Integrand size = 37, antiderivative size = 680 \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {8 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {15}{64} b^2 d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {4 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {15 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {1+c^2 x^2}} \]
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Time = 0.73 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {5796, 5838, 5785, 5783, 5776, 327, 221, 5798, 5784, 455, 45, 5806, 5812} \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {c^2 x^2+1}}-\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {c^2 x^2+1}}+\frac {2 i d^2 \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {c^2 x^2+1}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {c^2 x^2+1}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {15 b^2 d^2 \text {arcsinh}(c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}{64 c \sqrt {c^2 x^2+1}}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {4 i b^2 d^2 \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x}}{27 c}+\frac {15}{64} b^2 d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {8 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c} \]
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Rule 45
Rule 221
Rule 327
Rule 455
Rule 5776
Rule 5783
Rule 5784
Rule 5785
Rule 5796
Rule 5798
Rule 5806
Rule 5812
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int (d+i c d x)^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (\sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \left (d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2+2 i c d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2-c^2 d^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (2 i c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (c^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{2} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {\left (d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (4 i b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (c^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (b c^3 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x^3 (a+b \text {arcsinh}(c x)) \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}+\frac {\left (b c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (4 i b^2 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x \left (1+\frac {c^2 x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (2 i b^2 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \text {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{32 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {15}{64} b^2 d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{4 c \sqrt {1+c^2 x^2}}-\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (2 i b^2 c d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+c^2 x}}+\frac {1}{3} \sqrt {1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}} \\ & = \frac {8 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {15}{64} b^2 d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {4 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {15 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {4 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {4 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 3.60 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.31 \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {-6912 i a b c d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}+4608 i a^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+6912 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2592 a^2 c d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+4608 i a^2 c^2 d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-1728 a^2 c^3 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1440 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-1728 a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+256 i b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+108 a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+4320 a^2 d^{5/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+864 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))-768 i a b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))-27 b^2 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (4 \text {arcsinh}(c x))+12 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-576 i b c x+576 i a \sqrt {1+c^2 x^2}-144 b \cosh (2 \text {arcsinh}(c x))+192 i a \cosh (3 \text {arcsinh}(c x))+9 b \cosh (4 \text {arcsinh}(c x))+288 a \sinh (2 \text {arcsinh}(c x))-64 i b \sinh (3 \text {arcsinh}(c x))-36 a \sinh (4 \text {arcsinh}(c x))\right )+72 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (60 a+48 i b \sqrt {1+c^2 x^2}+16 i b \cosh (3 \text {arcsinh}(c x))+24 b \sinh (2 \text {arcsinh}(c x))-3 b \sinh (4 \text {arcsinh}(c x))\right )}{6912 c \sqrt {1+c^2 x^2}} \]
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\[\int \left (i c d x +d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {-i c f x +f}d x\]
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\[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {5}{2}} \sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \]
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Exception generated. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}} \,d x \]
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