Integrand size = 12, antiderivative size = 247 \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=-\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4} \]
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Time = 0.48 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5777, 5812, 5783, 5819, 3393, 3388, 2211, 2235, 2236} \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}-\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}-\frac {5 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {15 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{64 a^3}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5777
Rule 5783
Rule 5812
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {1}{8} (5 a) \int \frac {x^4 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}+\frac {15}{64} \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx+\frac {15 \int \frac {x^2 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{32 a} \\ & = \frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3}-\frac {45 \int x \sqrt {\text {arcsinh}(a x)} \, dx}{128 a^2}-\frac {1}{512} (15 a) \int \frac {x^4}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\sinh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}+\frac {45 \int \frac {x^2}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx}{512 a} \\ & = -\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}+\frac {45 \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4} \\ & = -\frac {45 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4096 a^4}+\frac {15 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1024 a^4}-\frac {45 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8192 a^4}-\frac {15 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8192 a^4}+\frac {15 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4}+\frac {15 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4}+\frac {45 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1024 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4096 a^4}-\frac {15 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4096 a^4}+\frac {15 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4}+\frac {15 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4}+\frac {45 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4}+\frac {45 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}+\frac {45 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4}+\frac {45 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.40 \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-4 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {16 \sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-16 \sqrt {2} \Gamma \left (\frac {7}{2},2 \text {arcsinh}(a x)\right )+\Gamma \left (\frac {7}{2},4 \text {arcsinh}(a x)\right )}{2048 a^4} \]
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\[\int x^{3} \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\int x^{3} \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
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\[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\int { x^{3} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\int x^3\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]
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