Optimal. Leaf size=57 \[ \frac {a x^2}{2}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d}+\frac {b \left (c+d x^2\right ) \sin ^{-1}\left (c+d x^2\right )}{2 d} \]
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Rubi [A] time = 0.06, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6715, 4803, 4619, 261} \[ \frac {a x^2}{2}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d}+\frac {b \left (c+d x^2\right ) \sin ^{-1}\left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4803
Rule 6715
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b \operatorname {Subst}\left (\int \sin ^{-1}(c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {b \operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x^2\right )}{2 d}\\ &=\frac {a x^2}{2}+\frac {b \left (c+d x^2\right ) \sin ^{-1}\left (c+d x^2\right )}{2 d}-\frac {b \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x^2\right )}{2 d}\\ &=\frac {a x^2}{2}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d}+\frac {b \left (c+d x^2\right ) \sin ^{-1}\left (c+d x^2\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 70, normalized size = 1.23 \[ \frac {a x^2}{2}+\frac {b \left (\sqrt {-c^2-2 c d x^2-d^2 x^4+1}+c \sin ^{-1}\left (c+d x^2\right )\right )}{2 d}+\frac {1}{2} b x^2 \sin ^{-1}\left (c+d x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 57, normalized size = 1.00 \[ \frac {a d x^{2} + {\left (b d x^{2} + b c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} b}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.46, size = 49, normalized size = 0.86 \[ \frac {{\left (d x^{2} + c\right )} a + {\left ({\left (d x^{2} + c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1}\right )} b}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 0.88 \[ \frac {a \left (d \,x^{2}+c \right )+b \left (\left (d \,x^{2}+c \right ) \arcsin \left (d \,x^{2}+c \right )+\sqrt {1-\left (d \,x^{2}+c \right )^{2}}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 45, normalized size = 0.79 \[ \frac {1}{2} \, a x^{2} + \frac {{\left ({\left (d x^{2} + c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1}\right )} b}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 108, normalized size = 1.89 \[ \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asin}\left (d\,x^2+c\right )}{2}+\frac {b\,\sqrt {-c^2-2\,c\,d\,x^2-d^2\,x^4+1}}{2\,d}+\frac {b\,c\,\ln \left (\sqrt {-c^2-2\,c\,d\,x^2-d^2\,x^4+1}-\frac {d^2\,x^2+c\,d}{\sqrt {-d^2}}\right )}{2\,\sqrt {-d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 76, normalized size = 1.33 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b c \operatorname {asin}{\left (c + d x^{2} \right )}}{2 d} + \frac {b x^{2} \operatorname {asin}{\left (c + d x^{2} \right )}}{2} + \frac {b \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{2 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \operatorname {asin}{\relax (c )}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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