Integrand size = 29, antiderivative size = 238 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4861, 4847, 4741, 4737, 30, 4767} \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}} \]
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Rule 30
Rule 4737
Rule 4741
Rule 4767
Rule 4847
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int (f+g x) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\sqrt {d-c^2 d x^2} \int \left (f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 c \sqrt {1-c^2 x^2}} \\ & = \frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.55 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-9 b c f x^2-\frac {4 b g x \left (-3+c^2 x^2\right )}{c}+18 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {12 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{c^2}+\frac {9 f (a+b \arcsin (c x))^2}{b c}\right )}{36 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.64
method | result | size |
default | \(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) g \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) g \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(628\) |
parts | \(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) g \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) g \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(628\) |
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\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
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\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Exception generated. \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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