Optimal. Leaf size=28 \[ \frac{\text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{c+d x},x\right )}{e} \]
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Rubi [A] time = 0.103211, 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 = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{c e+d e x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{c e+d e x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^{3/2}}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^{3/2}}{x} \, dx,x,c+d x\right )}{d e}\\ \end{align*}
Mathematica [A] time = 1.41524, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{c e+d e x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dex+ce} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{3}{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 \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{d e x + c e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} \frac{\int \frac{a \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}{c + d x}\, dx + \int \frac{b \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} \operatorname{asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \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 \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{d e x + c e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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