Integrand size = 37, antiderivative size = 942 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\frac {2 i b^2 d^2 \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
[Out]
Time = 0.89 (sec) , antiderivative size = 942, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {5796, 5838, 5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197, 5789, 4265, 267, 5800, 5810, 294, 221} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=-\frac {c^2 d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 x^3}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^2}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 d^2 \left (c^2 x^2+1\right )^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \text {arcsinh}(c x)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}} \]
[In]
[Out]
Rule 197
Rule 221
Rule 267
Rule 294
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4265
Rule 5787
Rule 5788
Rule 5789
Rule 5796
Rule 5797
Rule 5798
Rule 5800
Rule 5810
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(d+i c d x)^2 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {\left (1+c^2 x^2\right )^{5/2} \int \left (\frac {d^2 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}}+\frac {2 i c d^2 x (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}}-\frac {c^2 d^2 x^2 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {\left (d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (c^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 i b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b c^3 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b^2 c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b^2 c^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i b^2 d^2 \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i b^2 d^2 \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (8 b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i b^2 d^2 \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i b^2 d^2 \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {2 i b^2 d^2 \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}
Time = 6.96 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.56 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {a^2 (2 i+c x)}{(i+c x)^2}-\frac {a b \left (i \cosh \left (\frac {3}{2} \text {arcsinh}(c x)\right ) \left (\text {arcsinh}(c x)-2 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\frac {1}{2} i \log \left (1+c^2 x^2\right )\right )+\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (-2+3 i \text {arcsinh}(c x)+6 i \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\frac {3}{2} \log \left (1+c^2 x^2\right )\right )+2 \left (i+\left (-1+\sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)+2 \left (2+\sqrt {1+c^2 x^2}\right ) \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-\frac {1}{2} i \left (2+\sqrt {1+c^2 x^2}\right ) \log \left (1+c^2 x^2\right )\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^3}-\frac {b^2 \left ((1+i) \text {arcsinh}(c x)^2-\frac {\text {arcsinh}(c x) (2 i+\text {arcsinh}(c x))}{i+c x}+2 (i \pi +2 \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+i \pi \left (3 \text {arcsinh}(c x)-4 \log \left (1+e^{\text {arcsinh}(c x)}\right )-2 \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )+4 \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\frac {2 \text {arcsinh}(c x)^2 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^3}-\frac {2 \left (-2+\text {arcsinh}(c x)^2\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}\right )}{\sqrt {1+c^2 x^2}}\right )}{3 c d f^3} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (-i c f x +f \right )^{\frac {5}{2}} \sqrt {i c d x +d}}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d} {\left (-i \, c f x + f\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {i d \left (c x - i\right )} \left (- i f \left (c x + i\right )\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d} {\left (-i \, c f x + f\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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