Optimal. Leaf size=45 \[ \frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \text {ArcSin}\left (c x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 45, 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 = {6847, 4715,
267} \begin {gather*} \frac {a x^2}{2}+\frac {1}{2} b x^2 \text {ArcSin}\left (c x^2\right )+\frac {b \sqrt {1-c^2 x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 6847
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \left (a+b \sin ^{-1}(c x)\right ) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b \text {Subst}\left (\int \sin ^{-1}(c x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.96 \begin {gather*} \frac {a x^2}{2}+\frac {1}{2} b \left (\frac {\sqrt {1-c^2 x^4}}{c}+x^2 \text {ArcSin}\left (c x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 39, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {c \,x^{2} a +b \left (c \,x^{2} \arcsin \left (c \,x^{2}\right )+\sqrt {-c^{2} x^{4}+1}\right )}{2 c}\) | \(39\) |
default | \(\frac {c \,x^{2} a +b \left (c \,x^{2} \arcsin \left (c \,x^{2}\right )+\sqrt {-c^{2} x^{4}+1}\right )}{2 c}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 37, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, a x^{2} + \frac {{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt {-c^{2} x^{4} + 1}\right )} b}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.44, size = 38, normalized size = 0.84 \begin {gather*} \frac {b c x^{2} \arcsin \left (c x^{2}\right ) + a c x^{2} + \sqrt {-c^{2} x^{4} + 1} b}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 42, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {asin}{\left (c x^{2} \right )}}{2} + \frac {b \sqrt {- c^{2} x^{4} + 1}}{2 c} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 38, normalized size = 0.84 \begin {gather*} \frac {a c x^{2} + {\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt {-c^{2} x^{4} + 1}\right )} b}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 37, normalized size = 0.82 \begin {gather*} \frac {a\,x^2}{2}+\frac {b\,\sqrt {1-c^2\,x^4}}{2\,c}+\frac {b\,x^2\,\mathrm {asin}\left (c\,x^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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