Optimal. Leaf size=54 \[ x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right ) \]
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Rubi [A]
time = 0.03, 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 = {531, 441, 440}
\begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 531
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] Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(54)=108\).
time = 0.23, size = 167, normalized size = 3.09 \begin {gather*} \frac {3 a x \left (a+c x^2\right )^p \left (1-e^2 x^2\right )^m F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right )}{3 a 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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, 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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.02 \begin {gather*} \int {\left (c\,x^2+a\right )}^p\,{\left (1-e\,x\right )}^m\,{\left (e\,x+1\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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