Integrand size = 35, antiderivative size = 187 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\frac {2 i b f^3 \left (1+c^2 x^2\right )^{5/2}}{3 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f^3 (1-i c x)^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f^3 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5796, 665, 5837, 12, 641, 45} \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\frac {i f^3 (1-i c x)^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b f^3 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f^3 \left (c^2 x^2+1\right )^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rule 12
Rule 45
Rule 641
Rule 665
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)^3 (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 {i f^3 (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 {\left (b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {i f^3 (1-i c x)^3}{3 c \left (1+c^2 x^2\right )^2} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {i f^3 (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 {\left (i b f^3 \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 {i f^3 (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 {\left (i b f^3 \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 {i f^3 (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 {\left (i b f^3 \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 {2 i b f^3 \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 {i f^3 (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 {b f^3 \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 = 1.51 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\left ((i+c x) \left (-i b+b c x+a \sqrt {1+c^2 x^2}\right )\right )-b (i+c x) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b (-i+c x)^2 \log (d+i c d x)\right )}{3 c d^3 (-i+c x)^2 \sqrt {1+c^2 x^2}} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {-i c f x +f}}{\left (i c d x +d \right )^{\frac {5}{2}}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (142) = 284\).
Time = 0.32 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.93 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=-\frac {4 \, \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b c x + 2 \, {\left (b c^{2} x^{2} + 2 i \, b c x - b\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (c^{4} d^{3} x^{3} - i \, c^{3} d^{3} x^{2} + c^{2} d^{3} x - i \, c d^{3}\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}} \log \left (-\frac {{\left (i \, b c^{6} x^{2} + 2 \, b c^{5} x - 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} + {\left (i \, c^{9} d^{3} x^{4} + 2 \, c^{8} d^{3} x^{3} + i \, c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}}}{8 \, {\left (b c^{3} x^{3} - i \, b c^{2} x^{2} + b c x - i \, b\right )}}\right ) + {\left (c^{4} d^{3} x^{3} - i \, c^{3} d^{3} x^{2} + c^{2} d^{3} x - i \, c d^{3}\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}} \log \left (-\frac {{\left (i \, b c^{6} x^{2} + 2 \, b c^{5} x - 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} + {\left (-i \, c^{9} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{3} - i \, c^{7} d^{3} x^{2} - 2 \, c^{6} d^{3} x\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}}}{8 \, {\left (b c^{3} x^{3} - i \, b c^{2} x^{2} + b c x - i \, b\right )}}\right ) + 2 \, {\left (a c^{2} x^{2} + 2 i \, a c x - a\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{6 \, {\left (c^{4} d^{3} x^{3} - i \, c^{3} d^{3} x^{2} + c^{2} d^{3} x - i \, c d^{3}\right )}} \]
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\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int \frac {\sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (i d \left (c x - i\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\frac {1}{3} \, b c {\left (\frac {6 \, \sqrt {f}}{3 i \, c^{3} d^{\frac {5}{2}} x + 3 \, c^{2} d^{\frac {5}{2}}} + \frac {\sqrt {f} \log \left (c x - i\right )}{c^{2} d^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {2 i \, \sqrt {c^{2} d f x^{2} + d f}}{c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}} + \frac {3 i \, \sqrt {c^{2} d f x^{2} + d f}}{3 i \, c^{2} d^{3} x + 3 \, c d^{3}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 i \, \sqrt {c^{2} d f x^{2} + d f}}{c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}} + \frac {3 i \, \sqrt {c^{2} d f x^{2} + d f}}{3 i \, c^{2} d^{3} x + 3 \, c d^{3}}\right )} \]
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\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{5/2}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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