Optimal. Leaf size=124 \[ \frac {5 b d^2 x \sqrt {1-c^2 x^2}}{96 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2}}{144 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}+\frac {5 b d^2 \text {ArcSin}(c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))}{6 c^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4767, 201, 222}
\begin {gather*} -\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))}{6 c^2}+\frac {5 b d^2 \text {ArcSin}(c x)}{96 c^2}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2}}{144 c}+\frac {5 b d^2 x \sqrt {1-c^2 x^2}}{96 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 )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c^2}+\frac {\left (b d^2\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{6 c}\\ &=\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c^2}+\frac {\left (5 b d^2\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{36 c}\\ &=\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2}}{144 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c^2}+\frac {\left (5 b d^2\right ) \int \sqrt {1-c^2 x^2} \, dx}{48 c}\\ &=\frac {5 b d^2 x \sqrt {1-c^2 x^2}}{96 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2}}{144 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c^2}+\frac {\left (5 b d^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c}\\ &=\frac {5 b d^2 x \sqrt {1-c^2 x^2}}{96 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2}}{144 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}+\frac {5 b d^2 \sin ^{-1}(c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 94, normalized size = 0.76 \begin {gather*} \frac {d^2 \left (48 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )+3 b \left (-11+48 c^2 x^2-48 c^4 x^4+16 c^6 x^6\right ) \text {ArcSin}(c x)\right )}{288 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 127, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {\frac {d^{2} \left (c^{2} x^{2}-1\right )^{3} a}{6}+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\) | \(127\) |
default | \(\frac {\frac {d^{2} \left (c^{2} x^{2}-1\right )^{3} a}{6}+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 237 vs.
\(2 (107) = 214\).
time = 0.48, size = 237, normalized size = 1.91 \begin {gather*} \frac {1}{6} \, a c^{4} d^{2} x^{6} - \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{2} - \frac {1}{16} \, {\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^{2} + \frac {1}{2} \, a d^{2} 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^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.07, size = 137, normalized size = 1.10 \begin {gather*} \frac {48 \, a c^{6} d^{2} x^{6} - 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} d^{2} x^{6} - 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} - 11 \, b d^{2}\right )} \arcsin \left (c x\right ) + {\left (8 \, b c^{5} d^{2} x^{5} - 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 190, normalized size = 1.53 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{6}}{6} - \frac {a c^{2} d^{2} x^{4}}{2} + \frac {a d^{2} x^{2}}{2} + \frac {b c^{4} d^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{36} - \frac {b c^{2} d^{2} x^{4} \operatorname {asin}{\left (c x \right )}}{2} - \frac {13 b c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{144} + \frac {b d^{2} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {11 b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{96 c} - \frac {11 b d^{2} \operatorname {asin}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{2} 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 = 157, normalized size = 1.27 \begin {gather*} \frac {1}{6} \, a c^{4} d^{2} x^{6} - \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{36 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} \arcsin \left (c x\right )}{6 \, c^{2}} + \frac {5 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} x}{144 \, c} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{96 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2}}{2 \, c^{2}} + \frac {5 \, b d^{2} \arcsin \left (c x\right )}{96 \, 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 )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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