Optimal. Leaf size=68 \[ \frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {\sqrt {c-a c x}}{2 x^2}+\frac {7 a \sqrt {c-a c x}}{4 x} \]
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Rubi [A] time = 0.21, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6167, 6130, 21, 78, 51, 63, 208} \[ \frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {\sqrt {c-a c x}}{2 x^2}+\frac {7 a \sqrt {c-a c x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 21
Rule 51
Rule 63
Rule 78
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\\ &=-\left (c \int \frac {1+a x}{x^3 \sqrt {c-a c x}} \, dx\right )\\ &=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {1}{4} (7 a c) \int \frac {1}{x^2 \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{2 x^2}+\frac {7 a \sqrt {c-a c x}}{4 x}-\frac {1}{8} \left (7 a^2 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{2 x^2}+\frac {7 a \sqrt {c-a c x}}{4 x}+\frac {1}{4} (7 a) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=\frac {\sqrt {c-a c x}}{2 x^2}+\frac {7 a \sqrt {c-a c x}}{4 x}+\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.81 \[ \frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {(7 a x+2) \sqrt {c-a c x}}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 117, normalized size = 1.72 \[ \left [\frac {7 \, a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c} {\left (7 \, a x + 2\right )}}{8 \, x^{2}}, -\frac {7 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c} {\left (7 \, a x + 2\right )}}{4 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 76, normalized size = 1.12 \[ -\frac {\frac {7 \, a^{3} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{3} c - 9 \, \sqrt {-a c x + c} a^{3} c^{2}}{a^{2} c^{2} x^{2}}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 65, normalized size = 0.96 \[ 2 a^{2} c^{2} \left (\frac {-\frac {7 \left (-a c x +c \right )^{\frac {3}{2}}}{8 c}+\frac {9 \sqrt {-a c x +c}}{8}}{x^{2} a^{2} c^{2}}+\frac {7 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{8 c^{\frac {3}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 103, normalized size = 1.51 \[ -\frac {1}{8} \, a^{2} c^{2} {\left (\frac {2 \, {\left (7 \, {\left (-a c x + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {-a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} c + 2 \, {\left (a c x - c\right )} c^{2} + c^{3}} + \frac {7 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 54, normalized size = 0.79 \[ \frac {9\,\sqrt {c-a\,c\,x}}{4\,x^2}+\frac {7\,a^2\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )}{4}-\frac {7\,{\left (c-a\,c\,x\right )}^{3/2}}{4\,c\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 18.12, size = 270, normalized size = 3.97 \[ \frac {10 a^{2} c^{4} \sqrt {- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {6 a^{2} c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {3 a^{2} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} + \frac {3 a^{2} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} - \frac {a^{2} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {a^{2} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {a \sqrt {- a c x + c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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