Integrand size = 14, antiderivative size = 135 \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 x \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2} \]
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Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5778, 3388, 2211, 2236, 2235} \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5778
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {2 x \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2}+\frac {\text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {2 x \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}+\frac {2 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2} \\ & = -\frac {2 x \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {2 a}{b}} \left (\sqrt {2} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {2} e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-2 e^{\frac {2 a}{b}} \sinh (2 \text {arcsinh}(c x))\right )}{2 b c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int \frac {x}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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