Integrand size = 16, antiderivative size = 326 \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {3 b c \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{3/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {3 b^{3/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}-\frac {3 b^{3/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 b^{3/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2} \]
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Time = 0.74 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5859, 5830, 6873, 6874, 5433, 5432, 5407, 2236, 2235, 5406} \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^{3/2} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 \sqrt {\pi } b^{3/2} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}+\frac {3 b c \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}-\frac {3 b \sinh (2 \text {arcsinh}(c+d x)) \sqrt {a+b \text {arcsinh}(c+d x)}}{16 d^2}+\frac {\cosh (2 \text {arcsinh}(c+d x)) (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d^2} \]
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Rule 2235
Rule 2236
Rule 5406
Rule 5407
Rule 5432
Rule 5433
Rule 5830
Rule 5859
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right ) (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int (a+b x)^{3/2} \cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = -\frac {2 \text {Subst}\left (\int x^4 \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int x^4 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int \left (c x^4 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^4 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {\text {Subst}\left (\int x^4 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2}-\frac {(2 c) \text {Subst}\left (\int x^4 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{3/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {3 \text {Subst}\left (\int x^2 \cosh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^2}-\frac {(3 c) \text {Subst}\left (\int x^2 \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{d^2} \\ & = \frac {3 b c \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{3/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {3 b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2}-\frac {(3 b) \text {Subst}\left (\int \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{16 d^2}-\frac {(3 b c) \text {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 d^2} \\ & = \frac {3 b c \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{3/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {3 b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2}-\frac {(3 b) \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 )}{32 d^2}+\frac {(3 b) \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 )}{32 d^2}-\frac {(3 b c) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^2}-\frac {(3 b c) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^2} \\ & = \frac {3 b c \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{3/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {3 b^{3/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}-\frac {3 b^{3/2} 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 )}{64 d^2}-\frac {3 b^{3/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}+\frac {3 b^{3/2} 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 )}{64 d^2}-\frac {3 b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2} \\ \end{align*}
Time = 4.14 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.79 \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {-64 a c e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}}}\right )-16 \sqrt {b} c \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-3 \sqrt {1+(c+d x)^2}+2 (c+d x) \text {arcsinh}(c+d x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+4 a \left (8 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+\sqrt {b} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\sqrt {b} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )\right )+\sqrt {b} \left ((4 a+3 b) \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+(4 a-3 b) \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} (4 \text {arcsinh}(c+d x) \cosh (2 \text {arcsinh}(c+d x))-3 \sinh (2 \text {arcsinh}(c+d x)))\right )}{128 d^2} \]
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\[\int x \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} x \,d x } \]
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\[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} x \,d x } \]
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Timed out. \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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