Optimal. Leaf size=143 \[ \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p}{1+p}+\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} \sqrt {1+\frac {1}{a x}} (c-a c x)^p \, _2F_1\left (\frac {1}{2}-p,-p;1-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A]
time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96,
134} \begin {gather*} \frac {\sqrt {\frac {1}{a x}+1} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} (c-a c x)^p \, _2F_1\left (\frac {1}{2}-p,-p;1-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (p+1) \sqrt {1-\frac {1}{a x}}}+\frac {x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} (c-a c x)^p}{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 96
Rule 134
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int e^{\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^{\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 ) \text {Subst}\left (\int x^{-2-p} \left (1-\frac {x}{a}\right )^{-\frac {1}{2}+p} \sqrt {1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p}{1+p}-\frac {\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \text {Subst}\left (\int \frac {x^{-1-p} \left (1-\frac {x}{a}\right )^{-\frac {1}{2}+p}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a (1+p)}\\ &=\frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p}{1+p}+\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} \sqrt {1+\frac {1}{a x}} (c-a c x)^p \, _2F_1\left (\frac {1}{2}-p,-p;1-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 131, normalized size = 0.92 \begin {gather*} \frac {\sqrt {1+\frac {1}{a x}} \left (\frac {-1+a x}{1+a x}\right )^{-p} (c-a c x)^p \left (p (-1+a x) \left (\frac {-1+a x}{1+a x}\right )^p+\sqrt {\frac {-1+a x}{1+a x}} \, _2F_1\left (\frac {1}{2}-p,-p;1-p;\frac {2}{1+a x}\right )\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-a c x +c \right )^{p}}{\sqrt {\frac {a x -1}{a x +1}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (- c \left (a x - 1\right )\right )^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c-a\,c\,x\right )}^p}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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