Integrand size = 35, antiderivative size = 518 \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=-\frac {3 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Time = 0.28 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5796, 683, 685, 655, 221, 5837, 641, 45, 5783} \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\frac {5 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (c^2 x^2+1\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (c^2 x^2+1\right )^{3/2} \text {arcsinh}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (c^2 x^2+1\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {3 i b f^4 x \left (c^2 x^2+1\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rule 45
Rule 221
Rule 641
Rule 655
Rule 683
Rule 685
Rule 5783
Rule 5796
Rule 5837
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(f-i c f x)^4 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (\frac {15 i f^4}{2 c}+\frac {5 i f^4 (1-i c x)}{2 c}+\frac {2 i f^4 (1-i c x)^3}{c \left (1+c^2 x^2\right )}-\frac {15 f^4 \text {arcsinh}(c x)}{2 c \sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(1-i c x)^3}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (15 b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {\text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(1-i c x)^2}{1+i c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (-3+i c x+\frac {4}{1+i c x}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {3 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) (a+b \text {arcsinh}(c x))}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ \end{align*}
Time = 9.86 (sec) , antiderivative size = 779, normalized size of antiderivative = 1.50 \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\frac {\frac {4 a f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (24+7 i c x+c^2 x^2\right )}{d^2 (-i+c x)}-\frac {60 a f^{5/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{d^{3/2}}-\frac {4 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\text {arcsinh}(c x) \left (-4 i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-4 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \left (4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\log \left (1+c^2 x^2\right )\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{d^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}+\frac {16 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\left (c x-4 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-i \log \left (1+c^2 x^2\right )\right ) \left (-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\text {arcsinh}(c x) \left (i \left (2+\sqrt {1+c^2 x^2}\right ) \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\left (-2+\sqrt {1+c^2 x^2}\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{d^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}+\frac {b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-10 \text {arcsinh}(c x)^2 \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 (2 \text {arcsinh}(c x))+8 \left (2 i c x+4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\log \left (1+c^2 x^2\right )\right )\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \text {arcsinh}(c x) \left (\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (8-8 \sqrt {1+c^2 x^2}+i \sinh (2 \text {arcsinh}(c x))\right )+\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (8 i \left (1+\sqrt {1+c^2 x^2}\right )+\sinh (2 \text {arcsinh}(c x))\right )\right )\right )}{d^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}}{8 c} \]
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\[\int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{\left (i c d x +d \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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