Integrand size = 20, antiderivative size = 322 \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {\sqrt {b} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {\sqrt {b} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{48 c^3} \]
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Time = 0.61 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5793, 5772, 5819, 3389, 2211, 2236, 2235, 5777, 3393} \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=-\frac {\sqrt {\pi } \sqrt {b} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{48 c^3}+\frac {\sqrt {\pi } \sqrt {b} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{48 c^3}+\frac {\sqrt {\pi } \sqrt {b} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rule 5772
Rule 5777
Rule 5793
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \int \left (d \sqrt {a+b \text {arcsinh}(c x)}+e x^2 \sqrt {a+b \text {arcsinh}(c x)}\right ) \, dx \\ & = d \int \sqrt {a+b \text {arcsinh}(c x)} \, dx+e \int x^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} (b c d) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx-\frac {1}{6} (b c e) \int \frac {x^3}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {d \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 c}+\frac {e \text {Subst}\left (\int \frac {\sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{6 c^3} \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {d \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 c}-\frac {d \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 c}+\frac {(i e) \text {Subst}\left (\int \left (-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {3 i \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{6 c^3} \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {d \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}-\frac {d \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}+\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{24 c^3}-\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 c^3} \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {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 )}{48 c^3}-\frac {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 )}{48 c^3}-\frac {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 )}{16 c^3}+\frac {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 )}{16 c^3} \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {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 )}{24 c^3}-\frac {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 )}{24 c^3}-\frac {e \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 c^3}+\frac {e \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 c^3} \\ & = d x \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {\sqrt {b} 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 )}{48 c^3}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {\sqrt {b} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} 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 )}{48 c^3} \\ \end{align*}
Time = 2.10 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=\frac {d e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}}\right )}{2 c}+\frac {e e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}} \]
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\[\int \left (e \,x^{2}+d \right ) \sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}d x\]
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Exception generated. \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \left (d + e x^{2}\right )\, dx \]
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\[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int { {\left (e x^{2} + d\right )} \sqrt {b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int { {\left (e x^{2} + d\right )} \sqrt {b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int \sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}\,\left (e\,x^2+d\right ) \,d x \]
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