Optimal. Leaf size=317 \[ \frac{17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{96 a^4}+\frac{475 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac{475 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt{2} a^4}+\frac{4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}} \]
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Rubi [A] time = 0.24137, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {6126, 97, 153, 147, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{96 a^4}+\frac{475 (1-a x)^{3/4} \sqrt [4]{a x+1}}{64 a^4}+\frac{475 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt{2} a^4}+\frac{4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 97
Rule 153
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{5}{2} \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1+a x)^{5/4}}{(1-a x)^{5/4}} \, dx\\ &=\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}-\frac{4 \int \frac{x^2 \sqrt [4]{1+a x} \left (3+\frac{17 a x}{4}\right )}{\sqrt [4]{1-a x}} \, dx}{a}\\ &=\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{\int \frac{x \sqrt [4]{1+a x} \left (-\frac{17 a}{2}-\frac{113 a^2 x}{8}\right )}{\sqrt [4]{1-a x}} \, dx}{a^3}\\ &=\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac{475 \int \frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx}{64 a^3}\\ &=\frac{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac{475 \int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{128 a^3}\\ &=\frac{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac{475 \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{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac{475 \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{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac{475 \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{475 \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{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac{475 \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{475 \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{475 \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{475 \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{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}+\frac{475 \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{475 \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{475 \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{475 \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{475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac{4 x^3 (1+a x)^{5/4}}{a \sqrt [4]{1-a x}}+\frac{17 x^2 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^2}+\frac{(1-a x)^{3/4} (1+a x)^{5/4} (521+452 a x)}{96 a^4}-\frac{475 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt{2} a^4}+\frac{475 \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{475 \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 = 0.0455598, size = 74, normalized size = 0.23 \[ \frac{(a x+1)^{9/4} \left (-6 a^2 x^2-5 a x+59\right )-380 \sqrt [4]{2} (a x-1) \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{3}{4},\frac{7}{4},\frac{1}{2} (1-a x)\right )}{24 a^4 \sqrt [4]{1-a x}} \]
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}^{3}\, 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^{3} \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 = 1.83958, size = 1400, normalized size = 4.42 \begin{align*} \frac{5700 \, \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 ) + 5700 \, \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 ) - 1425 \, \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 ) + 1425 \, \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 (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} + 521 \, a x - 2467\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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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