Integrand size = 27, antiderivative size = 756 \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {6 b^2 c^2 d^2 x^{3+m}}{(3+m)^2 (5+m)^2}+\frac {2 b^2 c^2 d^2 x^{3+m}}{(3+m) (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m}}{(3+m)^3 (5+m)}-\frac {2 b^2 c^4 d^2 x^{5+m}}{(5+m)^3}-\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \arcsin (c x))^2}{(5+m) \left (3+4 m+m^2\right )}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}-\frac {8 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)}-\frac {6 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m)^2 \left (6+5 m+m^2\right )}-\frac {16 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {6 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^3 (5+m)}+\frac {16 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(3+m)^2 (5+m) \left (2+3 m+m^2\right )} \]
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Time = 0.55 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4787, 4723, 4805, 4783, 30, 14} \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {16 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+3)^2 (m+5) \left (m^2+3 m+2\right )}+\frac {8 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^3 (m+5)}+\frac {6 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^2 (m+5)^2}-\frac {6 b c d^2 x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+5)^2 \left (m^2+5 m+6\right )}-\frac {16 b c d^2 x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+5) \left (m^3+6 m^2+11 m+6\right )}-\frac {8 b c d^2 x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+2) (m+3)^2 (m+5)}+\frac {4 d^2 \left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m^2+8 m+15}+\frac {d^2 \left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}-\frac {2 b c d^2 \left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{(m+5)^2}-\frac {8 b c d^2 \sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{(m+3)^2 (m+5)}-\frac {6 b c d^2 \sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{(m+3) (m+5)^2}+\frac {8 d^2 x^{m+1} (a+b \arcsin (c x))^2}{(m+5) \left (m^2+4 m+3\right )}-\frac {2 b^2 c^4 d^2 x^{m+5}}{(m+5)^3}+\frac {8 b^2 c^2 d^2 x^{m+3}}{(m+3)^3 (m+5)}+\frac {2 b^2 c^2 d^2 x^{m+3}}{(m+3) (m+5)^2}+\frac {6 b^2 c^2 d^2 x^{m+3}}{(m+3)^2 (m+5)^2} \]
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Rule 14
Rule 30
Rule 4723
Rule 4783
Rule 4787
Rule 4805
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}+\frac {(4 d) \int x^m \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx}{5+m}-\frac {\left (2 b c d^2\right ) \int x^{1+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx}{5+m} \\ & = -\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}-\frac {\left (6 b c d^2\right ) \int x^{1+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{(5+m)^2}+\frac {\left (2 b^2 c^2 d^2\right ) \int x^{2+m} \left (1-c^2 x^2\right ) \, dx}{(5+m)^2}+\frac {\left (8 d^2\right ) \int x^m (a+b \arcsin (c x))^2 \, dx}{15+8 m+m^2}-\frac {\left (8 b c d^2\right ) \int x^{1+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{15+8 m+m^2} \\ & = -\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \arcsin (c x))^2}{15+23 m+9 m^2+m^3}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}+\frac {\left (2 b^2 c^2 d^2\right ) \int \left (x^{2+m}-c^2 x^{4+m}\right ) \, dx}{(5+m)^2}-\frac {\left (6 b c d^2\right ) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{(3+m) (5+m)^2}+\frac {\left (6 b^2 c^2 d^2\right ) \int x^{2+m} \, dx}{(3+m) (5+m)^2}-\frac {\left (8 b c d^2\right ) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{(3+m)^2 (5+m)}+\frac {\left (8 b^2 c^2 d^2\right ) \int x^{2+m} \, dx}{(3+m)^2 (5+m)}-\frac {\left (16 b c d^2\right ) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{15+23 m+9 m^2+m^3} \\ & = \frac {6 b^2 c^2 d^2 x^{3+m}}{(3+m)^2 (5+m)^2}+\frac {2 b^2 c^2 d^2 x^{3+m}}{(3+m) (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m}}{(3+m)^3 (5+m)}-\frac {2 b^2 c^4 d^2 x^{5+m}}{(5+m)^3}-\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \arcsin (c x))^2}{15+23 m+9 m^2+m^3}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}-\frac {6 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)}-\frac {16 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) \left (15+23 m+9 m^2+m^3\right )}+\frac {6 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^3 (5+m)}+\frac {16 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^2 \left (5+6 m+m^2\right )} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.53 \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=d^2 x^{1+m} \left (\frac {(a+b \arcsin (c x))^2}{1+m}-\frac {2 c^2 x^2 (a+b \arcsin (c x))^2}{3+m}+\frac {c^4 x^4 (a+b \arcsin (c x))^2}{5+m}+\frac {2 b c x \left (-\left ((3+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)}-\frac {4 b c^3 x^3 \left (-\left ((5+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {5}{2}+\frac {m}{2},\frac {5}{2}+\frac {m}{2};3+\frac {m}{2},\frac {7}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(3+m) (4+m) (5+m)}+\frac {2 b c^5 x^5 \left (-\left ((7+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {7}{2}+\frac {m}{2},\frac {7}{2}+\frac {m}{2};4+\frac {m}{2},\frac {9}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(5+m) (6+m) (7+m)}\right ) \]
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\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
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\[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int { {\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]
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\[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=d^{2} \left (\int a^{2} x^{m}\, dx + \int b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{m} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 2 a^{2} c^{2} x^{2} x^{m}\right )\, dx + \int a^{2} c^{4} x^{4} x^{m}\, dx + \int \left (- 2 b^{2} c^{2} x^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{4} x^{4} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 4 a b c^{2} x^{2} x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{4} x^{4} x^{m} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
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\[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int { {\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]
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\[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int { {\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]
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Timed out. \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]
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