Optimal. Leaf size=305 \[ \frac{(1-a x)^{3/4} (a x+1)^{9/4}}{3 a^3}+\frac{2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{11 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a^3}+\frac{55 (1-a x)^{3/4} \sqrt [4]{a x+1}}{8 a^3}+\frac{55 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}-\frac{55 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}-\frac{55 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{8 \sqrt{2} a^3}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^3} \]
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Rubi [A] time = 0.225932, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {6126, 89, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{(1-a x)^{3/4} (a x+1)^{9/4}}{3 a^3}+\frac{2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{11 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a^3}+\frac{55 (1-a x)^{3/4} \sqrt [4]{a x+1}}{8 a^3}+\frac{55 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}-\frac{55 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}-\frac{55 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{8 \sqrt{2} a^3}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^3} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 89
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{5}{2} \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 (1+a x)^{5/4}}{(1-a x)^{5/4}} \, dx\\ &=\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac{2 \int \frac{(1+a x)^{5/4} \left (\frac{5 a}{2}+\frac{a^2 x}{2}\right )}{\sqrt [4]{1-a x}} \, dx}{a^3}\\ &=\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}-\frac{11 \int \frac{(1+a x)^{5/4}}{\sqrt [4]{1-a x}} \, dx}{2 a^2}\\ &=\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}-\frac{55 \int \frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx}{8 a^2}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}-\frac{55 \int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{16 a^2}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}+\frac{55 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{4 a^3}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}+\frac{55 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^3}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}-\frac{55 \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 )}{8 a^3}+\frac{55 \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 )}{8 a^3}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}+\frac{55 \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 )}{16 a^3}+\frac{55 \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 )}{16 a^3}+\frac{55 \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 )}{16 \sqrt{2} a^3}+\frac{55 \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 )}{16 \sqrt{2} a^3}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}+\frac{55 \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 )}{16 \sqrt{2} a^3}-\frac{55 \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 )}{16 \sqrt{2} a^3}+\frac{55 \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 )}{8 \sqrt{2} a^3}-\frac{55 \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 )}{8 \sqrt{2} a^3}\\ &=\frac{55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac{11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac{2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac{(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}-\frac{55 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^3}+\frac{55 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^3}+\frac{55 \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 )}{16 \sqrt{2} a^3}-\frac{55 \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 )}{16 \sqrt{2} a^3}\\ \end{align*}
Mathematica [C] time = 0.0311002, size = 66, normalized size = 0.22 \[ \frac{(7-a x) (a x+1)^{9/4}-44 \sqrt [4]{2} (a x-1) \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{3}{4},\frac{7}{4},\frac{1}{2} (1-a x)\right )}{3 a^3 \sqrt [4]{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{5}{2}}}{x}^{2}\, 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^{2} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87446, size = 1364, normalized size = 4.47 \begin{align*} \frac{660 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{9} \sqrt{\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} +{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - 1\right ) + 660 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{9} \sqrt{-\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} -{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + 1\right ) - 165 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} +{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) + 165 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} -{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \,{\left (8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 61 \, a x - 287\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}} \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^{2} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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