Integrand size = 14, antiderivative size = 179 \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3 b^{3/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^2} \]
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Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5777, 5812, 5783, 5780, 5556, 12, 3389, 2211, 2236, 2235} \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^2}-\frac {3 b x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5777
Rule 5780
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{4} (3 b c) \int \frac {x^2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{16} \left (3 b^2\right ) \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx+\frac {(3 b) \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx}{8 c} \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {(3 b) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 c^2} \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {(3 b) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 c^2} \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {(3 b) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 c^2} \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {(3 b) \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 x)\right )}{64 c^2}+\frac {(3 b) \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 x)\right )}{64 c^2} \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\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 x)}\right )}{32 c^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 x)}\right )}{32 c^2} \\ & = -\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\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 x)}}{\sqrt {b}}\right )}{64 c^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 x)}}{\sqrt {b}}\right )}{64 c^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.64 \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {e^{-\frac {2 a}{b}} \left (b^2 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {5}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+b^2 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {5}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{16 \sqrt {2} c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int x \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x \,d x } \]
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\[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x \,d x } \]
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Timed out. \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2} \,d x \]
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