Optimal. Leaf size=65 \[ -\frac {2 \sqrt {2} (c-a c x)^{p+1} \, _2F_1\left (-\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\frac {1}{2} (1-a x)\right )}{a c (2 p+1) \sqrt {1-a x}} \]
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Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6130, 23, 69} \[ -\frac {2 \sqrt {2} (c-a c x)^{p+1} \, _2F_1\left (-\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\frac {1}{2} (1-a x)\right )}{a c (2 p+1) \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac {\sqrt {1+a x} (c-a c x)^p}{\sqrt {1-a x}} \, dx\\ &=\frac {\sqrt {c-a c x} \int \sqrt {1+a x} (c-a c x)^{-\frac {1}{2}+p} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {2} (c-a c x)^{1+p} \, _2F_1\left (-\frac {1}{2},\frac {1}{2}+p;\frac {3}{2}+p;\frac {1}{2} (1-a x)\right )}{a c (1+2 p) \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.82 \[ -\frac {2 \sqrt {2-2 a x} (c-a c x)^p \, _2F_1\left (-\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\frac {1}{2}-\frac {a x}{2}\right )}{2 a p+a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a c x + c\right )}^{p}}{a x - 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (-a c x + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) \left (-a c x +c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (-a c x + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-a\,c\,x\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{p} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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