Integrand size = 23, antiderivative size = 182 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5790, 5778, 3388, 2211, 2235, 2236} \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {2 \sqrt {2 \pi } \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}-\frac {8 x \sqrt {a^2 c x^2+c}}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{3 a \text {arcsinh}(a x)^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5778
Rule 5790
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {\left (4 a \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx}{3 \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {\left (8 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {\left (4 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (4 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {\left (8 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (8 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {c+a^2 c x^2} \left (1+a^2 x^2+4 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+\sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )}{3 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \]
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\[\int \frac {\sqrt {a^{2} c \,x^{2}+c}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {c\,a^2\,x^2+c}}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]
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