Integrand size = 18, antiderivative size = 411 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3} \]
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Time = 0.67 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5859, 5830, 6873, 6874, 5407, 2236, 2235, 5406, 5737} \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {\sqrt {\pi } c^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\pi } c^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} c e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{2}} c e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{3}} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3} \]
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Rule 2235
Rule 2236
Rule 5406
Rule 5407
Rule 5737
Rule 5830
Rule 5859
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-\frac {c}{d}+\frac {x}{d}\right )^2}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right )^2}{\sqrt {a+b x}} \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = \frac {2 \text {Subst}\left (\int \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int \left (c^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+c \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+\cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3}+\frac {(2 c) \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 )}{b d^3}+\frac {\left (2 c^2\right ) \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 )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {1}{4} \cosh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3}+\frac {c \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 )}{b d^3}-\frac {c \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 )}{b d^3}+\frac {c^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3}+\frac {c^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c 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 )}{2 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c 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 )}{2 \sqrt {b} d^3}+\frac {\text {Subst}\left (\int \cosh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 b d^3}-\frac {\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 b d^3} \\ & = \frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c 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 )}{2 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c 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 )}{2 \sqrt {b} d^3}+\frac {\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 )}{4 b d^3}-\frac {\text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 b d^3}-\frac {\text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 b d^3}+\frac {\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 )}{4 b d^3} \\ & = -\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c 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 )}{2 \sqrt {b} d^3}+\frac {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 )}{8 \sqrt {b} d^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c 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 )}{2 \sqrt {b} d^3}+\frac {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 )}{8 \sqrt {b} d^3} \\ \end{align*}
Time = 1.60 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {\sqrt {\frac {\pi }{6}} \left (\sqrt {2} \cosh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\sqrt {6} \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+4 \sqrt {6} c^2 \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-4 \sqrt {3} c \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2} \cosh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {6} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-4 \sqrt {6} c^2 \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+\sqrt {6} \left (-1+4 c^2\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+4 \sqrt {3} c \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+4 \sqrt {3} c \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {2} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )-\sqrt {2} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )\right )}{8 \sqrt {b} d^3} \]
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\[\int \frac {x^{2}}{\sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}}d x\]
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Exception generated. \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {x^{2}}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {x^{2}}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {x^{2}}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {x^2}{\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}} \,d x \]
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