Optimal. Leaf size=42 \[ \frac {c^2 x^{m+1} F_1\left (m+1;\frac {n-4}{2},-\frac {n}{2}-2;m+2;a x,-a x\right )}{m+1} \]
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Rubi [A] time = 0.09, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 133} \[ \frac {c^2 x^{m+1} F_1\left (m+1;\frac {n-4}{2},-\frac {n}{2}-2;m+2;a x,-a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6150
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^m (1-a x)^{2-\frac {n}{2}} (1+a x)^{2+\frac {n}{2}} \, dx\\ &=\frac {c^2 x^{1+m} F_1\left (1+m;\frac {1}{2} (-4+n),-2-\frac {n}{2};2+m;a x,-a x\right )}{1+m}\\ \end {align*}
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Mathematica [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int e^{n \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} x^{m} \left (-a^{2} c \,x^{2}+c \right )^{2}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} x^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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