Integrand size = 35, antiderivative size = 295 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\frac {i b f^2 \left (1+c^2 x^2\right )^{5/2}}{3 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^2 (1-i c x) \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f^2 \left (1+c^2 x^2\right )^{5/2} \arctan (c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^2 \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 295, 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, 667, 197, 5837, 641, 46, 209, 266} \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\frac {f^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^2 (1-i c x) \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f^2 \left (c^2 x^2+1\right )^{5/2} \arctan (c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b f^2 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^2 \left (c^2 x^2+1\right )^{5/2} \log \left (c^2 x^2+1\right )}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rule 46
Rule 197
Rule 209
Rule 266
Rule 641
Rule 667
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)^2 (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^2 (1-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 {f^2 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 {2 i f^2 (1-i c x)}{3 c \left (1+c^2 x^2\right )^2}+\frac {f^2 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 {2 i f^2 (1-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 {f^2 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 (2 i b f^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1-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 (b c f^2 \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 {2 i f^2 (1-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 {f^2 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 f^2 \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{(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 {2 i f^2 (1-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 {f^2 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 f^2 \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {1}{2 (-i+c x)^2}+\frac {1}{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 f^2 \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 {2 i f^2 (1-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 {f^2 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 f^2 \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i b f^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {i b f^2 \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 {2 i f^2 (1-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 {f^2 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 f^2 \left (1+c^2 x^2\right )^{5/2} \arctan (c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^2 \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.48 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left ((-2 i+c x) \left (-i b+b c x+a \sqrt {1+c^2 x^2}\right )+b (-2 i+c x) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-b (-i+c x)^2 \log (d+i c d x)\right )}{3 c d^3 f (-i+c x)^2 \sqrt {1+c^2 x^2}} \]
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\[\int \frac {a +b \,\operatorname {arcsinh}\left (c x \right )}{\left (i c d x +d \right )^{\frac {5}{2}} \sqrt {-i c f x +f}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (228) = 456\).
Time = 0.34 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=-\frac {2 \, \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b c x - 2 \, {\left (b c^{2} x^{2} - i \, b c x + 2 \, b\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (c^{4} d^{3} f x^{3} - i \, c^{3} d^{3} f x^{2} + c^{2} d^{3} f x - i \, c d^{3} f\right )} \sqrt {\frac {b^{2}}{c^{2} d^{5} f}} \log \left (-\frac {{\left (i \, b c^{6} x^{2} + 2 \, b c^{5} x - 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} + {\left (i \, c^{9} d^{3} f x^{4} + 2 \, c^{8} d^{3} f x^{3} + i \, c^{7} d^{3} f x^{2} + 2 \, c^{6} d^{3} f x\right )} \sqrt {\frac {b^{2}}{c^{2} d^{5} f}}}{8 \, {\left (b c^{3} x^{3} - i \, b c^{2} x^{2} + b c x - i \, b\right )}}\right ) - {\left (c^{4} d^{3} f x^{3} - i \, c^{3} d^{3} f x^{2} + c^{2} d^{3} f x - i \, c d^{3} f\right )} \sqrt {\frac {b^{2}}{c^{2} d^{5} f}} \log \left (-\frac {{\left (i \, b c^{6} x^{2} + 2 \, b c^{5} x - 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} + {\left (-i \, c^{9} d^{3} f x^{4} - 2 \, c^{8} d^{3} f x^{3} - i \, c^{7} d^{3} f x^{2} - 2 \, c^{6} d^{3} f x\right )} \sqrt {\frac {b^{2}}{c^{2} d^{5} f}}}{8 \, {\left (b c^{3} x^{3} - i \, b c^{2} x^{2} + b c x - i \, b\right )}}\right ) - 2 \, {\left (a c^{2} x^{2} - i \, a c x + 2 \, a\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{6 \, {\left (c^{4} d^{3} f x^{3} - i \, c^{3} d^{3} f x^{2} + c^{2} d^{3} f x - i \, c d^{3} f\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (i d \left (c x - i\right )\right )^{\frac {5}{2}} \sqrt {- i f \left (c x + i\right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\frac {1}{3} \, b c {\left (\frac {3}{3 i \, c^{3} d^{\frac {5}{2}} \sqrt {f} x + 3 \, c^{2} d^{\frac {5}{2}} \sqrt {f}} - \frac {\log \left (c x - i\right )}{c^{2} d^{\frac {5}{2}} \sqrt {f}}\right )} - \frac {1}{3} \, b {\left (\frac {i \, \sqrt {c^{2} d f x^{2} + d f}}{c^{3} d^{3} f x^{2} - 2 i \, c^{2} d^{3} f x - c d^{3} f} - \frac {3 i \, \sqrt {c^{2} d f x^{2} + d f}}{3 i \, c^{2} d^{3} f x + 3 \, c d^{3} f}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {i \, \sqrt {c^{2} d f x^{2} + d f}}{c^{3} d^{3} f x^{2} - 2 i \, c^{2} d^{3} f x - c d^{3} f} - \frac {3 i \, \sqrt {c^{2} d f x^{2} + d f}}{3 i \, c^{2} d^{3} f x + 3 \, c d^{3} f}\right )} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{\frac {5}{2}} \sqrt {-i \, c f x + f}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{5/2} \sqrt {f-i c f x}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \]
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