Optimal. Leaf size=357 \[ d^3 \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {5 b e^3 \sin ^{-1}(c x)}{96 c^6}-\frac {9 b d e^2 \sin ^{-1}(c x)}{32 c^4}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}-\frac {1}{2} i b d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i b d^3 \sin ^{-1}(c x)^2+b d^3 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^3 \log (x) \sin ^{-1}(c x) \]
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Rubi [A] time = 0.48, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {266, 43, 4731, 12, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i b d^3 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+d^3 \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {9 b d e^2 \sin ^{-1}(c x)}{32 c^4}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}-\frac {5 b e^3 \sin ^{-1}(c x)}{96 c^6}-\frac {1}{2} i b d^3 \sin ^{-1}(c x)^2+b d^3 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^3 \log (x) \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 12
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 )^3 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)}{12 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{12} (b c) \int \frac {18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{12} (b c) \int \left (\frac {18 d^2 e x^2}{\sqrt {1-c^2 x^2}}+\frac {9 d e^2 x^4}{\sqrt {1-c^2 x^2}}+\frac {2 e^3 x^6}{\sqrt {1-c^2 x^2}}+\frac {12 d^3 \log (x)}{\sqrt {1-c^2 x^2}}\right ) \, dx\\ &=\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (b c d^3\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (3 b c d^2 e\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{4} \left (3 b c d e^2\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{6} \left (b c e^3\right ) \int \frac {x^6}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )-b d^3 \sin ^{-1}(c x) \log (x)+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\left (b d^3\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx-\frac {\left (3 b d^2 e\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c}-\frac {\left (9 b d e^2\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c}-\frac {\left (5 b e^3\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{36 c}\\ &=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )-b d^3 \sin ^{-1}(c x) \log (x)+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\left (b d^3\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {\left (9 b d e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}-\frac {\left (5 b e^3\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{48 c^3}\\ &=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}-\frac {9 b d e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {1}{2} i b d^3 \sin ^{-1}(c x)^2+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )-b d^3 \sin ^{-1}(c x) \log (x)+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (2 i b d^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )-\frac {\left (5 b e^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}-\frac {9 b d e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {5 b e^3 \sin ^{-1}(c x)}{96 c^6}-\frac {1}{2} i b d^3 \sin ^{-1}(c x)^2+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+b d^3 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^3 \sin ^{-1}(c x) \log (x)+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (b d^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}-\frac {9 b d e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {5 b e^3 \sin ^{-1}(c x)}{96 c^6}-\frac {1}{2} i b d^3 \sin ^{-1}(c x)^2+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+b d^3 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^3 \sin ^{-1}(c x) \log (x)+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{2} \left (i b d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac {3 b d^2 e x \sqrt {1-c^2 x^2}}{4 c}+\frac {9 b d e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {5 b e^3 x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {3 b d e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {5 b e^3 x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^3 x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {3 b d^2 e \sin ^{-1}(c x)}{4 c^2}-\frac {9 b d e^2 \sin ^{-1}(c x)}{32 c^4}-\frac {5 b e^3 \sin ^{-1}(c x)}{96 c^6}-\frac {1}{2} i b d^3 \sin ^{-1}(c x)^2+\frac {3}{2} d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )+b d^3 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d^3 \sin ^{-1}(c x) \log (x)+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{2} i b d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.38, size = 278, normalized size = 0.78 \[ a d^3 \log (x)+\frac {3}{2} a d^2 e x^2+\frac {3}{4} a d e^2 x^4+\frac {1}{6} a e^3 x^6+\frac {3 b d^2 e \left (c x \sqrt {1-c^2 x^2}-\sin ^{-1}(c x)\right )}{4 c^2}+\frac {3 b d e^2 \left (c x \sqrt {1-c^2 x^2} \left (2 c^2 x^2+3\right )-3 \sin ^{-1}(c x)\right )}{32 c^4}+\frac {b e^3 \left (c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4+10 c^2 x^2+15\right )-15 \sin ^{-1}(c x)\right )}{288 c^6}-\frac {1}{2} i b d^3 \left (\sin ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )+b d^3 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{2} b d^2 e x^2 \sin ^{-1}(c x)+\frac {3}{4} b d e^2 x^4 \sin ^{-1}(c x)+\frac {1}{6} b e^3 x^6 \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \arcsin \left (c x\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 391, normalized size = 1.10 \[ \frac {a \,e^{3} x^{6}}{6}+\frac {3 a d \,e^{2} x^{4}}{4}+\frac {3 a \,d^{2} e \,x^{2}}{2}+d^{3} a \ln \left (c x \right )-\frac {i b \,d^{3} \arcsin \left (c x \right )^{2}}{2}+d^{3} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {b \arcsin \left (c x \right ) e^{3} \cos \left (6 \arcsin \left (c x \right )\right )}{192 c^{6}}+\frac {b \,e^{3} \sin \left (6 \arcsin \left (c x \right )\right )}{1152 c^{6}}+\frac {3 b \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) d \,e^{2}}{32 c^{4}}+\frac {b \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) e^{3}}{32 c^{6}}-\frac {3 b \sin \left (4 \arcsin \left (c x \right )\right ) d \,e^{2}}{128 c^{4}}-\frac {b \sin \left (4 \arcsin \left (c x \right )\right ) e^{3}}{128 c^{6}}-\frac {3 b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) d^{2} e}{4 c^{2}}-\frac {3 b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) d \,e^{2}}{8 c^{4}}-\frac {5 b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) e^{3}}{64 c^{6}}+\frac {3 b \sin \left (2 \arcsin \left (c x \right )\right ) d^{2} e}{8 c^{2}}+\frac {3 b \sin \left (2 \arcsin \left (c x \right )\right ) d \,e^{2}}{16 c^{4}}+\frac {5 b \sin \left (2 \arcsin \left (c x \right )\right ) e^{3}}{128 c^{6}} \]
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 \[ \frac {1}{6} \, a e^{3} x^{6} + \frac {3}{4} \, a d e^{2} x^{4} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} \log \relax (x) + \int \frac {{\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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