Integrand size = 31, antiderivative size = 450 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \]
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Time = 0.36 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4861, 4847, 4741, 4737, 30, 4767, 4783, 4795} \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}} \]
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Rule 30
Rule 4737
Rule 4741
Rule 4767
Rule 4783
Rule 4795
Rule 4847
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int (f+g x)^2 \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^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+2 f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (f^2 \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 (2 f 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 {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f 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^2 \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^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b f 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 {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {1-c^2 x^2}} \\ & = \frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f 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^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {1-c^2 x^2}} \\ & = \frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 f 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^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.53 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-36 b c f^2 x^2-9 b c g^2 x^4-\frac {32 b f g x \left (-3+c^2 x^2\right )}{c}+72 f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+36 g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {96 f g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{c^2}+\frac {36 f^2 (a+b \arcsin (c x))^2}{b c}+\frac {9 g^2 \left (b c^2 x^2-2 c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{b}\right )}{c^3}\right )}{144 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 973, normalized size of antiderivative = 2.16
method | result | size |
default | \(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} \left (4 c^{2} f^{2}+g^{2}\right )}{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 ) g^{2} \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 (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 ) f g \left (i+3 \arcsin \left (c x \right )\right )}{36 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^{2} \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 ) f g \left (\arcsin \left (c x \right )+i\right )}{4 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 ) f g \left (\arcsin \left (c x \right )-i\right )}{4 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^{2} \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 ) f g \left (-i+3 \arcsin \left (c x \right )\right )}{36 c^{2} \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 ) g^{2} \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(973\) |
parts | \(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} \left (4 c^{2} f^{2}+g^{2}\right )}{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 ) g^{2} \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 (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 ) f g \left (i+3 \arcsin \left (c x \right )\right )}{36 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^{2} \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 ) f g \left (\arcsin \left (c x \right )+i\right )}{4 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 ) f g \left (\arcsin \left (c x \right )-i\right )}{4 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^{2} \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 ) f g \left (-i+3 \arcsin \left (c x \right )\right )}{36 c^{2} \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 ) g^{2} \left (-i+4 \arcsin \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(973\) |
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\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]
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\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Exception generated. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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