Integrand size = 37, antiderivative size = 436 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=-\frac {4 i b^2 f^2 \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 f^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Time = 0.46 (sec) , antiderivative size = 436, 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, 5843, 3398, 3377, 2718, 3392, 32, 2715, 8} \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\frac {f^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 f^2 \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i b^2 f^2 \left (c^2 x^2+1\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 f^2 x \left (c^2 x^2+1\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rule 8
Rule 32
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)^2 (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))^2 \, dx,x,\text {arcsinh}(c x)\right )}{c^3 \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^2 f^2 (a+b x)^2-2 i c^2 f^2 (a+b x)^2 \sinh (x)-c^2 f^2 (a+b x)^2 \sinh ^2(x)\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {f^2 \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^2 \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 (f^2 \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 {b c f^2 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 f^2 \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 {f^2 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 {f^2 \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 (f^2 \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 (4 i b f^2 \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 (b^2 f^2 \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 {b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 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 {b c f^2 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 f^2 \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 {f^2 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 {f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (4 i b^2 f^2 \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 (b^2 f^2 \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 {4 i b^2 f^2 \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 f^2 \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 {4 i b f^2 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 {b c f^2 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 f^2 \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 {f^2 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 {f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ \end{align*}
Time = 8.52 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.22 \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\frac {32 i a b c f x \sqrt {d+i c d x} \sqrt {f-i c f x}-16 i a^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-32 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 a^2 c f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+4 b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3+2 a b f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+2 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (4 i \left (4 b c x+a (-4+i c x) \sqrt {1+c^2 x^2}\right )+b \cosh (2 \text {arcsinh}(c x))\right )+12 a^2 \sqrt {d} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+2 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (6 a-8 i b \sqrt {1+c^2 x^2}-b \sinh (2 \text {arcsinh}(c x))\right )}{8 c d \sqrt {1+c^2 x^2}} \]
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\[\int \frac {\left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\sqrt {i c d x +d}}d x\]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d}} \,d x } \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\int \frac {\left (- i f \left (c x + i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {i d \left (c x - i\right )}}\, dx \]
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Exception generated. \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\int { \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d}} \,d x } \]
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Timed out. \[ \int \frac {(f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \]
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