Optimal. Leaf size=113 \[ -\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^2 \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6133, 25, 514, 446, 80, 50, 63, 208} \[ -\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^2 \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 25
Rule 50
Rule 63
Rule 80
Rule 208
Rule 446
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^3 (1+a x)} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x^2 (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^3} \, dx}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x \left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right )\\ &=-4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\left (4 a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\left (8 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 79, normalized size = 0.70 \[ 4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )-\frac {2 \left (38 a^2 x^2-11 a x+3\right ) \sqrt {c-\frac {c}{a x}}}{15 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 185, normalized size = 1.64 \[ \left [\frac {2 \, {\left (15 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (38 \, a^{2} x^{2} - 11 \, a x + 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, x^{2}}, -\frac {2 \, {\left (30 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (38 \, a^{2} x^{2} - 11 \, a x + 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.18, size = 278, normalized size = 2.46 \[ -\frac {4 \, \sqrt {2} a^{3} c \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\relax (x)} - \frac {2 \, {\left (60 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{5} c - 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{4} c^{\frac {3}{2}} {\left | a \right |} + 35 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{5} c^{2} - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{4} c^{\frac {5}{2}} {\left | a \right |} + 3 \, a^{5} c^{3}\right )}}{15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{2} {\left | a \right |} \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 278, normalized size = 2.46 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-90 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{4}+30 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{4}+60 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}+45 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{4} a^{3}-30 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{4}-45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{4} a^{3}-16 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+6 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{15 x^{3} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}} \]
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 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 96, normalized size = 0.85 \[ -4\,a^2\,\sqrt {c-\frac {c}{a\,x}}-\frac {2\,a^2\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-\frac {2\,a^2\,{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{5\,c^2}-\sqrt {2}\,a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]
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 \[ - \int \left (- \frac {\sqrt {c - \frac {c}{a x}}}{a x^{4} + x^{3}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{4} + x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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