Optimal. Leaf size=583 \[ \frac{3 d^2 \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}},x\right )}{m^2+6 m+8}-\frac{2 b^2 c^2 d (3 m+10) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right )}{f^3 (m+2) (m+3) (m+4)^3 (1-c x) (c x+1)}-\frac{6 b^2 c^2 d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right )}{f^3 (m+2)^2 (m+3) (m+4) (1-c x) (c x+1)}-\frac{2 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2) (m+4) \sqrt{c x-1} \sqrt{c x+1}}-\frac{6 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2} (f x)^{m+4} \left (a+b \cosh ^{-1}(c x)\right )}{f^4 (m+4)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 d \sqrt{d-c^2 d x^2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f \left (m^2+6 m+8\right )}+\frac{\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (m+4)}-\frac{2 b^2 c^2 d \sqrt{d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^3} \]
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Rubi [A] time = 0.523117, 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 = {} \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int (f x)^m (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.513126, size = 0, normalized size = 0. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.082, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arccosh} \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{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{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 (-{\left (a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d} \left (f x\right )^{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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