Integrand size = 35, antiderivative size = 266 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\frac {2 i b f^2 x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {1+c^2 x^2}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Time = 0.33 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5796, 5838, 5783, 5798, 8, 5812, 30} \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\frac {3 f^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {c^2 x^2+1}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b f^2 x \sqrt {c^2 x^2+1}}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rule 8
Rule 30
Rule 5783
Rule 5796
Rule 5798
Rule 5812
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} \int \frac {(f-i c f x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\sqrt {1+c^2 x^2} \int \left (\frac {f^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}-\frac {2 i c f^2 x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}-\frac {c^2 f^2 x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\left (f^2 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i c f^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (c^2 f^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {2 i f^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (f^2 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (2 i b f^2 \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (b c f^2 \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {2 i b f^2 x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {1+c^2 x^2}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ \end{align*}
Time = 5.47 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.29 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\frac {16 i b c f x \sqrt {d+i c d x} \sqrt {f-i c f x}-16 i a f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 a c f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 b f (4 i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+6 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2+b f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+12 a \sqrt {d} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{8 c d \sqrt {1+c^2 x^2}} \]
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\[\int \frac {\left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{\sqrt {i c d x +d}}d x\]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {i \, c d x + d}} \,d x } \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int \frac {\left (- i f \left (c x + i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {i d \left (c x - i\right )}}\, dx \]
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Exception generated. \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {i \, c d x + d}} \,d x } \]
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Timed out. \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \]
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