Optimal. Leaf size=95 \[ \frac {10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac {(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}-5 a \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-5 a \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6261, 96, 95,
218, 212, 209} \begin {gather*} -5 a \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(a x+1)^{5/4}}{x \sqrt [4]{1-a x}}+\frac {10 a \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-5 a \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 209
Rule 212
Rule 218
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{\frac {5}{2} \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {(1+a x)^{5/4}}{x^2 (1-a x)^{5/4}} \, dx\\ &=-\frac {(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}+\frac {1}{2} (5 a) \int \frac {\sqrt [4]{1+a x}}{x (1-a x)^{5/4}} \, dx\\ &=\frac {10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac {(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}+\frac {1}{2} (5 a) \int \frac {1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac {10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac {(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}+(10 a) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac {(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}-(5 a) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-(5 a) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac {(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}-5 a \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-5 a \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 74, normalized size = 0.78 \begin {gather*} \frac {3 \left (-1+8 a x+9 a^2 x^2\right )+10 a x (-1+a x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-a x}{1+a x}\right )}{3 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 125, normalized size = 1.32 \begin {gather*} -\frac {10 \, a x \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 5 \, a x \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 5 \, a x \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (9 \, a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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