Integrand size = 25, antiderivative size = 328 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {3 b^{3/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{96 d}-\frac {3 b^{3/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{96 d} \]
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Time = 0.60 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5859, 12, 5777, 5812, 5798, 5774, 3388, 2211, 2236, 2235, 5780, 5556} \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^{3/2} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{96 d}-\frac {3 \sqrt {\pi } b^{3/2} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{96 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {b e^2 \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {b e^2 \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5774
Rule 5777
Rule 5780
Rule 5798
Rule 5812
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d} \\ & = -\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{12 d} \\ & = \frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{12 d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{6 d} \\ & = \frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{12 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{6 d} \\ & = \frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{48 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{48 d}-\frac {\left (b e^2\right ) \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 )}{12 d}-\frac {\left (b e^2\right ) \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 )}{12 d} \\ & = \frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {\left (b e^2\right ) \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 )}{96 d}-\frac {\left (b e^2\right ) \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 )}{96 d}+\frac {\left (b e^2\right ) \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 )}{96 d}+\frac {\left (b e^2\right ) \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 )}{96 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{6 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{6 d} \\ & = \frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {b^{3/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 d}-\frac {b^{3/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 d}+\frac {\left (b e^2\right ) \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 )}{48 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{48 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{48 d}+\frac {\left (b e^2\right ) \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 )}{48 d} \\ & = \frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {3 b^{3/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} 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 )}{96 d}-\frac {3 b^{3/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} 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 )}{96 d} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.73 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {b e^2 e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-27 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-27 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{216 d \sqrt {-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}}} \]
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\[\int \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=e^{2} \left (\int a c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 a c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 b c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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