Integrand size = 31, antiderivative size = 878 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}+\frac {16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \]
[Out]
Time = 0.61 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {4861, 4847, 4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1813, 1864} \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {2 b c^5 d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^7}{49 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^5}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^3}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{6} d^2 f \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {5}{16} d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {5}{24} d^2 f \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {2 b d^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x}{7 c \sqrt {1-c^2 x^2}}-\frac {1}{108} b^2 d^2 f \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} x-\frac {245 b^2 d^2 f \sqrt {d-c^2 d x^2} x}{1152}-\frac {65 b^2 d^2 f \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x}{1728}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2} \]
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Rule 12
Rule 200
Rule 201
Rule 222
Rule 327
Rule 1813
Rule 1864
Rule 4723
Rule 4737
Rule 4739
Rule 4741
Rule 4743
Rule 4767
Rule 4847
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2+g x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x)) \, dx}{7 c \sqrt {1-c^2 x^2}} \\ & = \frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{12 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x (a+b \arcsin (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \sqrt {1-c^2 x^2}} \\ & = -\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}} \\ & = -\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}-\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1-c^2 x}}+8 \sqrt {1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}} \\ & = \frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}+\frac {16 b^2 d^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {65 b^2 d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {12 b^2 d^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {2 b^2 d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.54 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (3087000 a^3 c f+88200 a^2 b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )-840 a b^2 c x \left (245 c^2 f x \left (99-39 c^2 x^2+8 c^4 x^4\right )+288 g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )\right )+b^3 \sqrt {1-c^2 x^2} \left (-8575 c^2 f x \left (897-194 c^2 x^2+32 c^4 x^4\right )-2304 g \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )\right )+105 b \left (88200 a^2 c f+1680 a b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )+b^2 c \left (-2304 g x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )-245 f \left (-299+792 c^2 x^2-312 c^4 x^4+64 c^6 x^6\right )\right )\right ) \arcsin (c x)+88200 b^2 \left (105 a c f+b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)^2+3087000 b^3 c f \arcsin (c x)^3\right )}{29635200 b c^2 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 2852, normalized size of antiderivative = 3.25
method | result | size |
default | \(\text {Expression too large to display}\) | \(2852\) |
parts | \(\text {Expression too large to display}\) | \(2852\) |
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\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
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\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int \left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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