Integrand size = 35, antiderivative size = 647 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]
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Time = 0.65 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {4823, 4787, 4783, 4803, 4268, 2611, 2320, 6724, 4715, 267, 4739, 455, 45} \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {2 d e \sqrt {c d x+d} \sqrt {e-c e x} \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {2 b c^3 d e x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2-\frac {2 a b c d e x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d e x \arcsin (c x) \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}-\frac {22}{9} b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \]
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Rule 45
Rule 267
Rule 455
Rule 2320
Rule 2611
Rule 4268
Rule 4715
Rule 4739
Rule 4783
Rule 4787
Rule 4803
Rule 4823
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e 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 {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \arcsin (c x) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e 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 {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e 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 {\left (2 b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}} \\ & = -\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \\ & = -\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 5.08 (sec) , antiderivative size = 632, normalized size of antiderivative = 0.98 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {1}{3} a^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (-4+c^2 x^2\right )+\frac {2 a b d e \sqrt {d+c d x} \sqrt {e-c e x} \left (-3 c x+c^3 x^3+3 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)\right )}{9 \sqrt {1-c^2 x^2}}+a^2 d^{3/2} e^{3/2} \log (c x)-a^2 d^{3/2} e^{3/2} \log \left (d e+\sqrt {d} \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x}\right )-\frac {2 a b d e \sqrt {d+c d x} \sqrt {e-c e x} \left (c x-\sqrt {1-c^2 x^2} \arcsin (c x)-\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (2 \sqrt {1-c^2 x^2}+2 c x \arcsin (c x)-\sqrt {1-c^2 x^2} \arcsin (c x)^2-\arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )+\arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )-2 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+2 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )-2 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (27 \sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )+\left (-2+9 \arcsin (c x)^2\right ) \cos (3 \arcsin (c x))-6 \arcsin (c x) (9 c x+\sin (3 \arcsin (c x)))\right )}{108 \sqrt {1-c^2 x^2}} \]
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\[\int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{x}d x\]
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\[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}}{x} \,d x \]
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