Integrand size = 25, antiderivative size = 245 \[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d}+\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d} \]
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Time = 0.42 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5859, 12, 5777, 5819, 3393, 3389, 2211, 2236, 2235} \[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d}+\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d}+\frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rule 5777
Rule 5819
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{6 d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{6 d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}+\frac {\left (i e^2\right ) \text {Subst}\left (\int \left (-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {3 i \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{6 d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{24 d}-\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}+\frac {e^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{48 d}-\frac {e^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{48 d}-\frac {e^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 d}+\frac {e^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}+\frac {e^2 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{24 d}-\frac {e^2 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{24 d}-\frac {e^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 d}+\frac {e^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 d} \\ & = \frac {e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d}+\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.97 \[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{72 d \sqrt {-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}}} \]
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\[\int \left (d e x +c e \right )^{2} \sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}d x\]
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Exception generated. \[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=e^{2} \left (\int c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 2 c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{2} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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\[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{2} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int (c e+d e x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]
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