Optimal. Leaf size=57 \[ \frac {a x^2}{2}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d}+\frac {b \left (c+d x^2\right ) \text {ArcSin}\left (c+d x^2\right )}{2 d} \]
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Rubi [A]
time = 0.05, 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 = {6847, 4887,
4715, 267} \begin {gather*} \frac {a x^2}{2}+\frac {b \left (c+d x^2\right ) \text {ArcSin}\left (c+d x^2\right )}{2 d}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 4887
Rule 6847
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=\frac {1}{2} \text {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 \text {Subst}\left (\int \sin ^{-1}(c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {b \text {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 \text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(183\) vs. \(2(57)=114\).
time = 0.11, size = 183, normalized size = 3.21 \begin {gather*} \frac {a x^2}{2}+\frac {1}{2} b x^2 \text {ArcSin}\left (c+d x^2\right )+\frac {b \left (2 d \sqrt {1-c^2-2 c d x^2-d^2 x^4}+2 c d \text {ArcTan}\left (\frac {\sqrt {-d^2} x^2-\sqrt {1-c^2-2 c d x^2-d^2 x^4}}{c}\right )+c \sqrt {-d^2} \log \left (-1+2 c d x^2+2 d^2 x^4+2 \sqrt {-d^2} x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}\right )\right )}{4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 50, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\left (d \,x^{2}+c \right ) a +b \left (\left (d \,x^{2}+c \right ) \arcsin \left (d \,x^{2}+c \right )+\sqrt {1-\left (d \,x^{2}+c \right )^{2}}\right )}{2 d}\) | \(50\) |
default | \(\frac {\left (d \,x^{2}+c \right ) a +b \left (\left (d \,x^{2}+c \right ) \arcsin \left (d \,x^{2}+c \right )+\sqrt {1-\left (d \,x^{2}+c \right )^{2}}\right )}{2 d}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 45, normalized size = 0.79 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.24, size = 57, normalized size = 1.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 76, normalized size = 1.33 \begin {gather*} \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}{\left (c \right )}\right )}{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 = 49, normalized size = 0.86 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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