Optimal. Leaf size=91 \[ \frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} x^m \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2},\frac{m+2}{2},a^2 x^2\right )}{a m (m+1)}+\frac{x^m}{a m (m+1)}+\frac{x^{m+1} e^{\text{sech}^{-1}(a x)}}{m+1} \]
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Rubi [A] time = 0.0395333, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6335, 30, 125, 364} \[ \frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} x^m \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{m+2}{2};a^2 x^2\right )}{a m (m+1)}+\frac{x^m}{a m (m+1)}+\frac{x^{m+1} e^{\text{sech}^{-1}(a x)}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 125
Rule 364
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^m \, dx &=\frac{e^{\text{sech}^{-1}(a x)} x^{1+m}}{1+m}+\frac{\int x^{-1+m} \, dx}{a (1+m)}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x^{-1+m}}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a (1+m)}\\ &=\frac{x^m}{a m (1+m)}+\frac{e^{\text{sech}^{-1}(a x)} x^{1+m}}{1+m}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x^{-1+m}}{\sqrt{1-a^2 x^2}} \, dx}{a (1+m)}\\ &=\frac{x^m}{a m (1+m)}+\frac{e^{\text{sech}^{-1}(a x)} x^{1+m}}{1+m}+\frac{x^m \sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{2+m}{2};a^2 x^2\right )}{a m (1+m)}\\ \end{align*}
Mathematica [A] time = 0.302242, size = 145, normalized size = 1.59 \[ -\frac{2^{m+1} x^m (a x)^{-m} e^{2 \text{sech}^{-1}(a x)} \left (\frac{e^{\text{sech}^{-1}(a x)}}{e^{2 \text{sech}^{-1}(a x)}+1}\right )^m \left (e^{2 \text{sech}^{-1}(a x)}+1\right )^m \left ((m+2) e^{2 \text{sech}^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{m}{2}+2,m+2,\frac{m}{2}+3,-e^{2 \text{sech}^{-1}(a x)}\right )-(m+4) \text{Hypergeometric2F1}\left (\frac{m}{2}+1,m+2,\frac{m}{2}+2,-e^{2 \text{sech}^{-1}(a x)}\right )\right )}{a (m+2) (m+4)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ){x}^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (\frac{a x x^{m} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + x^{m}}{a x}, 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*} \frac{\int \frac{x^{m}}{x}\, dx + \int a x^{m} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \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^{m}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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