Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=-\frac {2 (a+b \text {arccosh}(c+d x))^4}{5 d e (e (c+d x))^{5/2}}+\frac {8 b \text {Int}\left (\frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}},x\right )}{5 e} \]
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Not integrable
Time = 0.21 (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 = {} \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^4}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \text {arccosh}(c+d x))^4}{5 d e (e (c+d x))^{5/2}}+\frac {(8 b) \text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^3}{\sqrt {-1+x} (e x)^{5/2} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e} \\ \end{align*}
Not integrable
Time = 102.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx \]
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Not integrable
Time = 0.89 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {7}{2}}}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 5.08 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 1.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {7}{2}}} \,d x } \]
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Not integrable
Time = 2.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \]
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