Integrand size = 35, antiderivative size = 282 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {i b d \left (1+c^2 x^2\right )^{5/2}}{6 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (i-c x) \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (1+c^2 x^2\right )^{5/2} \arctan (c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {5796, 653, 197, 5837, 641, 46, 209, 266} \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {2 d x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (-c x+i) \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (c^2 x^2+1\right )^{5/2} \arctan (c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (c^2 x^2+1\right )^{5/2}}{6 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (c^2 x^2+1\right )^{5/2} \log \left (c^2 x^2+1\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rule 46
Rule 197
Rule 209
Rule 266
Rule 641
Rule 653
Rule 5796
Rule 5837
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(d+i c d x) (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 {d (i-c x) \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 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (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 {d (i-c x)}{3 c \left (1+c^2 x^2\right )^2}+\frac {2 d x}{3 \left (1+c^2 x^2\right )}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {d (i-c x) \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 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {i-c x}{\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 b c d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {d (i-c x) \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 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{(-i-c x)^2 (i-c x)} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {d (i-c x) \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 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {i}{2 (i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {i b d \left (1+c^2 x^2\right )^{5/2}}{6 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (i-c x) \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 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (i b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {i b d \left (1+c^2 x^2\right )^{5/2}}{6 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (i-c x) \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 d x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (1+c^2 x^2\right )^{5/2} \arctan (c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {\sqrt {f-i c f x} \left (4 i a-8 a c x+8 i a c^2 x^2-2 b \sqrt {1+c^2 x^2}+4 i b \left (1+2 i c x+2 c^2 x^2\right ) \text {arcsinh}(c x)+5 b (1-i c x) \sqrt {1+c^2 x^2} \log (d (-1+i c x))+3 b \sqrt {1+c^2 x^2} \log (d+i c d x)-3 i b c x \sqrt {1+c^2 x^2} \log (d+i c d x)\right )}{12 c d f^3 (i+c x)^2 \sqrt {d+i c d x}} \]
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\[\int \frac {a +b \,\operatorname {arcsinh}\left (c x \right )}{\left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (-i \, c f x + f\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\text {Timed out} \]
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none
Time = 0.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {1}{12} \, b c {\left (\frac {2 i \, \sqrt {d} \sqrt {f}}{c^{3} d^{2} f^{3} x + i \, c^{2} d^{2} f^{3}} - \frac {5 \, \log \left (c x + i\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}} - \frac {3 \, \log \left (c x - i\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {3 i}{-3 i \, \sqrt {c^{2} d f x^{2} + d f} c^{2} d f^{2} x + 3 \, \sqrt {c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {c^{2} d f x^{2} + d f} d f^{2}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {3 i}{-3 i \, \sqrt {c^{2} d f x^{2} + d f} c^{2} d f^{2} x + 3 \, \sqrt {c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {c^{2} d f x^{2} + d f} d f^{2}}\right )} \]
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Exception generated. \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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