Integrand size = 35, antiderivative size = 364 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\frac {4 i b f^4 \left (1+c^2 x^2\right )^{5/2}}{3 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^4 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Time = 0.27 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5796, 683, 667, 221, 5837, 641, 45, 31, 5783} \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=-\frac {2 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (c^2 x^2+1\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^4 \left (c^2 x^2+1\right )^{5/2} \text {arcsinh}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b f^4 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^4 \left (c^2 x^2+1\right )^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rule 31
Rule 45
Rule 221
Rule 641
Rule 667
Rule 683
Rule 5783
Rule 5796
Rule 5837
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(f-i c f x)^4 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (\frac {2 i f^4 (1-i c x)^3}{3 c \left (1+c^2 x^2\right )^2}-\frac {2 i f^4 (1-i c x)}{c \left (1+c^2 x^2\right )}+\frac {f^4 \text {arcsinh}(c x)}{c \sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(1-i c x)^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1-i c x}{1+c^2 x^2} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {\text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1-i c x}{(1+i c x)^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{1+i c x} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b f^4 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {2}{(-i+c x)^2}+\frac {i}{-i+c x}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {4 i b f^4 \left (1+c^2 x^2\right )^{5/2}}{3 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^4 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}
Time = 9.81 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.94 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\frac {-\frac {16 a f (-i+2 c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}{d^3 (-i+c x)^2}+\frac {12 a f^{3/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{d^{5/2}}-\frac {b f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right ) \left (\cosh \left (\frac {3}{2} \text {arcsinh}(c x)\right ) \left ((-14+3 i \text {arcsinh}(c x)) \text {arcsinh}(c x)-28 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+7 i \log \left (1+c^2 x^2\right )\right )+\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (84 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-i \left (8-6 i \text {arcsinh}(c x)+9 \text {arcsinh}(c x)^2+21 \log \left (1+c^2 x^2\right )\right )\right )+2 \left (4-4 i \text {arcsinh}(c x)+6 \text {arcsinh}(c x)^2+56 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+14 \log \left (1+c^2 x^2\right )+\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x) (-14 i+3 \text {arcsinh}(c x))+28 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+7 \log \left (1+c^2 x^2\right )\right )\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{d^3 (i+c x) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^4}+\frac {2 i b f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right ) \left (-i \cosh \left (\frac {3}{2} \text {arcsinh}(c x)\right ) \left (\text {arcsinh}(c x)-2 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-\frac {1}{2} i \log \left (1+c^2 x^2\right )\right )+\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (4+3 i \text {arcsinh}(c x)-6 i \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\frac {3}{2} \log \left (1+c^2 x^2\right )\right )+2 \left (\left (2+\sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)+2 \left (2+\sqrt {1+c^2 x^2}\right ) \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\frac {1}{2} i \left (4+\left (2+\sqrt {1+c^2 x^2}\right ) \log \left (1+c^2 x^2\right )\right )\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{d^3 (i+c x) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^4}}{12 c} \]
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\[\int \frac {\left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{\left (i c d x +d \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int \frac {\left (- i f \left (c x + i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (i d \left (c x - i\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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