Optimal. Leaf size=140 \[ \frac {x \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6182, 6179, 89, 80, 63, 208} \[ \frac {x \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 89
Rule 208
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\frac {\left (c-\frac {c}{a x}\right )^{3/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {\left (c-\frac {c}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2} x}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {\left (c-\frac {c}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {-\frac {5}{2 a}+\frac {x}{a^2}}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2} x}{\left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\left (5 \left (c-\frac {c}{a x}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2} x}{\left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\left (5 \left (c-\frac {c}{a x}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2} x}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.50 \[ \frac {c \sqrt {c-\frac {c}{a x}} \left (\sqrt {\frac {1}{a x}+1} (a x-2)-5 \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )\right )}{a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 315, normalized size = 2.25 \[ \left [\frac {5 \, {\left (a c x - c\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {5 \, {\left (a c x - c\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
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 = 0.07, size = 118, normalized size = 0.84 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x a -4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{2 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}} \]
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 \[ \int {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
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 \[ \int {\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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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