Optimal. Leaf size=71 \[ \frac {a \sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}+\frac {a \sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac {1}{5} a x^2 \sqrt {a x^6}-\frac {a \sqrt {a x^6}}{x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {15, 302, 212, 206, 203} \[ -\frac {1}{5} a x^2 \sqrt {a x^6}-\frac {a \sqrt {a x^6}}{x^2}+\frac {a \sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}+\frac {a \sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 203
Rule 206
Rule 212
Rule 302
Rubi steps
\begin {align*} \int \frac {\left (a x^6\right )^{3/2}}{x \left (1-x^4\right )} \, dx &=\frac {\left (a \sqrt {a x^6}\right ) \int \frac {x^8}{1-x^4} \, dx}{x^3}\\ &=\frac {\left (a \sqrt {a x^6}\right ) \int \left (-1-x^4+\frac {1}{1-x^4}\right ) \, dx}{x^3}\\ &=-\frac {a \sqrt {a x^6}}{x^2}-\frac {1}{5} a x^2 \sqrt {a x^6}+\frac {\left (a \sqrt {a x^6}\right ) \int \frac {1}{1-x^4} \, dx}{x^3}\\ &=-\frac {a \sqrt {a x^6}}{x^2}-\frac {1}{5} a x^2 \sqrt {a x^6}+\frac {\left (a \sqrt {a x^6}\right ) \int \frac {1}{1-x^2} \, dx}{2 x^3}+\frac {\left (a \sqrt {a x^6}\right ) \int \frac {1}{1+x^2} \, dx}{2 x^3}\\ &=-\frac {a \sqrt {a x^6}}{x^2}-\frac {1}{5} a x^2 \sqrt {a x^6}+\frac {a \sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}+\frac {a \sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.62 \[ -\frac {a \sqrt {a x^6} \left (4 x^5+20 x+5 \log (1-x)-5 \log (x+1)-10 \tan ^{-1}(x)\right )}{20 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 41, normalized size = 0.58 \[ -\frac {\sqrt {a x^{6}} {\left (4 \, a x^{5} + 20 \, a x - 10 \, a \arctan \relax (x) - 5 \, a \log \left (\frac {x + 1}{x - 1}\right )\right )}}{20 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 42, normalized size = 0.59 \[ -\frac {1}{20} \, {\left (4 \, x^{5} \mathrm {sgn}\relax (x) + 20 \, x \mathrm {sgn}\relax (x) - 10 \, \arctan \relax (x) \mathrm {sgn}\relax (x) - 5 \, \log \left ({\left | x + 1 \right |}\right ) \mathrm {sgn}\relax (x) + 5 \, \log \left ({\left | x - 1 \right |}\right ) \mathrm {sgn}\relax (x)\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.54 \[ -\frac {\left (a \,x^{6}\right )^{\frac {3}{2}} \left (4 x^{5}+20 x -10 \arctan \relax (x )+5 \ln \left (x -1\right )-5 \ln \left (x +1\right )\right )}{20 x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.09, size = 40, normalized size = 0.56 \[ -\frac {1}{5} \, a^{\frac {3}{2}} x^{5} - a^{\frac {3}{2}} x + \frac {1}{2} \, a^{\frac {3}{2}} \arctan \relax (x) + \frac {1}{4} \, a^{\frac {3}{2}} \log \left (x + 1\right ) - \frac {1}{4} \, a^{\frac {3}{2}} \log \left (x - 1\right ) \]
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 \[ -\int \frac {{\left (a\,x^6\right )}^{3/2}}{x\,\left (x^4-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\left (a x^{6}\right )^{\frac {3}{2}}}{x^{5} - x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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