Optimal. Leaf size=37 \[ \frac {(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6130, 23, 32} \[ \frac {(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \]
Antiderivative was successfully verified.
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Rule 23
Rule 32
Rule 6130
Rubi steps
\begin {align*} \int e^{2 p \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int (1-a x)^{-p} (1+a x)^p (c-a c x)^p \, dx\\ &=\left ((1-a x)^{-p} (c-a c x)^p\right ) \int (1+a x)^p \, dx\\ &=\frac {(1-a x)^{-p} (1+a x)^{1+p} (c-a c x)^p}{a (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.86 \[ \frac {(a x+1) (c-a c x)^p e^{2 p \tanh ^{-1}(a x)}}{a (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 37, normalized size = 1.00 \[ \frac {{\left (a x + 1\right )} {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}}{a p + a} \]
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 \[ \int {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 32, normalized size = 0.86 \[ \frac {\left (a x +1\right ) {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (-a c x +c \right )^{p}}{a \left (1+p \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 30, normalized size = 0.81 \[ \frac {{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )} {\left (a x + 1\right )}^{p}}{a {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 37, normalized size = 1.00 \[ \frac {{\left (c-a\,c\,x\right )}^p\,{\left (a\,x+1\right )}^{p+1}}{a\,{\left (1-a\,x\right )}^p\,\left (p+1\right )} \]
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 \[ \begin {cases} \frac {x}{c} & \text {for}\: a = 0 \wedge p = -1 \\c^{p} x & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{2 \operatorname {atanh}{\left (a x \right )}} - e^{2 \operatorname {atanh}{\left (a x \right )}}}\, dx}{c} & \text {for}\: p = -1 \\\frac {a x \left (- a c x + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{a p + a} + \frac {\left (- a c x + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{a p + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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