Optimal. Leaf size=94 \[ \frac {x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {3}{2}} (c-a c x)^p \, _2F_1\left (-p-\frac {3}{2},-p-1;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(p+1) \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.13, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6176, 6181, 132} \[ \frac {x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {3}{2}} (c-a c x)^p \, _2F_1\left (-p-\frac {3}{2},-p-1;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(p+1) \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 132
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p x^p \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \operatorname {Subst}\left (\int \frac {x^{-2-p} \left (1-\frac {x}{a}\right )^{\frac {3}{2}+p}}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{3/2} x (c-a c x)^p \, _2F_1\left (-\frac {3}{2}-p,-1-p;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(1+p) \sqrt {1+\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 96, normalized size = 1.02 \[ \frac {\sqrt {1-\frac {1}{a x}} (a x+1) \left (\frac {a x-1}{a x+1}\right )^{-p-\frac {1}{2}} (c-a c x)^p \, _2F_1\left (-p-\frac {3}{2},-p-1;-p;\frac {2}{a x+1}\right )}{a (p+1) \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x - 1\right )} {\left (-a c x + c\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1}, x\right ) \]
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 [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \left (-a c x +c \right )^{p} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (-a c x + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (c-a\,c\,x\right )}^p\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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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