Integrand size = 16, antiderivative size = 271 \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3} \]
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Time = 0.59 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5779, 5818, 5780, 5556, 3388, 2211, 2236, 2235, 5774} \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {\sqrt {3 \pi } e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}-\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {\sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5774
Rule 5779
Rule 5780
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {4 \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}} \, dx}{3 b c}+\frac {(2 c) \int \frac {x^3}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}} \, dx}{b} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {12 \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b^2}+\frac {8 \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {8 \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 )}{3 b^3 c^3}+\frac {12 \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^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {4 \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 )}{3 b^3 c^3}+\frac {4 \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 )}{3 b^3 c^3}+\frac {12 \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^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {8 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{3 b^3 c^3}+\frac {8 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{3 b^3 c^3}+\frac {3 \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 )}{b^3 c^3}-\frac {3 \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^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}-\frac {3 \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^3 c^3}-\frac {3 \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^3 c^3}+\frac {3 \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 )}{2 b^3 c^3}+\frac {3 \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 )}{2 b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {3 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^3 c^3}-\frac {3 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^3 c^3}-\frac {3 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^3 c^3}+\frac {3 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.25 \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\frac {e^{-3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (2 e^{\frac {4 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )-6 \sqrt {3} b e^{3 \text {arcsinh}(c x)} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+2 b e^{\frac {2 a}{b}+3 \text {arcsinh}(c x)} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-e^{\frac {3 a}{b}} \left (\left (-1+e^{2 \text {arcsinh}(c x)}\right ) \left (b \left (-1+e^{4 \text {arcsinh}(c x)}\right )+a \left (6+4 e^{2 \text {arcsinh}(c x)}+6 e^{4 \text {arcsinh}(c x)}\right )+2 b \left (3+2 e^{2 \text {arcsinh}(c x)}+3 e^{4 \text {arcsinh}(c x)}\right ) \text {arcsinh}(c x)\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{12 b^2 c^3 (a+b \text {arcsinh}(c x))^{3/2}} \]
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\[\int \frac {x^{2}}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2}} \,d x \]
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