Integrand size = 14, antiderivative size = 64 \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{6} b c^3 \sqrt {1-\frac {c^2}{x^2}} x+\frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4926, 12, 277, 197} \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \arcsin \left (\frac {c}{x}\right )\right )+\frac {1}{12} b c x^3 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 x \sqrt {1-\frac {c^2}{x^2}} \]
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Rule 12
Rule 197
Rule 277
Rule 4926
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \left (a+b \arcsin \left (\frac {c}{x}\right )\right )+\frac {1}{4} b \int \frac {c x^2}{\sqrt {1-\frac {c^2}{x^2}}} \, dx \\ & = \frac {1}{4} x^4 \left (a+b \arcsin \left (\frac {c}{x}\right )\right )+\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {1-\frac {c^2}{x^2}}} \, dx \\ & = \frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \arcsin \left (\frac {c}{x}\right )\right )+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}}} \, dx \\ & = \frac {1}{6} b c^3 \sqrt {1-\frac {c^2}{x^2}} x+\frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {a x^4}{4}+b \sqrt {\frac {-c^2+x^2}{x^2}} \left (\frac {c^3 x}{6}+\frac {c x^3}{12}\right )+\frac {1}{4} b x^4 \arcsin \left (\frac {c}{x}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05
method | result | size |
parts | \(\frac {a \,x^{4}}{4}-b \,c^{4} \left (-\frac {x^{4} \arcsin \left (\frac {c}{x}\right )}{4 c^{4}}-\frac {x^{3} \sqrt {1-\frac {c^{2}}{x^{2}}}}{12 c^{3}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{6 c}\right )\) | \(67\) |
derivativedivides | \(-c^{4} \left (-\frac {a \,x^{4}}{4 c^{4}}+b \left (-\frac {x^{4} \arcsin \left (\frac {c}{x}\right )}{4 c^{4}}-\frac {x^{3} \sqrt {1-\frac {c^{2}}{x^{2}}}}{12 c^{3}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{6 c}\right )\right )\) | \(71\) |
default | \(-c^{4} \left (-\frac {a \,x^{4}}{4 c^{4}}+b \left (-\frac {x^{4} \arcsin \left (\frac {c}{x}\right )}{4 c^{4}}-\frac {x^{3} \sqrt {1-\frac {c^{2}}{x^{2}}}}{12 c^{3}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{6 c}\right )\right )\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \arcsin \left (\frac {c}{x}\right ) + \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (2 \, b c^{3} x + b c x^{3}\right )} \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} \]
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Time = 1.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.67 \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {a x^{4}}{4} + \frac {b c \left (\begin {cases} \frac {2 c^{3} \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3} + \frac {c x^{2} \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\\frac {2 i c^{3} \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3} + \frac {i c x^{2} \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3} & \text {otherwise} \end {cases}\right )}{4} + \frac {b x^{4} \operatorname {asin}{\left (\frac {c}{x} \right )}}{4} \]
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arcsin \left (\frac {c}{x}\right ) + {\left (x^{3} {\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, c^{2} x \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )} c\right )} b \]
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Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 340, normalized size of antiderivative = 5.31 \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\frac {3 \, b c x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {c}{x}\right ) + 3 \, a c x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4} + 2 \, b c^{2} x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3} + 12 \, b c^{3} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {c}{x}\right ) + 12 \, a c^{3} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} + 18 \, b c^{4} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} + 18 \, b c^{5} \arcsin \left (\frac {c}{x}\right ) + 18 \, a c^{5} - \frac {18 \, b c^{6}}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} + \frac {12 \, b c^{7} \arcsin \left (\frac {c}{x}\right )}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a c^{7}}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b c^{8}}{x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b c^{9} \arcsin \left (\frac {c}{x}\right )}{x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a c^{9}}{x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4}}}{192 \, c} \]
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Timed out. \[ \int x^3 \left (a+b \arcsin \left (\frac {c}{x}\right )\right ) \, dx=\int x^3\,\left (a+b\,\mathrm {asin}\left (\frac {c}{x}\right )\right ) \,d x \]
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