Optimal. Leaf size=82 \[ -\frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} (c-a c x)^{p+1} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2} (1-a x)\right )}{a c (-n+2 p+2)} \]
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Rubi [A] time = 0.06, antiderivative size = 82, 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} \[ -\frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} (c-a c x)^{p+1} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2} (1-a x)\right )}{a c (-n+2 p+2)} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int (1-a x)^{-n/2} (1+a x)^{n/2} (c-a c x)^p \, dx\\ &=\left ((1-a x)^{-n/2} (c-a c x)^{n/2}\right ) \int (1+a x)^{n/2} (c-a c x)^{-\frac {n}{2}+p} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} (c-a c x)^{1+p} \, _2F_1\left (-\frac {n}{2},1-\frac {n}{2}+p;2-\frac {n}{2}+p;\frac {1}{2} (1-a x)\right )}{a c (2-n+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 0.94 \[ \frac {2^{\frac {n}{2}+1} (1-a x)^{1-\frac {n}{2}} (c-a c x)^p \, _2F_1\left (-\frac {n}{2},-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2}-\frac {a x}{2}\right )}{a (n-2 (p+1))} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a c x +c \right )^{p}\, dx \]
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 \[ \int {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \]
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 \[ \int \left (- c \left (a x - 1\right )\right )^{p} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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