Integrand size = 35, antiderivative size = 416 \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 i b f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {1}{4} c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {2 i f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {5 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \]
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Time = 0.39 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {5796, 5838, 5785, 5783, 30, 5798, 5806, 5812} \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {1}{4} c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))+\frac {5 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {c^2 x^2+1}}-\frac {2 i f^2 \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 c}+\frac {3}{8} f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {3 b c f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {c^2 x^2+1}}+\frac {2 i b f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {c^2 x^2+1}}+\frac {2 i b c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {c^2 x^2+1}}+\frac {b c^3 f^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {c^2 x^2+1}} \]
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Rule 30
Rule 5783
Rule 5785
Rule 5796
Rule 5798
Rule 5806
Rule 5812
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int (f-i c f x)^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (\sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \left (f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-2 i c f^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-c^2 f^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (2 i c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (c^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{2} f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {1}{4} c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {2 i f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {\left (f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (2 i b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (c^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (b c^3 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x^3 \, dx}{4 \sqrt {1+c^2 x^2}} \\ & = \frac {2 i b f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {1}{4} c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {2 i f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}+\frac {\left (f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}+\frac {\left (b c f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {2 i b f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {1}{4} c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {2 i f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {5 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 2.57 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.36 \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {576 i b c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-768 i a f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+432 a c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-768 i a c^2 f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-288 a c^3 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+360 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2-144 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+9 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+720 a \sqrt {d} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+64 i b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+12 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-48 i \sqrt {1+c^2 x^2}-16 i \cosh (3 \text {arcsinh}(c x))+24 \sinh (2 \text {arcsinh}(c x))-3 \sinh (4 \text {arcsinh}(c x))\right )}{1152 c \sqrt {1+c^2 x^2}} \]
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\[\int \left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {i c d x +d}d x\]
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\[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { \sqrt {i \, c d x + d} {\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \]
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Exception generated. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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