Optimal. Leaf size=79 \[ \frac{2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e}-\frac{2 b \text{Unintegrable}\left (\frac{(e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt{(c+d x)^2+1}},x\right )}{e} \]
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Rubi [A] time = 0.202466, 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 = {} \[ \int \sqrt{c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \sqrt{c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{e x} \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{3/2} \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}
Mathematica [A] time = 89.1327, size = 0, normalized size = 0. \[ \int \sqrt{c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.263, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}\sqrt{dex+ce}\, 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 (b^{3} \operatorname{arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arsinh}\left (d x + c\right ) + a^{3}\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \left (c + d x\right )} \left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{3}\, dx \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 \sqrt{d e x + c e}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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