Optimal. Leaf size=114 \[ -\frac {3 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6160, 6140, 43} \[ -\frac {a x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6160
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{3 \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^2}{1-a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (-3-a x+\frac {4}{1-a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{\sqrt {1-a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^3}{2 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1-a x)}{a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.48 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {a x^2}{2}-\frac {4 \log (1-a x)}{a}-3 x\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 387, normalized size = 3.39 \[ \left [\frac {4 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x - {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) + {\left (a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} x^{2} - a^{2}\right )}}, \frac {8 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right ) + {\left (a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} x^{2} - a^{2}\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}} x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 69, normalized size = 0.61 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}+6 a x +8 \ln \left (a x -1\right )\right )}{2 \left (a^{2} x^{2}-1\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.47, size = 149, normalized size = 1.31 \[ -\frac {1}{2} \, a^{3} {\left (-\frac {i \, \sqrt {c} x^{2}}{a^{3}} - \frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{5}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{5}}\right )} - \frac {3}{2} \, a^{2} {\left (-\frac {2 i \, \sqrt {c} x}{a^{3}} + \frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{4}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{4}}\right )} - \frac {3}{2} \, a {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{3}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{3}}\right )} - \frac {i \, \sqrt {c} \log \left (a x + 1\right )}{2 \, a^{2}} + \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{2 \, a^{2}} \]
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 \[ \int \frac {x\,\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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