Optimal. Leaf size=202 \[ -\frac {3 \sqrt {\frac {1}{a x}+1} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3}{2}-p} (c-a c x)^p \, _2F_1\left (1-p,\frac {3}{2}-p;2-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a^2 p \left (1-p^2\right ) x \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {x \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^p}{(p+1) \sqrt {1-\frac {1}{a x}}}+\frac {3 \sqrt {\frac {1}{a x}+1} (c-a c x)^p}{a p (p+1) \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.21, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6176, 6181, 94, 132} \[ -\frac {3 \sqrt {\frac {1}{a x}+1} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3}{2}-p} (c-a c x)^p \, _2F_1\left (1-p,\frac {3}{2}-p;2-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a^2 p \left (1-p^2\right ) x \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {x \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^p}{(p+1) \sqrt {1-\frac {1}{a x}}}+\frac {3 \sqrt {\frac {1}{a x}+1} (c-a c x)^p}{a p (p+1) \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 94
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 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 (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt {1-\frac {1}{a x}}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \operatorname {Subst}\left (\int x^{-1-p} \left (1-\frac {x}{a}\right )^{-\frac {3}{2}+p} \sqrt {1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{a (1+p)}\\ &=\frac {3 \sqrt {1+\frac {1}{a x}} (c-a c x)^p}{a p (1+p) \sqrt {1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt {1-\frac {1}{a x}}}-\frac {\left (3 \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^{-p} \left (1-\frac {x}{a}\right )^{-\frac {3}{2}+p}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2 p (1+p)}\\ &=\frac {3 \sqrt {1+\frac {1}{a x}} (c-a c x)^p}{a p (1+p) \sqrt {1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt {1-\frac {1}{a x}}}-\frac {3 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3}{2}-p} \sqrt {1+\frac {1}{a x}} (c-a c x)^p \, _2F_1\left (1-p,\frac {3}{2}-p;2-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a^2 p \left (1-p^2\right ) \left (1-\frac {1}{a x}\right )^{3/2} x}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 155, normalized size = 0.77 \[ \frac {\sqrt {\frac {1}{a x}+1} \left (\frac {a x-1}{a x+1}\right )^{-p} (c-a c x)^p \left (3 \sqrt {\frac {a x-1}{a x+1}} \, _2F_1\left (1-p,\frac {3}{2}-p;2-p;\frac {2}{a x+1}\right )+(p-1) (a x+1) (a p x+p+3) \left (\frac {a x-1}{a x+1}\right )^p\right )}{a (p-1) p (p+1) \sqrt {1-\frac {1}{a x}} (a x+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} {\left (-a c x + c\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a c x +c \right )^{p}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-a\,c\,x\right )}^p}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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