Integrand size = 16, antiderivative size = 327 \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=-\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{576 c^3} \]
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Time = 0.83 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5777, 5812, 5798, 5772, 5819, 3389, 2211, 2236, 2235, 3393} \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}-\frac {5 b x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {5 b \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rule 5772
Rule 5777
Rule 5798
Rule 5812
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {1}{6} (5 b c) \int \frac {x^3 (a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}+\frac {1}{12} \left (5 b^2\right ) \int x^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx+\frac {(5 b) \int \frac {x (a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {1+c^2 x^2}} \, dx}{9 c} \\ & = \frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {\left (5 b^2\right ) \int \sqrt {a+b \text {arcsinh}(c x)} \, dx}{6 c^2}-\frac {1}{72} \left (5 b^3 c\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx \\ & = -\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}+\frac {\left (5 b^2\right ) \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 )}{72 c^3}+\frac {\left (5 b^3\right ) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{12 c} \\ & = -\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}+\frac {\left (5 i b^2\right ) \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 )}{72 c^3}-\frac {\left (5 b^2\right ) \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 )}{12 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}+\frac {\left (5 b^2\right ) \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 )}{288 c^3}-\frac {\left (5 b^2\right ) \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 )}{96 c^3}-\frac {\left (5 b^2\right ) \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 )}{24 c^3}+\frac {\left (5 b^2\right ) \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 )}{24 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}+\frac {\left (5 b^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 )}{576 c^3}-\frac {\left (5 b^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 )}{576 c^3}-\frac {\left (5 b^2\right ) \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 )}{192 c^3}+\frac {\left (5 b^2\right ) \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 )}{192 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{12 c^3}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{12 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {5 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\left (5 b^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 )}{288 c^3}-\frac {\left (5 b^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 )}{288 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{96 c^3}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{96 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 b^{5/2} 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 )}{576 c^3}+\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} 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 )}{576 c^3} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.61 \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=-\frac {b^3 e^{-\frac {3 a}{b}} \left (-81 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {7}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {7}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-81 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {7}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {7}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{648 c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\int x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2} \,d x } \]
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Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2} \,d x \]
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