Optimal. Leaf size=42 \[ \frac {2 (c-a c x)^p}{a p}-\frac {(c-a c x)^{p+1}}{a c (p+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 42, 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 = {6167, 6130, 21, 43} \[ \frac {2 (c-a c x)^p}{a p}-\frac {(c-a c x)^{p+1}}{a c (p+1)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^p \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^p \, dx\\ &=-\int \frac {(1+a x) (c-a c x)^p}{1-a x} \, dx\\ &=-\left (c \int (1+a x) (c-a c x)^{-1+p} \, dx\right )\\ &=-\left (c \int \left (2 (c-a c x)^{-1+p}-\frac {(c-a c x)^p}{c}\right ) \, dx\right )\\ &=\frac {2 (c-a c x)^p}{a p}-\frac {(c-a c x)^{1+p}}{a c (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.67 \[ \frac {(a p x+p+2) (c-a c x)^p}{a p (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 28, normalized size = 0.67 \[ \frac {{\left (a p x + p + 2\right )} {\left (-a c x + c\right )}^{p}}{a p^{2} + a p} \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (-a c x + c\right )}^{p}}{a x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 0.69 \[ \frac {\left (a p x +p +2\right ) \left (-a c x +c \right )^{p}}{a p \left (1+p \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 49, normalized size = 1.17 \[ \frac {{\left (a c^{p} p x + c^{p}\right )} {\left (-a x + 1\right )}^{p}}{{\left (p^{2} + p\right )} a} + \frac {{\left (-a x + 1\right )}^{p} c^{p}}{a p} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 28, normalized size = 0.67 \[ \frac {{\left (c-a\,c\,x\right )}^p\,\left (p+a\,p\,x+2\right )}{a\,p\,\left (p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 124, normalized size = 2.95 \[ \begin {cases} - c^{p} x & \text {for}\: a = 0 \\- \frac {a x \log {\left (x - \frac {1}{a} \right )}}{a^{2} c x - a c} + \frac {\log {\left (x - \frac {1}{a} \right )}}{a^{2} c x - a c} + \frac {2}{a^{2} c x - a c} & \text {for}\: p = -1 \\x + \frac {2 \log {\left (x - \frac {1}{a} \right )}}{a} & \text {for}\: p = 0 \\\frac {a p x \left (- a c x + c\right )^{p}}{a p^{2} + a p} + \frac {p \left (- a c x + c\right )^{p}}{a p^{2} + a p} + \frac {2 \left (- a c x + c\right )^{p}}{a p^{2} + a p} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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