Integrand size = 16, antiderivative size = 194 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3} \]
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Time = 0.21 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5780, 5556, 3388, 2211, 2236, 2235} \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5780
Rubi steps \begin{align*} \text {integral}& = \frac {\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 )}{b c^3} \\ & = \frac {\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 )}{b c^3} \\ & = \frac {\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 )}{4 b c^3}-\frac {\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 )}{4 b c^3} \\ & = -\frac {\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 )}{8 b c^3}-\frac {\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 )}{8 b c^3}+\frac {\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 )}{8 b c^3}+\frac {\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 )}{8 b c^3} \\ & = \frac {\text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b c^3}-\frac {\text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b c^3}-\frac {\text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b c^3}+\frac {\text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b c^3} \\ & = -\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {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 )}{8 \sqrt {b} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {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 )}{8 \sqrt {b} c^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.01 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (3 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-3 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{24 c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int \frac {x^{2}}{\sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}}d x\]
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Exception generated. \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {x^{2}}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {x^{2}}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {x^{2}}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {x^2}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}} \,d x \]
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