Optimal. Leaf size=289 \[ -\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {38 a^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {50 a^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {32 a^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {8 a^4 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.29, antiderivative size = 289, 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 = {6182, 6180, 88} \[ -\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {38 a^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {50 a^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {32 a^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {8 a^4 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6180
Rule 6182
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^5} \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {x^3 \left (1-\frac {x}{a}\right )^2}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \left (-\frac {4 a^3}{\left (1+\frac {x}{a}\right )^{3/2}}+\frac {16 a^3}{\sqrt {1+\frac {x}{a}}}-25 a^3 \sqrt {1+\frac {x}{a}}+19 a^3 \left (1+\frac {x}{a}\right )^{3/2}-7 a^3 \left (1+\frac {x}{a}\right )^{5/2}+a^3 \left (1+\frac {x}{a}\right )^{7/2}\right ) \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 a^4 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {32 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {50 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {38 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 86, normalized size = 0.30 \[ -\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \left (656 a^5 x^5+328 a^4 x^4-82 a^3 x^3+41 a^2 x^2-20 a x+5\right ) \sqrt {c-\frac {c}{a x}}}{45 x^3 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 85, normalized size = 0.29 \[ -\frac {2 \, {\left (656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{45 \, {\left (a x^{5} - x^{4}\right )}} \]
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 = 86, normalized size = 0.30 \[ -\frac {2 \left (a x +1\right ) \left (656 x^{5} a^{5}+328 x^{4} a^{4}-82 x^{3} a^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 x^{4} \left (a x -1\right )^{2}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 108, normalized size = 0.37 \[ -\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}\,\left (656\,a^4\,x^4+984\,a^3\,x^3+902\,a^2\,x^2+943\,a\,x+923\right )}{45\,x^4}-\frac {1856\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{45\,x^4\,\left (a\,x-1\right )} \]
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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