Integrand size = 37, antiderivative size = 396 \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac {15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac {9 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \text {arcsinh}(c x)}{64 c \left (1+c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac {b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{8 \left (1+c^2 x^2\right )}+\frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c \left (1+c^2 x^2\right )^{3/2}} \]
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Time = 0.33 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {5796, 5786, 5785, 5783, 5776, 327, 221, 5798, 201} \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c \left (c^2 x^2+1\right )^{3/2}}+\frac {3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{8 \left (c^2 x^2+1\right )}-\frac {b \sqrt {c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {9 b^2 \text {arcsinh}(c x) (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 c \left (c^2 x^2+1\right )^{3/2}}+\frac {15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (c^2 x^2+1\right )}+\frac {1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \]
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Rule 201
Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5796
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx}{\left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\left (3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx}{2 \left (1+c^2 x^2\right )^{3/2}} \\ & = -\frac {b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{8 \left (1+c^2 x^2\right )}+\frac {\left (3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}-\frac {3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac {b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{8 \left (1+c^2 x^2\right )}+\frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{32 \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 c^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac {15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac {3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac {b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{8 \left (1+c^2 x^2\right )}+\frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac {15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac {9 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \text {arcsinh}(c x)}{64 c \left (1+c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac {b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{8 \left (1+c^2 x^2\right )}+\frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c \left (1+c^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 3.19 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.32 \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {160 a^2 c d f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+64 a^2 c^3 d f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+32 b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-64 a b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-4 a b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+96 a^2 d^{3/2} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+32 b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (4 \text {arcsinh}(c x))+8 b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 (12 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))-4 b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) (16 b \cosh (2 \text {arcsinh}(c x))+b \cosh (4 \text {arcsinh}(c x))-4 a (8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))))}{256 c \sqrt {1+c^2 x^2}} \]
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\[\int \left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}d x\]
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\[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \]
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Exception generated. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
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