Optimal. Leaf size=114 \[ \frac {2^{n/2} (2 a-n) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{b^2 (2-n)}-\frac {(-a-b x+1)^{1-\frac {n}{2}} (a+b x+1)^{\frac {n+2}{2}}}{2 b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6163, 80, 69} \[ \frac {2^{n/2} (2 a-n) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{b^2 (2-n)}-\frac {(-a-b x+1)^{1-\frac {n}{2}} (a+b x+1)^{\frac {n+2}{2}}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 80
Rule 6163
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a+b x)} x \, dx &=\int x (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx\\ &=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 b^2}-\frac {(2 a-n) \int (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx}{2 b}\\ &=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 b^2}+\frac {2^{n/2} (2 a-n) (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 )}{b^2 (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 96, normalized size = 0.84 \[ \frac {(-a-b x+1)^{1-\frac {n}{2}} \left (\frac {b 2^{\frac {n}{2}+1} (n-2 a) \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{n-2}-b (a+b x+1)^{\frac {n}{2}+1}\right )}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int x \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm e}^{n \arctanh \left (b x +a \right )} x\, dx \]
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 \[ \int x \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x e^{n \operatorname {atanh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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