Optimal. Leaf size=209 \[ \frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac {5 b d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}+\frac {5 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^2}+\frac {5}{288} b^2 c^2 d^2 x^4+\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}-\frac {25}{288} b^2 d^2 x^2 \]
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Rubi [A] time = 0.20, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4677, 4649, 4647, 4641, 30, 14, 261} \[ \frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac {5 b d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}+\frac {5 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^2}+\frac {5}{288} b^2 c^2 d^2 x^4+\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}-\frac {25}{288} b^2 d^2 x^2 \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 261
Rule 4641
Rule 4647
Rule 4649
Rule 4677
Rubi steps
\begin {align*} \int x \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}+\frac {\left (b d^2\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c}\\ &=\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}-\frac {1}{18} \left (b^2 d^2\right ) \int x \left (1-c^2 x^2\right )^2 \, dx+\frac {\left (5 b d^2\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{18 c}\\ &=\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}-\frac {1}{72} \left (5 b^2 d^2\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac {\left (5 b d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{24 c}\\ &=\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac {5 b d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}-\frac {1}{72} \left (5 b^2 d^2\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac {1}{48} \left (5 b^2 d^2\right ) \int x \, dx+\frac {\left (5 b d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{48 c}\\ &=-\frac {25}{288} b^2 d^2 x^2+\frac {5}{288} b^2 c^2 d^2 x^4+\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac {5 b d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{72 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{6 c^2}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 209, normalized size = 1.00 \[ \frac {d^2 \left (c x \left (144 a^2 c x \left (c^4 x^4-3 c^2 x^2+3\right )+6 a b \sqrt {1-c^2 x^2} \left (8 c^4 x^4-26 c^2 x^2+33\right )+b^2 c x \left (-8 c^4 x^4+39 c^2 x^2-99\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right )+b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-26 c^2 x^2+33\right )\right )+9 b^2 \left (16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right ) \sin ^{-1}(c x)^2\right )}{864 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 278, normalized size = 1.33 \[ \frac {8 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} d^{2} x^{6} - 3 \, {\left (144 \, a^{2} - 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \, {\left (48 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \, {\left (16 \, b^{2} c^{6} d^{2} x^{6} - 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} - 11 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (16 \, a b c^{6} d^{2} x^{6} - 48 \, a b c^{4} d^{2} x^{4} + 48 \, a b c^{2} d^{2} x^{2} - 11 \, a b d^{2}\right )} \arcsin \left (c x\right ) + 6 \, {\left (8 \, a b c^{5} d^{2} x^{5} - 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x + {\left (8 \, b^{2} c^{5} d^{2} x^{5} - 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 383, normalized size = 1.83 \[ \frac {1}{6} \, a^{2} c^{4} d^{2} x^{6} - \frac {1}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{18 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{6 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{18 \, c} + \frac {5 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{2} x \arcsin \left (c x\right )}{72 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {5 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{2} x}{72 \, c} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{48 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2}}{108 \, c^{2}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{48 \, c} + \frac {5 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2}}{288 \, c^{2}} + \frac {5 \, b^{2} d^{2} \arcsin \left (c x\right )^{2}}{96 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d^{2}}{2 \, c^{2}} - \frac {5 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2}}{96 \, c^{2}} + \frac {5 \, a b d^{2} \arcsin \left (c x\right )}{48 \, c^{2}} - \frac {245 \, b^{2} d^{2}}{6912 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 283, normalized size = 1.35 \[ \frac {d^{2} a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{2} c^{4} x^{4}+\frac {1}{2} c^{2} x^{2}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {5 \arcsin \left (c x \right )^{2}}{96}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}-\frac {11 \arcsin \left (c x \right )}{96}\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, a^{2} c^{4} d^{2} x^{6} - \frac {1}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac {1}{144} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} a b c^{4} d^{2} - \frac {1}{8} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b c^{2} d^{2} + \frac {1}{2} \, a^{2} d^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} + \frac {1}{6} \, {\left (b^{2} c^{4} d^{2} x^{6} - 3 \, b^{2} c^{2} d^{2} x^{4} + 3 \, b^{2} d^{2} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + \int \frac {{\left (b^{2} c^{5} d^{2} x^{6} - 3 \, b^{2} c^{3} d^{2} x^{4} + 3 \, b^{2} c d^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.23, size = 430, normalized size = 2.06 \[ \begin {cases} \frac {a^{2} c^{4} d^{2} x^{6}}{6} - \frac {a^{2} c^{2} d^{2} x^{4}}{2} + \frac {a^{2} d^{2} x^{2}}{2} + \frac {a b c^{4} d^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{3} + \frac {a b c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{18} - a b c^{2} d^{2} x^{4} \operatorname {asin}{\left (c x \right )} - \frac {13 a b c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname {asin}{\left (c x \right )} + \frac {11 a b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{48 c} - \frac {11 a b d^{2} \operatorname {asin}{\left (c x \right )}}{48 c^{2}} + \frac {b^{2} c^{4} d^{2} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{6} - \frac {b^{2} c^{4} d^{2} x^{6}}{108} + \frac {b^{2} c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{18} - \frac {b^{2} c^{2} d^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{2} + \frac {13 b^{2} c^{2} d^{2} x^{4}}{288} - \frac {13 b^{2} c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{72} + \frac {b^{2} d^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {11 b^{2} d^{2} x^{2}}{96} + \frac {11 b^{2} d^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{48 c} - \frac {11 b^{2} d^{2} \operatorname {asin}^{2}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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