Optimal. Leaf size=76 \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {3 b \sin ^{-1}(c x)}{32 c^4}+\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3} \]
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Rubi [A] time = 0.03, antiderivative size = 76, 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, 321, 216} \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {3 b \sin ^{-1}(c x)}{32 c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 4627
Rubi steps
\begin {align*} \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} (b c) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {(3 b) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c}\\ &=\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {(3 b) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b \sin ^{-1}(c x)}{32 c^4}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 1.07 \[ \frac {a x^4}{4}-\frac {3 b \sin ^{-1}(c x)}{32 c^4}+\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {1}{4} b x^4 \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 61, normalized size = 0.80 \[ \frac {8 \, a c^{4} x^{4} + {\left (8 \, b c^{4} x^{4} - 3 \, b\right )} \arcsin \left (c x\right ) + {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{32 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 95, normalized size = 1.25 \[ \frac {1}{4} \, a x^{4} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {5 \, b \arcsin \left (c x\right )}{32 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.95 \[ \frac {\frac {c^{4} x^{4} a}{4}+b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}-\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 70, normalized size = 0.92 \[ \frac {1}{4} \, a x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 80, normalized size = 1.05 \[ \begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b \operatorname {asin}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\\frac {a x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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