Optimal. Leaf size=64 \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c x^2 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4842, 12, 266, 51, 63, 208} \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c x^2 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 208
Rule 266
Rule 4842
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{3} b \int \frac {c x}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{3} (b c) \int \frac {x}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-c^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{6} b c \sqrt {1-\frac {c^2}{x^2}} x^2+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )-\frac {1}{12} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{6} b c \sqrt {1-\frac {c^2}{x^2}} x^2+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-\frac {c^2}{x^2}}\right )\\ &=\frac {1}{6} b c \sqrt {1-\frac {c^2}{x^2}} x^2+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c^3 \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 79, normalized size = 1.23 \[ \frac {a x^3}{3}+\frac {1}{6} b c x^2 \sqrt {\frac {x^2-c^2}{x^2}}+\frac {1}{6} b c^3 \log \left (x \left (\sqrt {\frac {x^2-c^2}{x^2}}+1\right )\right )+\frac {1}{3} b x^3 \sin ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 106, normalized size = 1.66 \[ -\frac {1}{6} \, b c^{3} \log \left (x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} - x\right ) + \frac {1}{6} \, b c x^{2} \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} + \frac {1}{3} \, a x^{3} + \frac {1}{3} \, {\left (b x^{3} - b\right )} \arcsin \left (\frac {c}{x}\right ) - \frac {2}{3} \, b \arctan \left (\frac {x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} - x}{c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.69, size = 298, normalized size = 4.66 \[ \frac {b c x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {c}{x}\right ) + a c x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3} + b c^{2} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} + 3 \, b c^{3} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} \arcsin \left (\frac {c}{x}\right ) + 3 \, a c^{3} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} + 4 \, b c^{4} \log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right ) - 4 \, b c^{4} \log \left (\frac {{\left | c \right |}}{{\left | x \right |}}\right ) + \frac {3 \, b c^{5} \arcsin \left (\frac {c}{x}\right )}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} + \frac {3 \, a c^{5}}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} - \frac {b c^{6}}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} + \frac {b c^{7} \arcsin \left (\frac {c}{x}\right )}{x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3}} + \frac {a c^{7}}{x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3}}}{24 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 1.06 \[ -c^{3} \left (-\frac {a \,x^{3}}{3 c^{3}}+b \left (-\frac {x^{3} \arcsin \left (\frac {c}{x}\right )}{3 c^{3}}-\frac {x^{2} \sqrt {1-\frac {c^{2}}{x^{2}}}}{6 c^{2}}-\frac {\arctanh \left (\frac {1}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )}{6}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 81, normalized size = 1.27 \[ \frac {1}{3} \, a x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \arcsin \left (\frac {c}{x}\right ) + {\left (c^{2} \log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right ) - c^{2} \log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} - 1\right ) + 2 \, x^{2} \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )} c\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (\frac {c}{x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.18, size = 107, normalized size = 1.67 \[ \frac {a x^{3}}{3} + \frac {b c \left (\begin {cases} \frac {c^{2} \operatorname {acosh}{\left (\frac {x}{c} \right )}}{2} + \frac {c x \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{2} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\- \frac {i c^{2} \operatorname {asin}{\left (\frac {x}{c} \right )}}{2} + \frac {i c x}{2 \sqrt {1 - \frac {x^{2}}{c^{2}}}} - \frac {i x^{3}}{2 c \sqrt {1 - \frac {x^{2}}{c^{2}}}} & \text {otherwise} \end {cases}\right )}{3} + \frac {b x^{3} \operatorname {asin}{\left (\frac {c}{x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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