Optimal. Leaf size=114 \[ \frac{2^{n/2} (2 a-n) (-a-b x+1)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (-a-b x+1)\right )}{b^2 (2-n)}-\frac{(-a-b x+1)^{1-\frac{n}{2}} (a+b x+1)^{\frac{n+2}{2}}}{2 b^2} \]
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Rubi [A] time = 0.0687108, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 6163
Rule 80
Rule 69
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*}
Mathematica [A] time = 0.0403405, size = 96, normalized size = 0.84 \[ \frac{(-a-b x+1)^{1-\frac{n}{2}} \left (\frac{b 2^{\frac{n}{2}+1} (n-2 a) \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (-a-b x+1)\right )}{n-2}-b (a+b x+1)^{\frac{n}{2}+1}\right )}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( bx+a \right ) }}x\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{b x + a + 1}{b x + a - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \left (\frac{b x + a + 1}{b x + a - 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x e^{n \operatorname{atanh}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{b x + a + 1}{b x + a - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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