Optimal. Leaf size=60 \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3} \]
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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 = {4627, 266, 43} \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4627
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 0.82 \[ \frac {1}{9} \left (3 a x^3+\frac {b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )}{c^3}+3 b x^3 \sin ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 53, normalized size = 0.88 \[ \frac {3 \, b c^{3} x^{3} \arcsin \left (c x\right ) + 3 \, a c^{3} x^{3} + {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {-c^{2} x^{2} + 1}}{9 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 74, normalized size = 1.23 \[ \frac {1}{3} \, a x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b x \arcsin \left (c x\right )}{3 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b}{3 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 64, normalized size = 1.07 \[ \frac {\frac {c^{3} x^{3} a}{3}+b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 59, normalized size = 0.98 \[ \frac {1}{3} \, a x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b \]
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 \[ \left \{\begin {array}{cl} b\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )+\frac {a\,x^3}{3} & \text {\ if\ \ }0<c\\ \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 65, normalized size = 1.08 \[ \begin {cases} \frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 b \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {a x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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