Optimal. Leaf size=84 \[ \frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{a^3 (1+2 p)}-\frac {\left (1-a^2 x^2\right )^{\frac {3}{2}+p}}{a^3 (3+2 p)}+\frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6284, 778, 371,
272, 45} \begin {gather*} \frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )+\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{a^3 (2 p+1)}-\frac {\left (1-a^2 x^2\right )^{p+\frac {3}{2}}}{a^3 (2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 371
Rule 778
Rule 6284
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} x^2 \left (1-a^2 x^2\right )^p \, dx &=\int x^2 (1-a x) \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=-\left (a \int x^3 \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\right )+\int x^2 \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=\frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )-\frac {1}{2} a \text {Subst}\left (\int x \left (1-a^2 x\right )^{-\frac {1}{2}+p} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )-\frac {1}{2} a \text {Subst}\left (\int \left (\frac {\left (1-a^2 x\right )^{-\frac {1}{2}+p}}{a^2}-\frac {\left (1-a^2 x\right )^{\frac {1}{2}+p}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{a^3 (1+2 p)}-\frac {\left (1-a^2 x^2\right )^{\frac {3}{2}+p}}{a^3 (3+2 p)}+\frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 75, normalized size = 0.89 \begin {gather*} \frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \left (2+a^2 (1+2 p) x^2\right )}{a^3 \left (3+8 p+4 p^2\right )}+\frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (-a^{2} x^{2}+1\right )^{p} \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx\]
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 \frac {x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{a x + 1}\, 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 \frac {x^2\,{\left (1-a^2\,x^2\right )}^p\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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