Optimal. Leaf size=70 \[ \frac {x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac {1}{2} (n-2 p),-\frac {n}{2}-p;m+2;a x,-a x\right )}{m+1} \]
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Rubi [A] time = 0.16, 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 = {6153, 6150, 133} \[ \frac {x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p F_1\left (m+1;\frac {1}{2} (n-2 p),-\frac {n}{2}-p;m+2;a x,-a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6150
Rule 6153
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 = 0.58, size = 0, normalized size = 0.00 \[ \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a^{2} c x^{2} + c\right )}^{p} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, 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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^p \,d x \]
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 \[ \int x^{m} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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