Optimal. Leaf size=165 \[ \frac{287 a^3 \sqrt [4]{a x+1}}{24 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{a x+1}}{24 x \sqrt [4]{1-a x}}-\frac{55}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{55}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{13 a \sqrt [4]{a x+1}}{12 x^2 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}} \]
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Rubi [A] time = 0.0787501, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6126, 98, 151, 155, 12, 93, 212, 206, 203} \[ \frac{287 a^3 \sqrt [4]{a x+1}}{24 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{a x+1}}{24 x \sqrt [4]{1-a x}}-\frac{55}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{55}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{13 a \sqrt [4]{a x+1}}{12 x^2 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{5}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1+a x)^{5/4}}{x^4 (1-a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{1}{3} \int \frac{-\frac{13 a}{2}-6 a^2 x}{x^3 (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}+\frac{1}{6} \int \frac{\frac{61 a^2}{4}+13 a^3 x}{x^2 (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{1+a x}}{24 x \sqrt [4]{1-a x}}-\frac{1}{6} \int \frac{-\frac{165 a^3}{8}-\frac{61 a^4 x}{4}}{x (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=\frac{287 a^3 \sqrt [4]{1+a x}}{24 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{1+a x}}{24 x \sqrt [4]{1-a x}}+\frac{\int \frac{165 a^4}{16 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{3 a}\\ &=\frac{287 a^3 \sqrt [4]{1+a x}}{24 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{1+a x}}{24 x \sqrt [4]{1-a x}}+\frac{1}{16} \left (55 a^3\right ) \int \frac{1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac{287 a^3 \sqrt [4]{1+a x}}{24 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{1+a x}}{24 x \sqrt [4]{1-a x}}+\frac{1}{4} \left (55 a^3\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{287 a^3 \sqrt [4]{1+a x}}{24 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{1+a x}}{24 x \sqrt [4]{1-a x}}-\frac{1}{8} \left (55 a^3\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}{8} \left (55 a^3\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{287 a^3 \sqrt [4]{1+a x}}{24 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{3 x^3 \sqrt [4]{1-a x}}-\frac{13 a \sqrt [4]{1+a x}}{12 x^2 \sqrt [4]{1-a x}}-\frac{61 a^2 \sqrt [4]{1+a x}}{24 x \sqrt [4]{1-a x}}-\frac{55}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{55}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0282137, size = 91, normalized size = 0.55 \[ \frac{110 a^3 x^3 (a x-1) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )+287 a^4 x^4+226 a^3 x^3-87 a^2 x^2-34 a x-8}{24 x^3 \sqrt [4]{1-a x} (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0049, size = 360, normalized size = 2.18 \begin{align*} -\frac{330 \, a^{3} x^{3} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 165 \, a^{3} x^{3} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 165 \, a^{3} x^{3} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \,{\left (287 \, a^{3} x^{3} - 61 \, a^{2} x^{2} - 26 \, a x - 8\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{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 \frac{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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