Optimal. Leaf size=70 \[ \frac {x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (1+m;\frac {1}{2} (n-2 p),-\frac {n}{2}-p;2+m;a x,-a x\right )}{1+m} \]
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Rubi [A]
time = 0.11, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6288, 6285,
138} \begin {gather*} \frac {x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac {1}{2} (n-2 p),-\frac {n}{2}-p;m+2;a x,-a x\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} x^m \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^{n \tanh ^{-1}(a x)} x^m \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 x^m (1-a x)^{-\frac {n}{2}+p} (1+a x)^{\frac {n}{2}+p} \, dx\\ &=\frac {x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (1+m;\frac {1}{2} (n-2 p),-\frac {n}{2}-p;2+m;a x,-a x\right )}{1+m}\\ \end {align*}
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Mathematica [F]
time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} x^{m} \left (-a^{2} c \,x^{2}+c \right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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