Optimal. Leaf size=229 \[ d^2 \log (x) \left (a+b \sin ^{-1}(c x)\right )+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {3 b e^2 \sin ^{-1}(c x)}{32 c^4}+\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {1}{2} i b d^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i b d^2 \sin ^{-1}(c x)^2+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \log (x) \sin ^{-1}(c x) \]
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Rubi [A] time = 0.33, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {266, 43, 4731, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i b d^2 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+d^2 \log (x) \left (a+b \sin ^{-1}(c x)\right )+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {1}{2} i b d^2 \sin ^{-1}(c x)^2+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \log (x) \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 216
Rule 266
Rule 321
Rule 2190
Rule 2279
Rule 2326
Rule 2391
Rule 3717
Rule 4625
Rule 4731
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {d e x^2+\frac {e^2 x^4}{4}+d^2 \log (x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \left (\frac {d e x^2}{\sqrt {1-c^2 x^2}}+\frac {e^2 x^4}{4 \sqrt {1-c^2 x^2}}+\frac {d^2 \log (x)}{\sqrt {1-c^2 x^2}}\right ) \, dx\\ &=d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (b c d^2\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx-(b c d e) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{4} \left (b c e^2\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\left (b d^2\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx-\frac {(b d e) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c}-\frac {\left (3 b e^2\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c}\\ &=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\left (b d^2\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {\left (3 b e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}-\frac {3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (2 i b d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}-\frac {3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (b d^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}-\frac {3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{2} \left (i b d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b d e \sin ^{-1}(c x)}{2 c^2}-\frac {3 b e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {1}{2} i b d^2 \sin ^{-1}(c x)^2+d e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^2 \sin ^{-1}(c x) \log (x)+d^2 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{2} i b d^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.38, size = 184, normalized size = 0.80 \[ a d^2 \log (x)+a d e x^2+\frac {1}{4} a e^2 x^4+\frac {b d e \left (c x \sqrt {1-c^2 x^2}-\sin ^{-1}(c x)\right )}{2 c^2}+\frac {b e^2 \left (c x \sqrt {1-c^2 x^2} \left (2 c^2 x^2+3\right )-3 \sin ^{-1}(c x)\right )}{32 c^4}-\frac {1}{2} i b d^2 \left (\sin ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )+b d^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b d e x^2 \sin ^{-1}(c x)+\frac {1}{4} b e^2 x^4 \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arcsin \left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 262, normalized size = 1.14 \[ \frac {a \,e^{2} x^{4}}{4}+a e d \,x^{2}+a \,d^{2} \ln \left (c x \right )-\frac {i b \,d^{2} \arcsin \left (c x \right )^{2}}{2}+b \,d^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+b \,d^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i b \,d^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i b \,d^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {b \arcsin \left (c x \right ) e^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{32 c^{4}}-\frac {b \,e^{2} \sin \left (4 \arcsin \left (c x \right )\right )}{128 c^{4}}-\frac {b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) d e}{2 c^{2}}-\frac {b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) e^{2}}{8 c^{4}}+\frac {b \sin \left (2 \arcsin \left (c x \right )\right ) d e}{4 c^{2}}+\frac {b \sin \left (2 \arcsin \left (c x \right )\right ) e^{2}}{16 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, a e^{2} x^{4} + a d e x^{2} + a d^{2} \log \relax (x) + \int \frac {{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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