Integrand size = 25, antiderivative size = 272 \[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {3 e^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {\sqrt {b} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {\sqrt {b} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {b} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {\sqrt {b} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d} \]
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Time = 0.43 (sec) , antiderivative size = 272, 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, 3388, 2211, 2236, 2235} \[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {\sqrt {\pi } \sqrt {b} e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {\pi } \sqrt {b} e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {3 e^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5777
Rule 5819
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^3 x^3 \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2} \sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{8 d} \\ & = \frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {e^3 \text {Subst}\left (\int \frac {\sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 d} \\ & = \frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {e^3 \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 d} \\ & = -\frac {3 e^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {e^3 \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{64 d}+\frac {e^3 \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 d} \\ & = -\frac {3 e^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {e^3 \text {Subst}\left (\int \frac {e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{128 d}-\frac {e^3 \text {Subst}\left (\int \frac {e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{128 d}+\frac {e^3 \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{32 d}+\frac {e^3 \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{32 d} \\ & = -\frac {3 e^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {e^3 \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{64 d}-\frac {e^3 \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{64 d}+\frac {e^3 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{16 d}+\frac {e^3 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{16 d} \\ & = -\frac {3 e^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}-\frac {\sqrt {b} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {\sqrt {b} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {b} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {\sqrt {b} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.82 \[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-4 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \left (-4 \sqrt {2} \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{128 d \sqrt {-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}}} \]
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\[\int \left (d e x +c e \right )^{3} \sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}d x\]
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Exception generated. \[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=e^{3} \left (\int c^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int d^{3} x^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 3 c d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 3 c^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{3} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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\[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{3} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int (c e+d e x)^3 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]
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