Optimal. Leaf size=39 \[ \frac {1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c x \sqrt {1-\frac {c^2}{x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4842, 12, 191} \[ \frac {1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c x \sqrt {1-\frac {c^2}{x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 4842
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b \int \frac {c}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{2} (b c) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{2} b c \sqrt {1-\frac {c^2}{x^2}} x+\frac {1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 1.21 \[ \frac {a x^2}{2}+\frac {1}{2} b c x \sqrt {\frac {x^2-c^2}{x^2}}+\frac {1}{2} b x^2 \sin ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 40, normalized size = 1.03 \[ \frac {1}{2} \, b x^{2} \arcsin \left (\frac {c}{x}\right ) + \frac {1}{2} \, b c x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} + \frac {1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 174, normalized size = 4.46 \[ \frac {b c x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {c}{x}\right ) + a c x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} + 2 \, b c^{2} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} + 2 \, b c^{3} \arcsin \left (\frac {c}{x}\right ) + 2 \, a c^{3} - \frac {2 \, b c^{4}}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} + \frac {b c^{5} \arcsin \left (\frac {c}{x}\right )}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} + \frac {a c^{5}}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}}}{8 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 1.31 \[ -c^{2} \left (-\frac {a \,x^{2}}{2 c^{2}}+b \left (-\frac {x^{2} \arcsin \left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{2 c}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 36, normalized size = 0.92 \[ \frac {1}{2} \, a x^{2} + \frac {1}{2} \, {\left (x^{2} \arcsin \left (\frac {c}{x}\right ) + c x \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 36, normalized size = 0.92 \[ \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asin}\left (\frac {c}{x}\right )}{2}+\frac {b\,c\,x\,\sqrt {1-\frac {c^2}{x^2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.93, size = 58, normalized size = 1.49 \[ \frac {a x^{2}}{2} + \frac {b c \left (\begin {cases} c \sqrt {-1 + \frac {x^{2}}{c^{2}}} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\i c \sqrt {1 - \frac {x^{2}}{c^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {b x^{2} \operatorname {asin}{\left (\frac {c}{x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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