Integrand size = 8, antiderivative size = 94 \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a} \]
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Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5772, 5798, 5819, 3389, 2211, 2235, 2236} \[ \int \text {arcsinh}(a x)^{5/2} \, dx=-\frac {5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15}{4} x \sqrt {\text {arcsinh}(a x)} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5772
Rule 5798
Rule 5819
Rubi steps \begin{align*} \text {integral}& = x \text {arcsinh}(a x)^{5/2}-\frac {1}{2} (5 a) \int \frac {x \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15}{4} \int \sqrt {\text {arcsinh}(a x)} \, dx \\ & = \frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}-\frac {1}{8} (15 a) \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = \frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a} \\ & = \frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a}-\frac {15 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a} \\ & = \frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a}-\frac {15 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a} \\ & = \frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.48 \[ \int \text {arcsinh}(a x)^{5/2} \, dx=-\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\Gamma \left (\frac {7}{2},\text {arcsinh}(a x)\right )}{2 a} \]
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Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {-16 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, a x +40 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}-60 \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, a x -15 \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+15 \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )}{16 \sqrt {\pi }\, a}\) | \(78\) |
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Exception generated. \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
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\[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int { \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]
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