Optimal. Leaf size=109 \[ -\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {64 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.22, antiderivative size = 109, 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 = {6178, 657, 649} \[ -\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {64 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^2}\\ &=-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {32 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {64 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 58, normalized size = 0.53 \[ \frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \left (23 a^2 x^2+10 a x-1\right ) \sqrt {c-\frac {c}{a x}}}{3 a^2 x^2-3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 59, normalized size = 0.54 \[ \frac {2 \, {\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 0.57 \[ \frac {2 \left (a x +1\right ) \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 x \left (a x -1\right )^{2}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 54, normalized size = 0.50 \[ \frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (23\,a^2\,x^2+10\,a\,x-1\right )}{3\,x\,\left (a\,x-1\right )} \]
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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