Integrand size = 32, antiderivative size = 366 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 b \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]
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Time = 0.32 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4763, 4747, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 197} \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=-\frac {2 i \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {4 b \left (1-c^2 x^2\right )^{5/2} \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
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Rule 197
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4745
Rule 4747
Rule 4763
Rule 4765
Rule 4767
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (b^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+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tan (x) \, dx,x,\arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 i b \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 b \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 b \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 i b^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {b^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 b \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ \end{align*}
Time = 10.85 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\frac {4 a^2 c x \left (3-2 c^2 x^2\right )+b^2 \left (c x+6 c x \arcsin (c x)^2+4 i \pi \arcsin (c x) \cos (3 \arcsin (c x))-2 i \arcsin (c x)^2 \cos (3 \arcsin (c x))+8 \pi \cos (3 \arcsin (c x)) \log \left (1+e^{-i \arcsin (c x)}\right )+2 \pi \cos (3 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )+4 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )-2 \pi \cos (3 \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )+4 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )-8 \pi \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 \pi \cos (3 \arcsin (c x)) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 \sqrt {1-c^2 x^2} \left (-3 i \arcsin (c x)^2+\arcsin (c x) \left (-2+6 i \pi +6 \log \left (1-i e^{i \arcsin (c x)}\right )+6 \log \left (1+i e^{i \arcsin (c x)}\right )\right )+3 \pi \left (4 \log \left (1+e^{-i \arcsin (c x)}\right )+\log \left (1-i e^{i \arcsin (c x)}\right )-\log \left (1+i e^{i \arcsin (c x)}\right )-4 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-\log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )\right )-2 \pi \cos (3 \arcsin (c x)) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-16 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-16 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+\sin (3 \arcsin (c x))+2 \arcsin (c x)^2 \sin (3 \arcsin (c x))\right )+4 a b \left (\sqrt {1-c^2 x^2} \left (-1+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 \cos (2 \arcsin (c x)) \left (\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )\right )+\arcsin (c x) (3 c x+\sin (3 \arcsin (c x)))\right )}{12 d^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (c-c^3 x^2\right )} \]
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\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {5}{2}} \left (-c e x +e \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^{5/2}\,{\left (e-c\,e\,x\right )}^{5/2}} \,d x \]
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