Optimal. Leaf size=198 \[ \frac{d \text{Unintegrable}\left (\frac{x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 d x^2+d}},x\right )}{m+2}+\frac{2 b^2 c^2 x^{m+3} \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right )}{(m+2)^2 (m+3) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{m+2}-\frac{2 b c x^{m+2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+2)^2 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.136242, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ \end{align*}
Mathematica [A] time = 0.29456, size = 0, normalized size = 0. \[ \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.852, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{{c}^{2}d{x}^{2}+d} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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