Optimal. Leaf size=49 \[ \frac {(1+a x)^{1+2 p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (1+2 p)} \]
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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6278, 6275, 32}
\begin {gather*} \frac {(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{2 p \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{2 p \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1+a x)^{2 p} \, dx\\ &=\frac {(1+a x)^{1+2 p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (1+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 36, normalized size = 0.73 \begin {gather*} \frac {e^{2 p \tanh ^{-1}(a x)} (1+a x) \left (c-a^2 c x^2\right )^p}{a+2 a p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 38, normalized size = 0.78
method | result | size |
gosper | \(\frac {\left (a x +1\right ) {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (1+2 p \right ) a}\) | \(38\) |
risch | \(\frac {\left (a x +1\right ) {\mathrm e}^{\frac {p \left (-2 i \pi \mathrm {csgn}\left (i \left (a x -1\right )\right )^{3}+i \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (a x -1\right )\right ) \pi -i \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i \left (a x +1\right )\right ) \mathrm {csgn}\left (i \left (a x -1\right )\right ) \pi -i \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{3} \pi +i \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (a x +1\right )\right ) \pi +i \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \pi -i \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i c \right ) \pi +i \pi \mathrm {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{3}+i \mathrm {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \mathrm {csgn}\left (i c \right ) \pi +2 i \pi \mathrm {csgn}\left (i \left (a x -1\right )\right )^{2}-2 i \pi \mathrm {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2}+4 \ln \left (a x +1\right )+2 \ln \left (c \right )\right )}{2}}}{\left (1+2 p \right ) a}\) | \(319\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 0.61 \begin {gather*} \frac {{\left (a c^{p} x + c^{p}\right )} {\left (a x + 1\right )}^{2 \, p}}{a {\left (2 \, 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 = 43, normalized size = 0.88 \begin {gather*} \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} \left (-\frac {a x + 1}{a x - 1}\right )^{p}}{2 \, 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}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x & \text {for}\: a = 0 \\\int \frac {e^{- \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (- a^{2} c x^{2} + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{2 a p + a} + \frac {\left (- a^{2} c x^{2} + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{2 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 = 1.06, size = 43, normalized size = 0.88 \begin {gather*} \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^{p+1}}{a\,\left (2\,p+1\right )\,{\left (1-a\,x\right )}^p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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