Integrand size = 16, antiderivative size = 282 \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \]
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Time = 0.54 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5777, 5812, 5798, 5774, 3388, 2211, 2236, 2235, 5780, 5556} \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {b x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {b \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5774
Rule 5777
Rule 5780
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} (b c) \int \frac {x^3 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{12} b^2 \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx+\frac {b \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx}{3 c} \\ & = \frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {b \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 x)\right )}{12 c^3}-\frac {b^2 \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{6 c^2} \\ & = \frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {b \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 x)\right )}{12 c^3}-\frac {b \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{6 c^3} \\ & = \frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {b \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 x)\right )}{48 c^3}-\frac {b \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{48 c^3}-\frac {b \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 x)\right )}{12 c^3}-\frac {b \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 x)\right )}{12 c^3} \\ & = \frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {b \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 x)\right )}{96 c^3}-\frac {b \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 x)\right )}{96 c^3}+\frac {b \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 x)\right )}{96 c^3}+\frac {b \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 x)\right )}{96 c^3}-\frac {b \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{6 c^3}-\frac {b \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{6 c^3} \\ & = \frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {b \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3}-\frac {b \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3}-\frac {b \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3}+\frac {b \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{48 c^3} \\ & = \frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.76 \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {b e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-27 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {5}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {5}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-27 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {5}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {5}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{216 c^3 \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}} \]
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\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]
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Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2} \,d x \]
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