Integrand size = 25, antiderivative size = 25 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {2 (e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x))^4}{5 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x))^3}{\sqrt {1+(c+d x)^2}},x\right )}{5 e} \]
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Not integrable
Time = 0.14 (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 = {} \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^{3/2} (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x))^4}{5 d e}-\frac {(8 b) \text {Subst}\left (\int \frac {(e x)^{5/2} (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d e} \\ \end{align*}
Not integrable
Time = 34.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \left (d e x +c e \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{4}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 5.20 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
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Not integrable
Time = 39.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int \left (e \left (c + d x\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}\, dx \]
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Exception generated. \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 2.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
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Not integrable
Time = 2.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (c e+d e x)^{3/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]
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