Optimal. Leaf size=170 \[ -\frac {2^{n/2} \left (6 a^2-6 a n+n^2+2\right ) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{3 b^3 (2-n)}+\frac {(4 a-n) (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{6 b^3}-\frac {x (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{3 b^2} \]
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Rubi [A] time = 0.16, 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 = {6163, 90, 80, 69} \[ -\frac {2^{n/2} \left (6 a^2-6 a n+n^2+2\right ) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{3 b^3 (2-n)}+\frac {(4 a-n) (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{6 b^3}-\frac {x (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 80
Rule 90
Rule 6163
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 = 0.11, size = 127, normalized size = 0.75 \[ \frac {(-a-b x+1)^{1-\frac {n}{2}} \left (\frac {2^{\frac {n}{2}+1} \left (6 a^2-6 a n+n^2+2\right ) \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{n-2}+(4 a-n) (a+b x+1)^{\frac {n}{2}+1}-2 b x (a+b x+1)^{\frac {n}{2}+1}\right )}{6 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int x^{2} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, 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 \[ \int x^{2} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{2} e^{n \operatorname {atanh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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