Optimal. Leaf size=168 \[ -\frac{15 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{64 x}+\frac{123}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{123}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{3 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{8 x^3}-\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4} \]
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Rubi [A] time = 0.0780924, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6126, 99, 151, 12, 93, 298, 203, 206} \[ -\frac{15 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}{64 x}+\frac{123}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{123}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{3 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{8 x^3}-\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 99
Rule 151
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\frac{3}{2} \tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac{(1+a x)^{3/4}}{x^5 (1-a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}+\frac{1}{4} \int \frac{\frac{9 a}{2}+3 a^2 x}{x^4 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{1}{12} \int \frac{-\frac{45 a^2}{4}-9 a^3 x}{x^3 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}+\frac{1}{24} \int \frac{\frac{189 a^3}{8}+\frac{45 a^4 x}{4}}{x^2 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 x}-\frac{1}{24} \int -\frac{369 a^4}{16 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 x}+\frac{1}{128} \left (123 a^4\right ) \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 x}+\frac{1}{32} \left (123 a^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 x}-\frac{1}{64} \left (123 a^4\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}{64} \left (123 a^4\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{\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac{3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac{15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}-\frac{63 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 x}+\frac{123}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{123}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0272951, size = 86, normalized size = 0.51 \[ -\frac{\sqrt [4]{1-a x} \left (246 a^4 x^4 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )+63 a^4 x^4+93 a^3 x^3+54 a^2 x^2+40 a x+16\right )}{64 x^4 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7373, size = 386, normalized size = 2.3 \begin{align*} \frac{246 \, a^{4} x^{4} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 123 \, a^{4} x^{4} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 123 \, a^{4} x^{4} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \,{\left (63 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 24 \, a x + 16\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{128 \, x^{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 \frac{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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