Integrand size = 37, antiderivative size = 508 \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {4 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {1}{4} b^2 d x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {2 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{4 c \sqrt {1+c^2 x^2}}-\frac {2 i b d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}-\frac {2 i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {1+c^2 x^2}} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {5796, 5838, 5785, 5783, 5776, 327, 221, 5798, 5784, 455, 45} \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{2 \sqrt {c^2 x^2+1}}-\frac {2 i b d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}+\frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {i d \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {2 i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {c^2 x^2+1}}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {b^2 d \text {arcsinh}(c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 c \sqrt {c^2 x^2+1}}+\frac {2 i b^2 d \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x}}{27 c}+\frac {1}{4} b^2 d x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {4 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c} \]
[In]
[Out]
Rule 45
Rule 221
Rule 327
Rule 455
Rule 5776
Rule 5783
Rule 5784
Rule 5785
Rule 5796
Rule 5798
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) \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 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2+i c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d \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 (i c d \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 {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \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 \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 (2 i b d \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 \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 {2 i b d 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 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 {2 i b c^2 d 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 {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \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 \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 (2 i b^2 c d \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 \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 {1}{4} b^2 d x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {2 i b d 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 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 {2 i b c^2 d 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 {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \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 \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 (b^2 d \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 (i b^2 c d \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 {1}{4} b^2 d x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {b^2 d \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 {2 i b d 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 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 {2 i b c^2 d 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 {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \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 \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 (i b^2 c d \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 {4 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {1}{4} b^2 d x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {2 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {b^2 d \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 {2 i b d 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 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 {2 i b c^2 d 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 {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \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 \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}} \\ \end{align*}
Time = 3.23 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.39 \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {-108 i a b c d x \sqrt {d+i c d x} \sqrt {f-i c f x}+72 i a^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+108 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+108 a^2 c d x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+72 i a^2 c^2 d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+36 b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-54 a b d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+4 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+108 a^2 d^{3/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+27 b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+18 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (6 a+3 i b \sqrt {1+c^2 x^2}+i b \cosh (3 \text {arcsinh}(c x))+3 b \sinh (2 \text {arcsinh}(c x))\right )-12 i a b d \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+6 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-9 b \cosh (2 \text {arcsinh}(c x))+2 \left (-9 i b c x+9 i a \sqrt {1+c^2 x^2}+3 i a \cosh (3 \text {arcsinh}(c x))+9 a \sinh (2 \text {arcsinh}(c x))-i b \sinh (3 \text {arcsinh}(c x))\right )\right )}{216 c \sqrt {1+c^2 x^2}} \]
[In]
[Out]
\[\int \left (i c d x +d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {-i c f x +f}d x\]
[In]
[Out]
\[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {3}{2}} \sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int \left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
[In]
[Out]
Exception generated. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Exception generated. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}} \,d x \]
[In]
[Out]