Optimal. Leaf size=255 \[ -\frac{(1-a x)^{3/4} (a x+1)^{5/4}}{2 a^2}-\frac{(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 a^2}-\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}+\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a^2} \]
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Rubi [A] time = 0.173493, 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, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{(1-a x)^{3/4} (a x+1)^{5/4}}{2 a^2}-\frac{(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 a^2}-\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}+\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a^2} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 80
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int e^{\frac{1}{2} \tanh ^{-1}(a x)} x \, dx &=\int \frac{x \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx\\ &=-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac{\int \frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx}{4 a}\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac{\int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{8 a}\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{2 a^2}\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a^2}\\ &=-\frac{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac{\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{\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{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac{\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{\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{\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{\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{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}-\frac{\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{\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{\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{\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{(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{3/4} (1+a x)^{5/4}}{2 a^2}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}-\frac{\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{\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 = 0.020352, size = 56, normalized size = 0.22 \[ -\frac{(1-a x)^{3/4} \left (2 \sqrt [4]{2} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{3}{4},\frac{7}{4},\frac{1}{2} (1-a x)\right )+3 (a x+1)^{5/4}\right )}{6 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}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 \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98196, size = 1303, normalized size = 5.11 \begin{align*} -\frac{4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{6} \sqrt{\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{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{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - 1\right ) + 4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{6} \sqrt{-\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{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{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + 1\right ) - \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} +{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) + \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} -{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \,{\left (2 \, a^{2} x^{2} + a x - 3\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx \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 \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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