Optimal. Leaf size=202 \[ \frac {3 \sqrt {1+\frac {1}{a x}} (c-a c x)^p}{a p (1+p) \sqrt {1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt {1-\frac {1}{a x}}}-\frac {3 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3}{2}-p} \sqrt {1+\frac {1}{a x}} (c-a c x)^p \, _2F_1\left (1-p,\frac {3}{2}-p;2-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a^2 p \left (1-p^2\right ) \left (1-\frac {1}{a x}\right )^{3/2} x} \]
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Rubi [A]
time = 0.14, 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 = {6311, 6316, 96,
134} \begin {gather*} -\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 96
Rule 134
Rule 6311
Rule 6316
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 ) \text {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 ) \text {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 ) \text {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 = 0.09, size = 155, normalized size = 0.77 \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 ((-1+p) \left (\frac {-1+a x}{1+a x}\right )^p (1+a x) (3+p+a p x)+3 \sqrt {\frac {-1+a x}{1+a x}} \, _2F_1\left (1-p,\frac {3}{2}-p;2-p;\frac {2}{1+a x}\right )\right )}{a (-1+p) p (1+p) \sqrt {1-\frac {1}{a x}} (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-a c x +c \right )^{p}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\, dx\]
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 \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}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \frac {{\left (c-a\,c\,x\right )}^p}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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