Optimal. Leaf size=57 \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {b \sin ^{-1}\left (c x^2\right )}{8 c^2}+\frac {b x^2 \sqrt {1-c^2 x^4}}{8 c} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4842, 12, 275, 321, 216} \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {b x^2 \sqrt {1-c^2 x^4}}{8 c}-\frac {b \sin ^{-1}\left (c x^2\right )}{8 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 275
Rule 321
Rule 4842
Rubi steps
\begin {align*} \int x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{4} b \int \frac {2 c x^5}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{2} (b c) \int \frac {x^5}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{4} (b c) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {b x^2 \sqrt {1-c^2 x^4}}{8 c}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x^2}} \, dx,x,x^2\right )}{8 c}\\ &=\frac {b x^2 \sqrt {1-c^2 x^4}}{8 c}-\frac {b \sin ^{-1}\left (c x^2\right )}{8 c^2}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 1.09 \[ \frac {a x^4}{4}-\frac {b \sin ^{-1}\left (c x^2\right )}{8 c^2}+\frac {b x^2 \sqrt {1-c^2 x^4}}{8 c}+\frac {1}{4} b x^4 \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 53, normalized size = 0.93 \[ \frac {2 \, a c^{2} x^{4} + \sqrt {-c^{2} x^{4} + 1} b c x^{2} + {\left (2 \, b c^{2} x^{4} - b\right )} \arcsin \left (c x^{2}\right )}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 59, normalized size = 1.04 \[ \frac {2 \, a c x^{4} + \frac {{\left (\sqrt {-c^{2} x^{4} + 1} c x^{2} + 2 \, {\left (c^{2} x^{4} - 1\right )} \arcsin \left (c x^{2}\right ) + \arcsin \left (c x^{2}\right )\right )} b}{c}}{8 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 74, normalized size = 1.30 \[ \frac {x^{4} a}{4}+\frac {b \,x^{4} \arcsin \left (c \,x^{2}\right )}{4}+\frac {b \,x^{2} \sqrt {-c^{2} x^{4}+1}}{8 c}-\frac {b \arctan \left (\frac {\sqrt {c^{2}}\, x^{2}}{\sqrt {-c^{2} x^{4}+1}}\right )}{8 c \sqrt {c^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 88, normalized size = 1.54 \[ \frac {1}{4} \, a x^{4} + \frac {1}{8} \, {\left (2 \, x^{4} \arcsin \left (c x^{2}\right ) + c {\left (\frac {\arctan \left (\frac {\sqrt {-c^{2} x^{4} + 1}}{c x^{2}}\right )}{c^{3}} + \frac {\sqrt {-c^{2} x^{4} + 1}}{{\left (c^{4} - \frac {{\left (c^{2} x^{4} - 1\right )} c^{2}}{x^{4}}\right )} x^{2}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 50, normalized size = 0.88 \[ \frac {a\,x^4}{4}+\frac {b\,\left (\frac {\mathrm {asin}\left (c\,x^2\right )\,\left (2\,c^2\,x^4-1\right )}{4}+\frac {c\,x^2\,\sqrt {1-c^2\,x^4}}{4}\right )}{2\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 60, normalized size = 1.05 \[ \begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {asin}{\left (c x^{2} \right )}}{4} + \frac {b x^{2} \sqrt {- c^{2} x^{4} + 1}}{8 c} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{8 c^{2}} & \text {for}\: c \neq 0 \\\frac {a x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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