Optimal. Leaf size=37 \[ \frac {(1-a x)^{-p} (1+a x)^{1+p} (c-a c x)^p}{a (1+p)} \]
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Rubi [A]
time = 0.02, 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 = {6265, 23, 32}
\begin {gather*} \frac {(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 32
Rule 6265
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 = 0.02, size = 32, normalized size = 0.86 \begin {gather*} \frac {e^{2 p \tanh ^{-1}(a x)} (1+a x) (c-a c x)^p}{a (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 32, normalized size = 0.86
method | result | size |
gosper | \(\frac {\left (a x +1\right ) {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (-c x a +c \right )^{p}}{\left (p +1\right ) a}\) | \(32\) |
risch | \(\frac {\left (a x +1\right ) {\mathrm e}^{\frac {p \left (i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (a x -1\right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (a x -1\right )\right ) \mathrm {csgn}\left (i \left (a x -1\right )\right )+i \pi \mathrm {csgn}\left (i c \left (a x -1\right )\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (a x -1\right )\right )^{2} \mathrm {csgn}\left (i \left (a x -1\right )\right )-2 i \pi \mathrm {csgn}\left (i \left (a x -1\right )\right )^{3}-2 i \pi \mathrm {csgn}\left (i c \left (a x -1\right )\right )^{2}+2 i \pi \mathrm {csgn}\left (i \left (a x -1\right )\right )^{2}+2 \ln \left (a x +1\right )+2 \ln \left (c \right )\right )}{2}}}{\left (p +1\right ) a}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 26, normalized size = 0.70 \begin {gather*} \frac {{\left (a c^{p} x + c^{p}\right )} {\left (a x + 1\right )}^{p}}{a {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 1.03 \begin {gather*} \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} \end {gather*}
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 {gather*} \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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