Optimal. Leaf size=262 \[ -\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+3 d^2 e \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sin ^{-1}(c x)}{32 c^4}+\frac {3 b e^2 x \sqrt {1-c^2 x^2} \left (8 c^2 d+e\right )}{32 c^3}-\frac {3}{2} i b d^2 e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \log (x) \sin ^{-1}(c x) \]
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Rubi [A] time = 0.78, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {266, 43, 4731, 12, 6742, 1807, 1584, 459, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {3}{2} i b d^2 e \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+3 d^2 e \log (x) \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 x \sqrt {1-c^2 x^2} \left (8 c^2 d+e\right )}{32 c^3}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sin ^{-1}(c x)}{32 c^4}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \log (x) \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 216
Rule 266
Rule 321
Rule 459
Rule 1584
Rule 1807
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^3} \, dx &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{4 x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \left (\frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {1-c^2 x^2}}+\frac {12 d^2 e \log (x)}{\sqrt {1-c^2 x^2}}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {1-c^2 x^2}} \, dx-\left (3 b c d^2 e\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{4} (b c) \int \frac {-6 d e^2 x^3-e^3 x^5}{x \sqrt {1-c^2 x^2}} \, dx+\left (3 b d^2 e\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{4} (b c) \int \frac {x^2 \left (-6 d e^2-e^3 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx+\left (3 b d^2 e\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (6 i b d^2 e\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 (3 b e^2 \left (8 c^2 d+e\right )\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c}\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\left (3 b d^2 e\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac {\left (3 b e^2 \left (8 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sin ^{-1}(c x)}{32 c^4}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{2} \left (3 i b d^2 e\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sin ^{-1}(c x)}{32 c^4}-\frac {3}{2} i b d^2 e \sin ^{-1}(c x)^2-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \sin ^{-1}(c x)\right )+3 b d^2 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-3 b d^2 e \sin ^{-1}(c x) \log (x)+3 d^2 e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {3}{2} i b d^2 e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.47, size = 220, normalized size = 0.84 \[ \frac {1}{32} \left (-\frac {16 a d^3}{x^2}+96 a d^2 e \log (x)+48 a d e^2 x^2+8 a e^3 x^4-\frac {16 b d^3 \left (c x \sqrt {1-c^2 x^2}+\sin ^{-1}(c x)\right )}{x^2}+\frac {24 b d e^2 \left (c x \sqrt {1-c^2 x^2}+\left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )}{c^2}+\frac {b e^3 \left (\left (8 c^4 x^4-3\right ) \sin ^{-1}(c x)+c x \sqrt {1-c^2 x^2} \left (2 c^2 x^2+3\right )\right )}{c^4}+96 b d^2 e \left (\sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i \left (\sin ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \arcsin \left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 360, normalized size = 1.37 \[ \frac {a \,e^{3} x^{4}}{4}+\frac {3 a \,x^{2} d \,e^{2}}{2}+3 a \,d^{2} e \ln \left (c x \right )-\frac {a \,d^{3}}{2 x^{2}}+3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {b \sin \left (4 \arcsin \left (c x \right )\right ) e^{3}}{128 c^{4}}+\frac {b \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) e^{3}}{32 c^{4}}-3 i b \,d^{2} e \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i b \,d^{2} e \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {3 b \sqrt {-c^{2} x^{2}+1}\, x d \,e^{2}}{4 c}+\frac {i c^{2} b \,d^{3}}{2}+3 b \,d^{2} e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 i b \,d^{2} e \arcsin \left (c x \right )^{2}}{2}-\frac {b c \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 x}+\frac {b \sqrt {-c^{2} x^{2}+1}\, x \,e^{3}}{8 c^{3}}-\frac {3 b \arcsin \left (c x \right ) d \,e^{2}}{4 c^{2}}-\frac {b \arcsin \left (c x \right ) e^{3}}{8 c^{4}}+\frac {b \arcsin \left (c x \right ) x^{2} e^{3}}{4 c^{2}}+\frac {3 b \arcsin \left (c x \right ) x^{2} d \,e^{2}}{2}-\frac {b \arcsin \left (c x \right ) d^{3}}{2 x^{2}} \]
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 \[ \frac {1}{4} \, a e^{3} x^{4} + \frac {3}{2} \, a d e^{2} x^{2} - \frac {1}{2} \, b d^{3} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} + \frac {\arcsin \left (c x\right )}{x^{2}}\right )} + 3 \, a d^{2} e \log \relax (x) - \frac {a d^{3}}{2 \, x^{2}} + \int \frac {{\left (b e^{3} x^{4} + 3 \, b d e^{2} x^{2} + 3 \, b d^{2} e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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