Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^4}{d e}-\frac {8 b \text {Int}\left (\frac {\sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^3}{\sqrt {1+(c+d x)^2}},x\right )}{e} \]
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Not integrable
Time = 0.13 (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 = {} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{\sqrt {e x}} \, dx,x,c+d x\right )}{d} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^4}{d e}-\frac {(8 b) \text {Subst}\left (\int \frac {\sqrt {e x} (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e} \\ \end{align*}
Not integrable
Time = 13.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{4}}{\sqrt {d e x +c e}}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{\sqrt {d e x + c e}} \,d x } \]
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Not integrable
Time = 5.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}}{\sqrt {e \left (c + d x\right )}}\, dx \]
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Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{\sqrt {d e x + c e}} \,d x } \]
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Not integrable
Time = 2.93 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{\sqrt {c e+d e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{\sqrt {c\,e+d\,e\,x}} \,d x \]
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