Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\frac {\text {Int}\left (\frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c+d x},x\right )}{e} \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^{3/2}}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^{3/2}}{x} \, dx,x,c+d x\right )}{d e} \\ \end{align*}
Not integrable
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}}{d e x +c e}d x\]
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Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\frac {\int \frac {a \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}{c + d x}\, dx + \int \frac {b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
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Not integrable
Time = 0.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{d e x + c e} \,d x } \]
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Not integrable
Time = 2.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{d e x + c e} \,d x } \]
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Not integrable
Time = 2.63 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2}}{c\,e+d\,e\,x} \,d x \]
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