Optimal. Leaf size=290 \[ -\frac{x^2 (1-a x)^{7/4} \sqrt [4]{a x+1}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{a x+1}}{32 a^4}-\frac{41 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac{123 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt{2} a^4} \]
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Rubi [A] time = 0.20771, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {6126, 100, 147, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{x^2 (1-a x)^{7/4} \sqrt [4]{a x+1}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{a x+1}}{32 a^4}-\frac{41 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac{123 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt{2} a^4} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 100
Rule 147
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{3}{2} \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1-a x)^{3/4}}{(1+a x)^{3/4}} \, dx\\ &=-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{\int \frac{x (1-a x)^{3/4} \left (-2+\frac{3 a x}{2}\right )}{(1+a x)^{3/4}} \, dx}{4 a^2}\\ &=-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac{41 \int \frac{(1-a x)^{3/4}}{(1+a x)^{3/4}} \, dx}{64 a^3}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac{123 \int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{128 a^3}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac{123 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{32 a^4}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac{123 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{32 a^4}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac{123 \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 )}{64 a^4}+\frac{123 \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 )}{64 a^4}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac{123 \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 )}{128 a^4}+\frac{123 \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 )}{128 a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}-\frac{123 \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 )}{128 \sqrt{2} a^4}+\frac{123 \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 )}{64 \sqrt{2} a^4}-\frac{123 \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 )}{64 \sqrt{2} a^4}\\ &=-\frac{41 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}-\frac{x^2 (1-a x)^{7/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(11-4 a x) (1-a x)^{7/4} \sqrt [4]{1+a x}}{32 a^4}-\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt{2} a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}-\frac{123 \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 )}{128 \sqrt{2} a^4}\\ \end{align*}
Mathematica [C] time = 0.103745, size = 116, normalized size = 0.4 \[ \frac{(1-a x)^{7/4} \left (12 \sqrt [4]{2} \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-a x)\right )-20 \sqrt [4]{2} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-a x)\right )+7 \sqrt [4]{2} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-a x)\right )-7 a^2 x^2 \sqrt [4]{a x+1}\right )}{28 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.107, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{3}{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 \frac{x^{3}}{\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 = 1.95388, size = 1388, normalized size = 4.79 \begin{align*} \frac{492 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{12} \sqrt{\frac{\sqrt{2}{\left (a^{5} x - a^{4}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} +{\left (a^{9} x - a^{8}\right )} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - \sqrt{2} a^{12} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - 1\right ) + 492 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{12} \sqrt{-\frac{\sqrt{2}{\left (a^{5} x - a^{4}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} -{\left (a^{9} x - a^{8}\right )} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - \sqrt{2} a^{12} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + 1\right ) - 123 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{5} x - a^{4}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} +{\left (a^{9} x - a^{8}\right )} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) + 123 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{5} x - a^{4}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} -{\left (a^{9} x - a^{8}\right )} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \,{\left (16 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 54 \, a^{2} x^{2} - 93 \, a x + 63\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{256 \, a^{4}} \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 \frac{x^{3}}{\left (\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\, 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 \frac{x^{3}}{\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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