Optimal. Leaf size=264 \[ \frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.30, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6197, 6193, 88} \[ \frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a x^5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^5} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1+a x)^2}{x^6 (-1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (-\frac {1}{x^6}-\frac {3 a}{x^5}-\frac {4 a^2}{x^4}-\frac {4 a^3}{x^3}-\frac {4 a^4}{x^2}-\frac {4 a^5}{x}+\frac {4 a^6}{-1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 90, normalized size = 0.34 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-4 a^5 \log (x)+4 a^5 \log (1-a x)+\frac {240 a^4 x^4+120 a^3 x^3+80 a^2 x^2+45 a x+12}{60 x^5}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 109, normalized size = 0.41 \[ \frac {240 \, a^{6} \sqrt {c} x^{5} \log \left (\frac {2 \, a^{3} c x^{2} - 2 \, a^{2} c x - \sqrt {a^{2} c} {\left (2 \, a x - 1\right )} \sqrt {c} + a c}{a x^{2} - x}\right ) + {\left (240 \, a^{4} x^{4} + 120 \, a^{3} x^{3} + 80 \, a^{2} x^{2} + 45 \, a x + 12\right )} \sqrt {a^{2} c}}{60 \, a^{2} x^{5}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{x^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 106, normalized size = 0.40 \[ -\frac {\left (240 a^{5} \ln \relax (x ) x^{5}-240 \ln \left (a x -1\right ) x^{5} a^{5}-240 x^{4} a^{4}-120 x^{3} a^{3}-80 a^{2} x^{2}-45 a x -12\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x -1\right )}{60 \left (a x +1\right )^{2} x^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{x^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}}{x^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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