Optimal. Leaf size=88 \[ \frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{5 d e}-\frac{8 b \text{Unintegrable}\left (\frac{(e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{\sqrt{c+d x-1} \sqrt{c+d x+1}},x\right )}{5 e} \]
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Rubi [A] time = 0.315273, 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 = {} \[ \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{3/2} \left (a+b \cosh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{5 d e}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{(e x)^{5/2} \left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{5 d e}\\ \end{align*}
Mathematica [A] time = 70.6157, size = 0, normalized size = 0. \[ \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.37, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{4}\, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} d e x + a^{4} c e +{\left (b^{4} d e x + b^{4} c e\right )} \operatorname{arcosh}\left (d x + c\right )^{4} + 4 \,{\left (a b^{3} d e x + a b^{3} c e\right )} \operatorname{arcosh}\left (d x + c\right )^{3} + 6 \,{\left (a^{2} b^{2} d e x + a^{2} b^{2} c e\right )} \operatorname{arcosh}\left (d x + c\right )^{2} + 4 \,{\left (a^{3} b d e x + a^{3} b c e\right )} \operatorname{arcosh}\left (d x + c\right )\right )} \sqrt{d e x + c e}, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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