Optimal. Leaf size=60 \[ \frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {b \sqrt {1-c^2 x^2}}{3 c^3} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4628, 266, 43} \[ \frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {b \sqrt {1-c^2 x^2}}{3 c^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4628
Rubi steps
\begin {align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{3} (b c) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \sqrt {1-c^2 x^2}}{3 c^3}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.92 \[ \frac {a x^3}{3}+b \left (-\frac {2}{9 c^3}-\frac {x^2}{9 c}\right ) \sqrt {1-c^2 x^2}+\frac {1}{3} b x^3 \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 54, normalized size = 0.90 \[ \frac {3 \, b c^{3} x^{3} \arccos \left (c x\right ) + 3 \, a c^{3} x^{3} - {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {-c^{2} x^{2} + 1}}{9 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 56, normalized size = 0.93 \[ \frac {1}{3} \, b x^{3} \arccos \left (c x\right ) + \frac {1}{3} \, a x^{3} - \frac {\sqrt {-c^{2} x^{2} + 1} b x^{2}}{9 \, c} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b}{9 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 1.07 \[ \frac {\frac {c^{3} x^{3} a}{3}+b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 60, normalized size = 1.00 \[ \frac {1}{3} \, a x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b \]
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 \[ \left \{\begin {array}{cl} \frac {a\,x^3}{3}-b\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}-\frac {x^3\,\mathrm {acos}\left (c\,x\right )}{3}\right ) & \text {\ if\ \ }0<c\\ \int x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 70, normalized size = 1.17 \[ \begin {cases} \frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {acos}{\left (c x \right )}}{3} - \frac {b x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {2 b \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {x^{3} \left (a + \frac {\pi b}{2}\right )}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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