Optimal. Leaf size=90 \[ \frac {3 b d x \sqrt {1-c^2 x^2}}{32 c}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}+\frac {3 b d \text {ArcSin}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{4 c^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4767, 201, 222}
\begin {gather*} -\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{4 c^2}+\frac {3 b d \text {ArcSin}(c x)}{32 c^2}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}+\frac {3 b d x \sqrt {1-c^2 x^2}}{32 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 4767
Rubi steps
\begin {align*} \int x \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}+\frac {(b d) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{4 c}\\ &=\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}+\frac {(3 b d) \int \sqrt {1-c^2 x^2} \, dx}{16 c}\\ &=\frac {3 b d x \sqrt {1-c^2 x^2}}{32 c}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}+\frac {(3 b d) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c}\\ &=\frac {3 b d x \sqrt {1-c^2 x^2}}{32 c}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}+\frac {3 b d \sin ^{-1}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 77, normalized size = 0.86 \begin {gather*} -\frac {d \left (c x \left (8 a c x \left (-2+c^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (-5+2 c^2 x^2\right )\right )+b \left (5-16 c^2 x^2+8 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{32 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 92, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {d \left (c^{2} x^{2}-1\right )^{2} a}{4}-d b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {5 \arcsin \left (c x \right )}{32}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{32}\right )}{c^{2}}\) | \(92\) |
default | \(\frac {-\frac {d \left (c^{2} x^{2}-1\right )^{2} a}{4}-d b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {5 \arcsin \left (c x \right )}{32}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{32}\right )}{c^{2}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 128, normalized size = 1.42 \begin {gather*} -\frac {1}{4} \, a c^{2} d 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 c^{2} d + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.46, size = 86, normalized size = 0.96 \begin {gather*} -\frac {8 \, a c^{4} d x^{4} - 16 \, a c^{2} d x^{2} + {\left (8 \, b c^{4} d x^{4} - 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \arcsin \left (c x\right ) + {\left (2 \, b c^{3} d x^{3} - 5 \, b c d x\right )} \sqrt {-c^{2} x^{2} + 1}}{32 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 117, normalized size = 1.30 \begin {gather*} \begin {cases} - \frac {a c^{2} d x^{4}}{4} + \frac {a d x^{2}}{2} - \frac {b c^{2} d x^{4} \operatorname {asin}{\left (c x \right )}}{4} - \frac {b c d x^{3} \sqrt {- c^{2} x^{2} + 1}}{16} + \frac {b d x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {5 b d x \sqrt {- c^{2} x^{2} + 1}}{32 c} - \frac {5 b d \operatorname {asin}{\left (c x \right )}}{32 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d 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 = 0.43, size = 100, normalized size = 1.11 \begin {gather*} -\frac {1}{4} \, a c^{2} d x^{4} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d x}{16 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d x}{32 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d}{2 \, c^{2}} + \frac {3 \, b d \arcsin \left (c x\right )}{32 \, c^{2}} \end {gather*}
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 \begin {gather*} \int x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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