Optimal. Leaf size=106 \[ \frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x} \]
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Rubi [A] time = 0.26, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6167, 6130, 21, 98, 151, 156, 63, 208, 206} \[ \frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 21
Rule 63
Rule 98
Rule 151
Rule 156
Rule 206
Rule 208
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx\\ &=-\int \frac {(1-a x) \sqrt {c-a c x}}{x^3 (1+a x)} \, dx\\ &=-\frac {\int \frac {(c-a c x)^{3/2}}{x^3 (1+a x)} \, dx}{c}\\ &=\frac {\sqrt {c-a c x}}{2 x^2}+\frac {\int \frac {\frac {9 a c^2}{2}-\frac {7}{2} a^2 c^2 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{2 c}\\ &=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {\int \frac {\frac {23 a^2 c^3}{4}-\frac {9}{4} a^3 c^3 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{2 c^2}\\ &=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {1}{8} \left (23 a^2 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx+\left (4 a^3 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {1}{4} (23 a) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )-\left (8 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 93, normalized size = 0.88 \[ \frac {23}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {(2-9 a x) \sqrt {c-a c x}}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 204, normalized size = 1.92 \[ \left [\frac {16 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 23 \, a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{8 \, x^{2}}, \frac {16 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 23 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{4 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 106, normalized size = 1.00 \[ \frac {4 \, \sqrt {2} a^{2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {23 \, a^{2} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} + \frac {9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c - 7 \, \sqrt {-a c x + c} a^{2} c^{2}}{4 \, a^{2} c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 95, normalized size = 0.90 \[ 2 a^{2} c^{2} \left (-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}-\frac {\frac {-\frac {9 \left (-a c x +c \right )^{\frac {3}{2}}}{8}+\frac {7 c \sqrt {-a c x +c}}{8}}{x^{2} a^{2} c^{2}}-\frac {23 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 152, normalized size = 1.43 \[ \frac {1}{8} \, a^{2} c^{2} {\left (\frac {2 \, {\left (9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} - 7 \, \sqrt {-a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} c + 2 \, {\left (a c x - c\right )} c^{2} + c^{3}} + \frac {16 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {3}{2}}} - \frac {23 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 88, normalized size = 0.83 \[ \frac {9\,{\left (c-a\,c\,x\right )}^{3/2}}{4\,c\,x^2}-\frac {a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,23{}\mathrm {i}}{4}-\frac {7\,\sqrt {c-a\,c\,x}}{4\,x^2}+\sqrt {2}\,a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.86, size = 352, normalized size = 3.32 \[ \frac {10 a^{2} c^{4} \sqrt {- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {6 a^{2} c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {3 a^{2} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} + \frac {3 a^{2} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} + \frac {3 a^{2} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} - \frac {3 a^{2} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} - \frac {8 a^{2} c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {4 \sqrt {2} a^{2} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {3 a \sqrt {- a c x + c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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