Optimal. Leaf size=107 \[ \frac {2 c^2 (1-a x)^{3/2} \sqrt {a x+1}}{(c-a c x)^{3/2}}+\frac {8 c^2 (1-a x)^{3/2}}{\sqrt {a x+1} (c-a c x)^{3/2}}-\frac {2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{(c-a c x)^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6130, 23, 87, 63, 208} \[ \frac {2 c^2 (1-a x)^{3/2} \sqrt {a x+1}}{(c-a c x)^{3/2}}+\frac {8 c^2 (1-a x)^{3/2}}{\sqrt {a x+1} (c-a c x)^{3/2}}-\frac {2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{(c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 63
Rule 87
Rule 208
Rule 6130
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx &=\int \frac {(1-a x)^{3/2} \sqrt {c-a c x}}{x (1+a x)^{3/2}} \, dx\\ &=\frac {(1-a x)^{3/2} \int \frac {(c-a c x)^2}{x (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=\frac {(1-a x)^{3/2} \int \left (-\frac {4 a c^2}{(1+a x)^{3/2}}+\frac {a c^2}{\sqrt {1+a x}}+\frac {c^2}{x \sqrt {1+a x}}\right ) \, dx}{(c-a c x)^{3/2}}\\ &=\frac {8 c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}+\frac {2 c^2 (1-a x)^{3/2} \sqrt {1+a x}}{(c-a c x)^{3/2}}+\frac {\left (c^2 (1-a x)^{3/2}\right ) \int \frac {1}{x \sqrt {1+a x}} \, dx}{(c-a c x)^{3/2}}\\ &=\frac {8 c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}+\frac {2 c^2 (1-a x)^{3/2} \sqrt {1+a x}}{(c-a c x)^{3/2}}+\frac {\left (2 c^2 (1-a x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {1+a x}\right )}{a (c-a c x)^{3/2}}\\ &=\frac {8 c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}+\frac {2 c^2 (1-a x)^{3/2} \sqrt {1+a x}}{(c-a c x)^{3/2}}-\frac {2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {1+a x}\right )}{(c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 51, normalized size = 0.48 \[ \frac {2 c \sqrt {1-a x} \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};a x+1\right )+a x+4\right )}{\sqrt {a x+1} \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 209, normalized size = 1.95 \[ \left [\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 5\right )}}{a^{2} x^{2} - 1}, -\frac {2 \, {\left ({\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 5\right )}\right )}}{a^{2} x^{2} - 1}\right ] \]
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 = 0.05, size = 78, normalized size = 0.73 \[ \frac {2 \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {c}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) \sqrt {c \left (a x +1\right )}-a c x -5 c \right )}{\left (a x -1\right ) c \left (a x +1\right )} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}{{\left (a x + 1\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}}{x\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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