Integrand size = 37, antiderivative size = 615 \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=-\frac {68 i b^2 f^3 \left (1+c^2 x^2\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 f^3 \left (1+c^2 x^2\right )^2}{27 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b^2 f^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {22 i b f^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Time = 0.54 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {5796, 5843, 3398, 3377, 2718, 3392, 32, 2715, 8, 2713} \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\frac {5 f^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {22 i b f^3 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b^2 f^3 \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 f^3 \left (c^2 x^2+1\right )^2}{27 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 b^2 f^3 x \left (c^2 x^2+1\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {68 i b^2 f^3 \left (c^2 x^2+1\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rule 8
Rule 32
Rule 2713
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 5796
Rule 5843
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} \int \frac {(f-i c f x)^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\sqrt {1+c^2 x^2} \text {Subst}\left (\int (a+b x)^2 (c f-i c f \sinh (x))^3 \, dx,x,\text {arcsinh}(c x)\right )}{c^4 \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\sqrt {1+c^2 x^2} \text {Subst}\left (\int \left (c^3 f^3 (a+b x)^2-3 i c^3 f^3 (a+b x)^2 \sinh (x)-3 c^3 f^3 (a+b x)^2 \sinh ^2(x)+i c^3 f^3 (a+b x)^2 \sinh ^3(x)\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^4 \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (i f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sinh ^3(x) \, dx,x,\text {arcsinh}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (3 i f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text {arcsinh}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (3 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\text {arcsinh}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {3 b c f^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 i f^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text {arcsinh}(c x)\right )}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (3 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\text {arcsinh}(c x)\right )}{2 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (6 i b f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \cosh (x) \, dx,x,\text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (2 i b^2 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \sinh ^3(x) \, dx,x,\text {arcsinh}(c x)\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (3 b^2 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {arcsinh}(c x)\right )}{2 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {6 i b f^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (4 i b f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \cosh (x) \, dx,x,\text {arcsinh}(c x))}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i b^2 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (6 i b^2 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int \sinh (x) \, dx,x,\text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (3 b^2 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int 1 \, dx,x,\text {arcsinh}(c x))}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {56 i b^2 f^3 \left (1+c^2 x^2\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 f^3 \left (1+c^2 x^2\right )^2}{27 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b^2 f^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {22 i b f^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (4 i b^2 f^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int \sinh (x) \, dx,x,\text {arcsinh}(c x))}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {68 i b^2 f^3 \left (1+c^2 x^2\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 f^3 \left (1+c^2 x^2\right )^2}{27 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b^2 f^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {22 i b f^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ \end{align*}
Time = 14.05 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.18 \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\frac {1620 i a b c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-792 i a^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-1620 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-324 a^2 c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+72 i a^2 c^2 f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+180 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3+162 a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+4 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+6 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (27 b \cosh (2 \text {arcsinh}(c x))+2 i \left (-4 b c x \left (-33+c^2 x^2\right )+27 a (-5+2 i c x) \sqrt {1+c^2 x^2}+3 a \cosh (3 \text {arcsinh}(c x))\right )\right )+540 a^2 \sqrt {d} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-81 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+18 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (30 a-45 i b \sqrt {1+c^2 x^2}+i b \cosh (3 \text {arcsinh}(c x))-9 b \sinh (2 \text {arcsinh}(c x))\right )-12 i a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))}{216 c d \sqrt {1+c^2 x^2}} \]
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\[\int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\sqrt {i c d x +d}}d x\]
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\[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d}} \,d x } \]
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Timed out. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \]
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