a^2}+\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}+\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}\\ \end{align*}
Mathematica [C] time = 0.0173623, size = 54, normalized size = 0.21 \[ -\frac{\sqrt [4]{1-a x} \left (6\ 2^{3/4} \text{Hypergeometric2F1}\left (-\frac{3}{4},\frac{1}{4},\frac{5}{4},\frac{1}{2} (1-a x)\right )+(a x+1)^{7/4}\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{3}{2}}}x\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8909, size = 1323, normalized size = 5.19 \begin{align*} -\frac{36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{2} \sqrt{\frac{\sqrt{2}{\left (a^{7} x - a^{6}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} +{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - 1\right ) + 36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{2} \sqrt{-\frac{\sqrt{2}{\left (a^{7} x - a^{6}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} -{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + 1\right ) + 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{7} x - a^{6}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} +{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{7} x - a^{6}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} -{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) + 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x + 5\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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