Integrand size = 22, antiderivative size = 608 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5} \]
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Time = 0.82 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5793, 5774, 3388, 2211, 2236, 2235, 5780, 5556} \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {\sqrt {\pi } e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}-\frac {\sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\frac {\pi }{5}} e^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}-\frac {\sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\frac {\pi }{5}} e^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}-\frac {\sqrt {\pi } d e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} d e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {\sqrt {\pi } d e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} d e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {\sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5774
Rule 5780
Rule 5793
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2}{\sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 d e x^2}{\sqrt {a+b \text {arcsinh}(c x)}}+\frac {e^2 x^4}{\sqrt {a+b \text {arcsinh}(c x)}}\right ) \, dx \\ & = d^2 \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx+(2 d e) \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx+e^2 \int \frac {x^4}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx \\ & = \frac {d^2 \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 )}{b c}+\frac {(2 d e) \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 {e^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^5} \\ & = \frac {d^2 \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 )}{2 b c}+\frac {d^2 \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 )}{2 b c}+\frac {(2 d e) \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 {e^2 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}+\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^5} \\ & = \frac {d^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b c}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b c}+\frac {(d e) \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 )}{2 b c^3}-\frac {(d e) \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 )}{2 b c^3}+\frac {e^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^5}+\frac {e^2 \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 )}{8 b c^5}-\frac {\left (3 e^2\right ) \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 )}{16 b c^5} \\ & = \frac {d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {(d e) \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 )}{4 b c^3}-\frac {(d e) \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 )}{4 b c^3}+\frac {(d e) \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 )}{4 b c^3}+\frac {(d e) \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 )}{4 b c^3}+\frac {e^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^5}+\frac {e^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^5}+\frac {e^2 \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 )}{16 b c^5}+\frac {e^2 \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 )}{16 b c^5}-\frac {\left (3 e^2\right ) \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 )}{32 b c^5}-\frac {\left (3 e^2\right ) \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 )}{32 b c^5} \\ & = \frac {d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {(d e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^3}-\frac {(d e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^3}-\frac {(d e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^3}+\frac {(d e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^3}+\frac {e^2 \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{16 b c^5}+\frac {e^2 \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{16 b c^5}+\frac {e^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 b c^5}+\frac {e^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 b c^5}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{16 b c^5}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{16 b c^5} \\ & = \frac {d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e 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 )}{4 \sqrt {b} c^3}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e 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 )}{4 \sqrt {b} c^3}-\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {5 a}{b}} \left (-30 \left (8 c^4 d^2-4 c^2 d e+e^2\right ) e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \sqrt {5} e^2 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+40 \sqrt {3} c^2 d e e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-15 \sqrt {3} e^2 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+240 c^4 d^2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-120 c^2 d e e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+30 e^2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-40 \sqrt {3} c^2 d e e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+15 \sqrt {3} e^2 e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {5} e^2 e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{480 c^5 \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int \frac {\left (e \,x^{2}+d \right )^{2}}{\sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}}d x\]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {\left (d + e x^{2}\right )^{2}}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx \]
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\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}} \,d x \]
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