Optimal. Leaf size=170 \[ \frac {(4 a-n) (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{6 b^3}-\frac {x (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{3 b^2}-\frac {2^{n/2} \left (2+6 a^2-6 a n+n^2\right ) (1-a-b x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a-b x)\right )}{3 b^3 (2-n)} \]
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Rubi [A]
time = 0.11, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6298, 92, 81,
71} \begin {gather*} -\frac {2^{n/2} \left (6 a^2-6 a n+n^2+2\right ) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{3 b^3 (2-n)}+\frac {(4 a-n) (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{6 b^3}-\frac {x (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 81
Rule 92
Rule 6298
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a+b x)} x^2 \, dx &=\int x^2 (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx\\ &=-\frac {x (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{3 b^2}-\frac {\int (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \left (-1+a^2+b (4 a-n) x\right ) \, dx}{3 b^2}\\ &=\frac {(4 a-n) (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{6 b^3}-\frac {x (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{3 b^2}+\frac {\left (2+6 a^2-6 a n+n^2\right ) \int (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx}{6 b^2}\\ &=\frac {(4 a-n) (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{6 b^3}-\frac {x (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{3 b^2}-\frac {2^{n/2} \left (2+6 a^2-6 a n+n^2\right ) (1-a-b x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a-b x)\right )}{3 b^3 (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 127, normalized size = 0.75 \begin {gather*} \frac {(1-a-b x)^{1-\frac {n}{2}} \left ((4 a-n) (1+a+b x)^{1+\frac {n}{2}}-2 b x (1+a+b x)^{1+\frac {n}{2}}+\frac {2^{1+\frac {n}{2}} \left (2+6 a^2-6 a n+n^2\right ) \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a-b x)\right )}{-2+n}\right )}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (b x +a \right )} x^{2}\, dx\]
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 \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 x^{2} e^{n \operatorname {atanh}{\left (a + b x \right )}}\, 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 x^2\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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