Integrand size = 29, antiderivative size = 303 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4783, 4795, 4737, 4723, 327, 222} \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}}-\frac {b^2 \arcsin (c x) \sqrt {d-c^2 d x^2}}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2} \]
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Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4783
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x^3 (a+b \arcsin (c x)) \, dx}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x (a+b \arcsin (c x)) \, dx}{4 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}} \\ & = \frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}} \\ & = \frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.81 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (8 a^3+3 b^3 c x \left (1-2 c^2 x^2\right ) \sqrt {1-c^2 x^2}-24 a b^2 c^2 x^2 \left (-1+c^2 x^2\right )+24 a^2 b c x \sqrt {1-c^2 x^2} \left (-1+2 c^2 x^2\right )-3 b \left (-8 a^2+16 a b c x \left (1-2 c^2 x^2\right ) \sqrt {1-c^2 x^2}+b^2 \left (1-8 c^2 x^2+8 c^4 x^4\right )\right ) \arcsin (c x)+24 b^2 \left (a+b c x \sqrt {1-c^2 x^2} \left (-1+2 c^2 x^2\right )\right ) \arcsin (c x)^2+8 b^3 \arcsin (c x)^3\right )}{192 b c^3 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.24
method | result | size |
default | \(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{24 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 \arcsin \left (c x \right )+i\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(678\) |
parts | \(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{24 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 \arcsin \left (c x \right )+i\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(678\) |
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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