Optimal. Leaf size=54 \[ x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 54, 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 = {517, 430, 429} \[ x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right ) \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 517
Rubi steps
\begin {align*} \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx &=\int \left (a+c x^2\right )^p \left (1-e^2 x^2\right )^m \, dx\\ &=\left (\left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (1-e^2 x^2\right )^m \, dx\\ &=x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right )\\ \end {align*}
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Mathematica [B] time = 0.20, size = 167, normalized size = 3.09 \[ \frac {3 a x \left (1-e^2 x^2\right )^m \left (a+c x^2\right )^p F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right )}{2 x^2 \left (c p F_1\left (\frac {3}{2};1-p,-m;\frac {5}{2};-\frac {c x^2}{a},e^2 x^2\right )-a e^2 m F_1\left (\frac {3}{2};-p,1-m;\frac {5}{2};-\frac {c x^2}{a},e^2 x^2\right )\right )+3 a F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )}^{p} {\left (e x + 1\right )}^{m} {\left (-e x + 1\right )}^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + 1\right )}^{m} {\left (-e x + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (-e x +1\right )^{m} \left (e x +1\right )^{m} \left (c \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + 1\right )}^{m} {\left (-e x + 1\right )}^{m}\,{d x} \]
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 \[ \int {\left (c\,x^2+a\right )}^p\,{\left (1-e\,x\right )}^m\,{\left (e\,x+1\right )}^m \,d x \]
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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