Integrand size = 12, antiderivative size = 143 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c} \]
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Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5773, 5818, 5774, 3388, 2211, 2236, 2235} \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\frac {2 \sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 \sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 \sqrt {c^2 x^2+1}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5773
Rule 5774
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {(2 c) \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}} \, dx}{3 b} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {4 \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{3 b^2} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {4 \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} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 \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}+\frac {2 \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} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {4 \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}+\frac {4 \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} \\ & = -\frac {2 \sqrt {1+c^2 x^2}}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 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} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\frac {e^{-\frac {a+b \text {arcsinh}(c x)}{b}} \left (-e^{a/b} \left (b+2 a \left (-1+e^{2 \text {arcsinh}(c x)}\right )-2 b \text {arcsinh}(c x)+b e^{2 \text {arcsinh}(c x)} (1+2 \text {arcsinh}(c x))\right )-2 e^{\frac {2 a}{b}+\text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )-2 b e^{\text {arcsinh}(c x)} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )}{3 b^2 c (a+b \text {arcsinh}(c x))^{3/2}} \]
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\[\int \frac {1}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2}} \,d x \]
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