Integrand size = 21, antiderivative size = 91 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))}{d e (1+m)}-\frac {b (e (c+d x))^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-(c+d x)^2\right )}{d e^2 (1+m) (2+m)} \]
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Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5859, 5776, 371} \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\frac {(e (c+d x))^{m+1} (a+b \text {arcsinh}(c+d x))}{d e (m+1)}-\frac {b (e (c+d x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-(c+d x)^2\right )}{d e^2 (m+1) (m+2)} \]
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Rule 371
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^m (a+b \text {arcsinh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))}{d e (1+m)}-\frac {b \text {Subst}\left (\int \frac {(e x)^{1+m}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e (1+m)} \\ & = \frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))}{d e (1+m)}-\frac {b (e (c+d x))^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-(c+d x)^2\right )}{d e^2 (1+m) (2+m)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.87 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {(c+d x) (e (c+d x))^m \left (-((2+m) (a+b \text {arcsinh}(c+d x)))+b (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-(c+d x)^2\right )\right )}{d (1+m) (2+m)} \]
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\[\int \left (d e x +c e \right )^{m} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )d x\]
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\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} {\left (d e x + c e\right )}^{m} \,d x } \]
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\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )\, dx \]
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\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} {\left (d e x + c e\right )}^{m} \,d x } \]
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\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} {\left (d e x + c e\right )}^{m} \,d x } \]
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Timed out. \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^m\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right ) \,d x \]
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