Optimal. Leaf size=65 \[ -\frac {\sqrt {2} \sqrt {1-a x} (c-a c x)^{p+1} \, _2F_1\left (\frac {1}{2},p+\frac {3}{2};p+\frac {5}{2};\frac {1}{2} (1-a x)\right )}{a c (2 p+3)} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6130, 23, 69} \[ -\frac {\sqrt {2} \sqrt {1-a x} (c-a c x)^{p+1} \, _2F_1\left (\frac {1}{2},p+\frac {3}{2};p+\frac {5}{2};\frac {1}{2} (1-a x)\right )}{a c (2 p+3)} \]
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 {1-a x} \int \frac {(c-a c x)^{\frac {1}{2}+p}}{\sqrt {1+a x}} \, dx}{\sqrt {c-a c x}}\\ &=-\frac {\sqrt {2} \sqrt {1-a x} (c-a c x)^{1+p} \, _2F_1\left (\frac {1}{2},\frac {3}{2}+p;\frac {5}{2}+p;\frac {1}{2} (1-a x)\right )}{a c (3+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 0.91 \[ \frac {\sqrt {2-2 a x} (a x-1) (c-a c x)^p \, _2F_1\left (\frac {1}{2},p+\frac {3}{2};p+\frac {5}{2};\frac {1}{2}-\frac {a x}{2}\right )}{a (2 p+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, 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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a c x + c\right )}^{p}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a c x +c \right )^{p} \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a c x + c\right )}^{p}}{a x + 1}\,{d x} \]
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 \[ \int \frac {\sqrt {1-a^2\,x^2}\,{\left (c-a\,c\,x\right )}^p}{a\,x+1} \,d x \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{p} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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