Optimal. Leaf size=136 \[ \frac{25 a^2 \sqrt [4]{a x+1}}{2 \sqrt [4]{1-a x}}-\frac{25}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{25}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(a x+1)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac{5 a (a x+1)^{5/4}}{4 x \sqrt [4]{1-a x}} \]
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Rubi [A] time = 0.0485649, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6126, 96, 94, 93, 212, 206, 203} \[ \frac{25 a^2 \sqrt [4]{a x+1}}{2 \sqrt [4]{1-a x}}-\frac{25}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{25}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(a x+1)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac{5 a (a x+1)^{5/4}}{4 x \sqrt [4]{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
Rule 94
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{5}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1+a x)^{5/4}}{x^3 (1-a x)^{5/4}} \, dx\\ &=-\frac{(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac{1}{4} (5 a) \int \frac{(1+a x)^{5/4}}{x^2 (1-a x)^{5/4}} \, dx\\ &=-\frac{5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac{(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac{1}{8} \left (25 a^2\right ) \int \frac{\sqrt [4]{1+a x}}{x (1-a x)^{5/4}} \, dx\\ &=\frac{25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac{5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac{(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac{1}{8} \left (25 a^2\right ) \int \frac{1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac{25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac{5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac{(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}+\frac{1}{2} \left (25 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac{5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac{(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac{1}{4} \left (25 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{1}{4} \left (25 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{25 a^2 \sqrt [4]{1+a x}}{2 \sqrt [4]{1-a x}}-\frac{5 a (1+a x)^{5/4}}{4 x \sqrt [4]{1-a x}}-\frac{(1+a x)^{9/4}}{2 x^2 \sqrt [4]{1-a x}}-\frac{25}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{25}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0235652, size = 86, normalized size = 0.63 \[ \frac{50 a^2 x^2 (a x-1) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )+3 \left (43 a^3 x^3+34 a^2 x^2-11 a x-2\right )}{12 x^2 \sqrt [4]{1-a x} (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{5}{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{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12449, size = 335, normalized size = 2.46 \begin{align*} -\frac{50 \, a^{2} x^{2} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 25 \, a^{2} x^{2} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 25 \, a^{2} x^{2} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \,{\left (43 \, a^{2} x^{2} - 9 \, a x - 2\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{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 \frac{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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